Faculty Introduction 2024

Graduate School of Mathematics, Nagoya University(April 1, 2024)

CONTENTS
ARANO, Yuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

AWATA, Hidetoshi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

FUJIE-OKAMOTO, Futaba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

FUJIWARA, Kazuhiro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

FURUSHO, Hidekazu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

HAMANAKA, Masashi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

HAYASHI, Masahito . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

HAYASHI, Takahiro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

HESSELHOLT, Lars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

HIRAI, Hiroshi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

HISHIDA, Toshiaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

ITO, Kentaro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

IZUMI, Keisuke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

KANNO, Hiroaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

KATO, Jun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

KAWAMURA, Tomomi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

KITA, Nanao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

LE GALL, François . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

MINAMI, Kazuhiko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

MORIYOSHI, Hitoshi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

NAGAO, Taro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

NAITO, Hisashi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

NAKANISHI, Tomoki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

NAKAOKA, Hiroyuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

NAKASHIMA, Makoto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

NAYATANI, Shin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

OHIRA, Toru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

OHKUBO, Shun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

OHTA, Hiroshi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

OKADA, Soichi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

SHIROMIZU, Tetsuya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

SUGIMOTO, Mitsuru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

SUZUKI, Hiroshi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

TAKAHASHI, Ryo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

TANIMOTO, Sho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

TERASAWA, Yutaka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

UEDA, Yoshimichi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

UZAWA, Tohru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

YANAGIDA, Shintaro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

YOSHIDA, Nobuo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

BOURNE, Chris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

ARANO, Yuki

Associate Professor
Oﬃce:

Rm 304 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 4877

Email:

y.arano@math.nagoya-u.ac.jp

Web-site:(ext. 4877)

https://www.math.nagoya-u.ac.jp/~y.arano/index.html

Membership of Academic Societies:
The Mathematical Society of Japan

Research Interest:• Operator algebra• Quantum group• Tensor category

Research Summary:An operator algebra is a certain subalgebra of the algebra of all bounded operators on a Hilbert
space. Such algebra can be seen as a “noncommutative version” of the algebra of all continuousfunctions on a topological space. In the operator algebraic quantum group theory, we study thegroup structure on such “noncommutative” space. Not only such quantum group gives important
examples of operator algebras, but also this appears as an analogue of the Galois group in thesubfactor theory.
My interest is the repretation theoretic/topological aspects of quantum groups. Along this line,
I also worked on the actions of quantum groups on operator algebras and more general tensor
categories etc.

Major Publications:[1] Y. Arano, Unitary spherical representations of Drinfeld doubles. J. Reine Angew. Math. 742(2018), 157–186[2] Y. Arano, Comparison of unitary duals of Drinfeld doubles and complex semisimple Lie groups.
Comm. Math. Phys. 351 (2017), no.3, 1137–1147.
[3] Y. Arano, Y. Isono. A. Marrakchi, Ergodic theory of aﬃne isometric actions on Hilbert spaces,
Geom. Funct. Anal. 31 (2021), no.5, 1013–1094.

Awards and Prizes:• 2017, The Takebe Katahiro Prize for Encouragement of Young Researchers, “Studies of operator algebraic quantum groups”

Education and Appointments:
2017
2023

Assitant professor, Kyoto University
Associate professor, Nagoya University

1Message to Prospective Students:
Standard textbooks on my ﬁeld are, for example,• Murphy, Gerard J. C*-algebras and operator theory. Academic Press, Inc., 1990
• Jantzen, Jens Carsten. Lectures on quantum groups. Graduate Studies in Mathematics, 6.
American Mathematical Society,, 1996
• Neshveyev, Sergey ; Tuset, Lars. Compact quantum groups and their representation categories. Cours Spécialisés, 20. Société Mathématique de France, 2013.
but you may study anything related to operator algebras and quantum groups.
During the seminar, I would recommend you not to see any memos. This does not mean you
need to memorize everything. You need to understand the contents as deep as, for example, thehigh school math. (I suppose you can talk about the deﬁnition and the basic properties of a
derivative of a single variable without seeing anything.)
You are required to understand calculus, linear algebras, topology, basic group theory, measure
theory, Fourier analysis and functional analysis before entering the graduate school. Undrstanding any further math such as representation theory, probability theory, algebraic topology andmathematical physics would be appreciated, but deep understanding of basic math is much more
important.

2

AWATA, Hidetoshi

Associate Professor

Oﬃce:

Rm 306 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 5601

Email:

awata@math.nagoya-u.ac.jp

Web-site:(ext. 5601)

http://www.math.nagoya-u.ac.jp/~awata/

Membership of Academic Societies:
The Mathematical Society of Japan

Research Interest:• Quantum ﬁeld theory• Representation theory of inﬁnite dimensional algebras• Symmetric function

Research Summary:
H. Awata works in the areas of representation theory and quantum ﬁeld theory, which involve
the representation theory of inﬁnite dimensional algebras such as Virasoro algebra, W algebra
and aﬃne Lie algebra and also involve the quantum ﬁeld theories possessing one of these algebras as a symmetry such as super string theory, conformal ﬁeld theory, two dimensional solvablemodel and topological ﬁeld theory. Much of his work focuses on studying (q-deformed) KnizhnikZamolodchikov equation, Jack polynomial, Macdonald polynomial and Nekrasov’s partition function. He is currently focusing on a study of the AGT conjecture, which reveals the relation between
the conformal ﬁeld theory and the Nekrasov’s partition function.

Major Publications:[1] with Akihiro Tsuchiya and Yasuhiko Yamada, “Integral formulas for the WZNW correlation
functions,” Nuclear Physics B365 (1991), 680–696.
d[2] with Satoru Odake and Jun’ichi Shiraishi, “Free Boson realization of Uq (slN ),” Communicationsin Mathematical Physics 162 (1994), 61–83[3] with Masafumi Fukuma, Yutaka Matsuo and Satoru Odake, “Representation Theory of theW1+∞ Algebra,” Progress of Theoretical Physics, Supplement 118 (1995), 343–373[4] with Harunobu Kubo, Satoru Odake and Jun’ichi Shiraishi, “Quantum WN Algebras and Macdonald Polynomials,” Communications in Mathematical Physics 179 (1996), 401–416[5] with Miao Li, Djordje Minic and Tamiaki Yoneya, “On the Quantization of Nambu Brackets,”
Journal of High Energy Physics 0102 (2001) 013[6] with Hiroaki Kanno, “Instanton counting, Macdonald function and the moduli space of Dbranes,” Journal of High Energy Physics 0505 (2005) 039.
[7] with Yasuhiko Yamada, “Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra,” Journal of High Energy Physics 1001 (2010) 125.
[8] with Boris Feigin and Jun’ichi Shiraishi, “Quantum Algebraic Approach to Reﬁned Topological
Vertex”, Journal of High Energy Physics 1203 (2012) 041.

3

Education and Appointments:
1993
1994
1995
1996
1998
1999

Soryushi-Syogakkai fellow at Yukawa Institute for Theoretical Physics, Kyoto University
JSPS fellow at Research Institute for Mathematical Science, Kyoto University
JSPS fellow at Yukawa Institute for Theoretical Physics, Kyoto University
Visiting Scholar at Enrico Fermi and James Frank Institutes of Chicago University
COE at Yukawa Institute for Theoretical Physics, Kyoto University
Associate Professor at Graduate School of Mathematics, Nagoya University

Message to Prospective Students:• The basic literature,
J. Polchinski, “String Theory”, Cambridge univ. press, 1998.
V. Kac, “Inﬁnite dimensional Lie algebras”, Cambridge univ. press, 1990.
I.G. Macdonald, “Symmetric functions and Hall polynomials”, Second Edition, Oxford University Press, 1995.

4

FUJIE-OKAMOTO, Futaba

Associate Professor

Oﬃce:

Rm 407 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 5603

Email:

futaba@math.nagoya-u.ac.jp(ext. 5603)

Membership of Academic Societies:
AMS (American Mathematical Society)
ICA (The Institute of Combinatorics and Its Applications)

Research Interest:• Graph Theory

Research Summary:Graphs have been studied through a number of aspects such as connectivity, colorings andlabelings, decompositions and factorizations, traversability, etc. In my research, I study traditional
concepts including those mentioned above as well as try coming up with new ways of looking at
graphs that may shed some light on other concepts and problems in mathematics.
The connectivity of a connected graph is the smallest number of vertices that disconnect thegraph when deleted. Menger’s Theorem suggests that studying the connectivity of a graph is closely
related to studying the number of internally-disjoint paths connecting each pair of vertices in thegraph. In [2], a new parameter of graphs was introduced. For a nontrivial connected graph G
and a nonempty non-singleton set S ⊆ V (G), the number κ(S) is the maximum number ℓ suchthat there is a collection T1 , T2 , . . . , Tℓ of ℓ pairwise edge-disjoint trees with V (Ti ) ∩ V (Tj ) = S
for 1 ≤ i ̸= j ≤ ℓ. The k-connectivity of G is then deﬁned to be min{κ(S)}, where the minimum
is taken over all k-subsets S of V (G). This generalizes the traditional concept of connectivity ofgraphs, where the standard connectivity is exactly the 2-connectivity. Roughly speaking, the kconnectivity of a graph can be seen as the number of “independent” ways for a set of k individuals
in a network to communicate with each other.
Another popular area of research in graph theory is graph coloring. For example, the famous
Four Color Theorem states that every planar graph has a proper 4-coloring. Colorings that deserve
our attention include those that distinguish either every two vertices or every two adjacent vertices
in a graph in some manner. Colorings possessing such properties are said to be vertex-distinguishingand neighbor-distinguishing, respectively. I have introduced and studied numerous new vertexdistinguishing colorings and neighbor-distinguishing colorings with my research partners. The
sigma coloring in [1] is an example of the latter.

Major Publications:[1] G. Chartrand, F. Okamoto, and P. Zhang, The sigma chromatic number of a graph, Graphs
Combin., 26:6 (2010) 755–773.
[2] G. Chartrand, F. Okamoto, and P. Zhang, Rainbow trees in graphs and generalized connectivity,
Networks, 55:4 (2010) 360–367.
[3] F. Fujie and P. Zhang, Covering Walks in Graphs, Springer Briefs in Mathematics, Springer,
2014.

5

Awards and Prizes:• The 2008 Kirkman Medal (The Institute of Combinatorics and Its Applications)

Education and Appointments:
2007
2007
2011
2012

Ph.D. in Mathematics, Western Michigan University
Assistant Professor, University of Wisconsin La Crosse
Associate Professor, University of Wisconsin La Crosse
Associate Professor, Nagoya University

Message to Prospective Students:
Graph theory is a relatively new area of mathematics and has increased in popularity, perhaps
partly due to the fact that introductory books usually contain many cute ﬁgures and interesting
real-world applications. It is possible that one can jump in and start studying graph theory without
having very deep background, so I welcome those students even if they are new to this ﬁeld. Of
course, however, that does not mean graph theory is easy! It is unlikely that one will be successful
without having strong interest and dedication. Being strong in reading and writing would be
another requirement; again, you do not have to be perfect from the beginning, but this is a headsup. For those that want to learn the basics, here are some good books that cover standard topics:[4] G. Chartrand, L. Lesniak, and P. Zhang, Graphs and Digraphs (CRC Press, 2010).
[5] J.A. Bondy and U.S.R. Murty, Graph Theory (Springer, 2008).
Finding good problems is as essential as becoming familiar with the subject. When one reads
literature, remember to ask oneself questions such as how the given concepts can be generalized
and what else can be explored.

6

FUJIWARA, Kazuhiro

Professor

Oﬃce:

Rm 321 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2818

Email:

fujiwara@math.nagoya-u.ac.jp

Web-site:(ext. 2818)

no webpage available

Membership of Academic Societies:
MSJ (The Mathematical Society of Japan)

Research Interest:• Number Theory• Arithmetic Algebraic Geometry• Automorphic forms and Shimura varieties

Research Summary:
I am interested in themes which relate geometric viewpoint (arithmetic geometry) and analytic
viewpoint (such as harmonic analysis on adeles). In particular, I am trying to understand nonabelian class ﬁeld theory, which is a vast generalization of classical class ﬁeld theory due to Teiji
Takagi and Emil Artin.
Non-abelian class ﬁeld theory is, after eﬀorts of many mathematicians, now formulated as acorrespondence between:
1. Galois representations (algebraic and geometric objects obtained mainly from algebraic varieties),
2. Automorphic representations (representation-theoretical interpretation of automorphic forms,
which admit large discrete symmetries)
which is called as Langlands correspondence. This correspondence is expected to preserve Lfunctions deﬁned on both sides (non-abelian reciprocity law), which yield highly non-trivial consequences in number theory.
Usually such a correspondence is obtained from a deep study of Shimura varieties. Now myinterest is focused on:
1. Arithmetic geometry of Shimura varieties
2. Galois representations and p-adic Hecke algebras
3. Application of non-abelian class ﬁeld theory to classical number theoretical problems
which are related to each other.

Major Publications:[1] K. Fujiwara, Rigid geometry, Lefschetz trace formula and Deligne s conjecture, Inv. Math. 127(1997), 489―533.
[2] K. Fujiwara, Galois deformations and arithmetic geometry of Shimura varieties, Proceedings ofthe International Congress of Mathematicians Madrid 2006 (2006), vol. 2, 347―371.
[3] K. Fujiwara and F. Kato, Rigid geometry and applications, Moduli spaces and Arithmetic
Geometry, Advanced Studies in Pure Math. 45, (2006), 327-386

7Awards and Prizes:• Algebra prize (1998), Mathematical Society of Japan

Education and Appointments:
1990
1994
1997
2001

Assistant professor, University of Tokyo
Lecturer, Nagoya University
Associate professor, Nagoya University
Professor, Nagoya University

Message to Prospective Students:
My main research area, number theory, has a long history. In any area which has a long history,
one needs to learn many ideas and insights from the existing literature, before starting an actual
research. The shortest way to understand it is, I think, to have a ﬁrm knowledge on basic notions.
So I expect you to read foundational textbooks in algebra, geometry, and analysis.
At the same time, I strongly recommend to read research papers. In doing so, you will feelsomething, think deeply, and get inspirations. This is a route to study mathematics, especially for
beginners.
My role is to oﬀer you a technical support in mathematics. You are recommended to ﬁnd yourown mathematics, not mine. Mathematics is full of freedom.
To get an impression on the basic literature, please look at the following books.
[1] N. Bourbaki, Commutative Algebra, Chapters 1-7, Springer[2] N. Bourbaki, General Topology, Chapters 1-4, Springer[3] H. Hida，Elementary theory of L-functions and Eisenstein series, LMS．
[4] A. W. Knapp, Elliptic curves, Princeton Univ. Press.
[5] N. Koblitz, Introduction to elliptic curves and modular forms, Springer.
[6] J. P. Serre，Abelian ℓ-adic representations and elliptic curves, Research notes in Mathematics ．

8

FURUSHO, Hidekazu

Professor

Oﬃce:

Rm 455 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2418

Email:

furusho@math.nagoya-u.ac.jp(ext. 2418)

Membership of Academic Societies:
MSJ (Mathematical Society of Japan)

Research Interest:• Various ﬁelds related to number theory (particularly Arithmetic Geometry and Arithmetic Topology):

Research Summary:My research is based on number theory. However it is not restricted to number theory, rather
I am working on various ﬁelds related to number theory:• Arithmetic Geometry is a part of number theory and motive theory is one of the mostimportant theory in arithmetic geometry. Lots of people tend to regard that motive theory is a quite abstract and much general theory. But actually I am working oppositely on
a very concrete side of motive theory, particularly on (p-adic) (multiple) zeta functions (padic) (multiple) polylogarithms and Knizhnik-Zamolodchikov equation. Teichmüller-Lego
philosophy was posed by Grothendieck in his mysterious note ‘Esquisse d’un programme’(’84). This philosophy is closely related to the above motive theory and also Drinfeld’s subsequent works (in 80’s) on quantum groups, where my research takes place. It has undergone agreat interests due to appearance in diﬀerent branches of mathematics, including motive theory, quantum group theory, deformation quantization theory, operad theory, analytic number
theory, conformal ﬁeld theory, diﬀerential geometry, low dimensional topology, mathematicalphysics, etc. It is one of the most exciting area to work today.
• Arithmetic Topology is a quite new area of mathematics, where I have started to work
recently. It detects and purses several conceptual analogies between algebraic number theory
and 3-dimensional topology (including quantum topology). One of the most impressive
and advertising analogies might be the ones between primes and knots, which sound very
mysterious and stimulating. Arithmetic topology is ‘baby-like’ because it has just started
and is waiting to be developed. It is really a good time to get started pioneering works foryounger generations like you!

Major Publications:[1] H. Furusho, Double shuﬄe relation for associators, Annals of Mathematics, Vol. 174 (2011),
No. 1, 341-360.
[2] H. Furusho, Pentagon and hexagon equations, Annals of Mathematics, Vol. 171 (2010), No. 1,
545-556.
[3] H. Furusho, p-adic multiple zeta values I – p-adic multiple polylogarithms and the p-adic KZ
equation, Inventiones Mathematicae, Volume 155, Number 2, 253-286, (2004).

9

Awards and Prizes:• 2014, Algebra Prize of Math. Soc. Japan.
• 2007, Inoue Research Award for Young Scientists.
• 2004, Takebe Prize of Math. Soc. Japan.

Education and Appointments:
2018 – Now
20172010 – 2018
2013
2013
2007 – 20092004 – 2005
2004 – 2010
2003

Professor at Nagoya University, Nagoya, Japan.
Institut de Rechérche Mathématique Avancee, Strasbourg university, Strasbourg,
France.
Associate Professor at Nagoya University, Nagoya, Japan.
Isaac Newton Institute for Mathematical Sciences, Cambridge, UK.
Max Planck Institute for Mathematics, Bonn, Germany.
École Normale Supérieure, Paris, France.
Institute for Advanced Study, Princeton, USA.
Assistant Professor at Nagoya University, Nagoya, Japan.
Ph.D from RIMS, Kyoto, Japan.

Message to Prospective Students:
For undergraduate students; please be familiar with the standard techniques in number theory
by reading, for example,• J.P. Serre, ”A Course in Arithmetic”, Graduate Texts in Mathematics, 67, Springer-Verlag.
For master course students; it is important to learn several theories other than number theory
if you want to work with me. The followings might be good to learn arithmetic topology.
• S. Chmutov, S. Duzhin, J. Mostovoy, ”Introduction to Vassiliev knot invariants”,Cambridge
University Press.
• C. Kassel, ”Quantum Groups”,Graduate Texts in Mathematics, 155. Springer-Verlag.
• T. Ohtsuki, ”Quantum Invariants”,Series on Knots and Everything, 29. World Scientiﬁc.
• M. Morishita, ”Knots and primes”, Universitext. Springer, London, 2012.
But before you will contact with me, please ﬁnd by yourself the literatures (other than the above
mentioned books if possible!) which you want to read with me during the master course and also
be prepared to explain me your mathematical perspectives.
For doctor course students; the following are the paper which I like most. Make a challenge toread one of the following papers:• P. Deligne, ”Le groupe fondamental de la droite projective moins trois points”, Galois groups
over Q, 79–297, Math. Sci. Res. Inst. Publ., 16, Springer, New York, (1989).
• V.G. Drinfel’d, ”On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)”, Leningrad Math. J. 2 (1991), no. 4, 829–860.
• M. Kontsevich, ”Operads and motives in deformation quantization”, Lett. Math. Phys. 48(1999), no. 1, 35–72.
Once you start to read, you will soon realized that it is really hard to read. But they are reallyenriched papers. I wish to educate perspective students for years to overcome them.
Good Luck!

10

HAMANAKA, Masashi

Associate Professor

Oﬃce:

Rm 327 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2408

Email:

hamanaka@math.nagoya-u.ac.jp

Web-site:(ext. 2408)

http://www2.yukawa.kyoto-u.ac.jp/~masashi.hamanaka/

Membership of Academic Societies:
The Physical Society of Japan (JPS),
The Mathematical Society of Japan (MSJ)

Research Interest:• Mathematical Physics• Gauge Theory related to String Theory• Noncommutative Solitons

Research Summary:
For the last several years, I have studied extension of soliton theories and integrable systems
to noncommutative spaces. In gauge theories, the noncommutative extension corresponds to introduction of background ﬂux and has been applied to the corresponding physical situations such as
the quantum Hall eﬀects in background magnetic ﬂux. Furthermore, in noncommutative spaces,
singularities could be resolved and therefore new physical (smooth) objects could appear. For example, U (1) instantons are one of such new objects which come from resolution of singularity inthe (instanton) moduli spaces of the anti-self-dual Yang-Mills equations in four dimensional noncommutative Euclidean spaces. This is also due to the fact that the ADHM construction can work
well in the noncommutative settings, and in this sense, integrability is also preserved (e.g. [1]).
Noncommutative solitons actually describe D-branes themselves in some D-brane conﬁgurations,which makes us to analyze various properties of D-branes by analyzing those of noncommutative
solitons. The latter is sometimes much easier than the former and long-standing problems on
D-branes such as Sen’s conjecture have been exactly solved.
After the development of noncommutative solitons in the eﬀective theories of D-branes which
are gauge theories, noncommutative extension of lower-dimensional soliton equations (e.g. KdV
eq. in scalar theories) had been also studied intensively (e.g. [2]). In [3], it is proved that many ofsuch lower-dimensional noncommutative soliton equations can be derived from the 4-dimensional
noncommutative anti-self-dual Yang-Mills equation by reduction. This result implies that the lowerdimensional soliton equations also belong to gauge theories in this context, and hence have thecorresponding physical situations with background ﬂux. These soliton equations can be embedded
to the N = 2 strings and can be applied to them via analysis of the exact soliton solutions.
In [4], we ﬁnd a Bäcklund transformation for the noncommutative anti-self-dual Yang-Mills
equation and construct wide class of new solutions including not only noncommutative instantons
but non-linear plane-wave (domain-wall type) solutions. In the construction of the solutions, thequasideterminants play cricial roles and simplify proofs in the commutative case. (For a survey ofquasideterminants, see, e.g. [math/0208146].) This is in common with the lower-dimensional soliton
equations (e.g. [5]). I look for a universal and essential structure and formulation of integrable
systems from the viewpoint of the quasideterminants.

11

Major Publications:[1] M. Hamanaka and T. Nakatsu, “Noncommutative Instantons and Reciprocity,” in preparation.
(For a survey, see [arXiv:1311.5227].)[2] M. Hamanaka, “Noncommutative Solitons and Integrable Systems,” hep-th/0504001.
[3] M. Hamanaka, “Noncommutative Ward’s Conjecture and Integrable Systems,”
Nuclear Physics B 741 (2006), 368 – 389 [hep-th/0601209].
[4] C. R. Gilson, M. Hamanaka and J. J. C. Nimmo, “Bäcklund Transformations and the AtiyahWard ansatz for Noncommutative Anti-Self-Dual Yang-Mills Equations,”
Proceedings of the Royal Society A 465 (2009), 2613 – 2632 [arXiv:0812.1222].
[5] M. Hamanaka, “Noncommutative Integrable Systems and Quasideterminants,”
AIP Conf. Proc. 1212, 122 (2010) [arXiv:1012.6043].

Education and Appointments:
2003/3
2003/4
2004/2

2016/4

Ph.D at the University of Tokyo, Department of Physics
JSPS postdoctoral fellow at the Univ. of Tokyo (Komaba)
Assistant Professor, Nagoya University(2005/8-2006/12: Visiting Researcher, Univ. of Oxford)(2008/10-2009/2: Visiting Researcher, Univ. of Glasgow)(2009/2-2009/3: Visiting Researcher, IHES, France)
Associate Professor, Nagoya University

Message to Prospective Students:
I would be interested in mathematical structure behind the fundamental law of nature, currently,
quantum ﬁeld theory and string theory. Any student is welcome to discuss with me. (Please notethat I am not a mathematician but a physicist.)
So far, I have been an adviser of seven graduate students. In the case of a main adviser, I have
had a meeting or an informal seminar to help them once per one or two weeks. The following lists
are a part of the relevant papers including collaborations with my students (underlined).
[1] P. Etingof, I. Gelfand and V. Retakh, “Factorization of diﬀerential operators, quasideterminants, and nonabelian Toda ﬁeld equations,” Math. Res. Lett. 4, 413 (1997) [q-alg/9701008].
[2] K. Ueno and K. Takasaki, “Toda lattice hierarchy,” Adv. Stud. Pure Math. 4, 1 (1984).
[3] S. A. Cherkis and R. S. Ward, “Moduli of Monopole Walls and Amoebas,”
JHEP 1205, 090 (2012) [arXiv:1202.1294].
[4] M. Hamanaka, H. Kanno and D. Muranaka, “Hyperkähler metrics from monopole walls,”
Phys. Rev. D 89, 065033 (2014) [arXiv:1311.7143].
[5] S. A. Cherkis, “Octonions, Monopoles, and Knots,” Lett. Math. Phys. 105, 641 (2015)[arXiv:1403.6836].
[6] M. Hamanaka and H. Okabe, “Soliton Scattering in Noncommutative Spaces,” Theor. Math.
Phys. 197, 1451-1468 (2018) [arXiv:1806.05188].
[7] L. J. Mason and N. M. Woodhouse, Integrability, Self-Duality, and Twistor Theory (Oxford
UP, 1996) [ISBN/0-19-853498-1].
[8] C. R. Gilson, M. Hamanaka, S. C. Huang and J. J. C. Nimmo, “Soliton solutions of noncommutative anti-self-dual Yang–Mills equations,” J. Phys. A 53, 404002 (2020) [arXiv:2004.01718].
[9] M. Hamanaka and S. C. Huang, “New Soliton Solutions of Anti-Self-Dual Yang-Mills equations,” JHEP 10, 101 (2020) [arXiv:2004.09248].
12[10] M. Hamanaka, S. C. Huang and H. Kanno, “Solitons in Open N = 2 String Theory,” PTEP
2023, 4, 043B03 (2023) [arXiv:2212.11800].

13

HAYASHI, Masahito

Professor

Oﬃce:

Rm 355 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2549

Email:

masahito@math.nagoya-u.ac.jp

Web-site:(ext. 2549)

http://www.math.nagoya-u.ac.jp/~masahito/index_e.html

Membership of Academic Societies:
IEEE (The Institute of Electrical and Electronics Engineers), The
Mathematical Society of Japan, JPS (The Physical Society of Japan),
IEICE (The Institute of Electronics, Information and Communication
Engineers)

Research Interest:• Quantum Information Theory• Quantum Cryptography• Information Theory• Non-equilibrium Statistical Physics for Closed System

Research Summary:My research area is mathematical theory for information and its application, especially, I have studied
mathematical theory for communication, statistical inference, and cryptography. These topics have diﬀerent
applied aspects, and have diﬀerent communities due to their historical reason. However, these topics havecommon mathematical aspects. Hence, a common mathematical treatment is possible for these topics. I
have investigated these topics based on the common mathematical properties. In particular, I have studies
these topics mainly for quantum systems but also for non-quantum (classical) system. Recently, using thismethod, I study the foundation of thermodynamics.
Recently, I am mainly studying the following points. One is mathematical treatment for quantum
information processing based on group representation theory. Group symmetry simpliﬁes many kinds ofproblems in quantum systems by removing the basis dependence. In fact, even if a given information
processing problem requires a diﬃcult analysis due to the complexity of the problem, the group symmetrysimpliﬁes the problem by reducing the complexity. Using the group symmetry, we can construct universal
protocols that works independently of the basis. Since the group-theoretical approach to quantum systems
are not ﬁnished, further developments are required. The second is mathematical theory for informationtheoretical secrecy. Recently, I have proposed several approaches for this topic, however, their relations are
not so clear and several problems still remain open. Thus, further researches are required for this topic. The
third is the information theory for Markov chain. This topic can be treated via Perron-Frobenius theorem.
Finally, I am also working on non-equilibrium statistical physics for closed system, which reﬁnes thermo
dynamics.

Major Publications:[1] M. Hayashi and H. Nagaoka, “General formulas for capacity of classical-quantum channels,” IEEE Transactions on Information Theory, 49 (2003), no. 7, 1753-1768.
[2] M. Hayashi, “Upper bounds of eavesdropper’s performances in ﬁnite-length code with the decoy method,”
Physical Review A, 76, (2007), 012329.
[3] M. Hayashi, “Universal coding for classical-quantum channel,” Communications in Mathematical Physics
289 (2009), no. 3, 1087-1098.
[4] M. Hayashi, “Information Spectrum Approach to Second-Order Coding Rate in Channel Coding,” IEEE
Transactions on Information Theory, 55 (2009), no. 11, 4947 - 4966.

14

Awards and Prizes:• 2015 JSPS PRIZE: “Information Theory and Quantum Information Theory for Finite-CodingLength”• 2011 IEEE Information Theory Society Paper Award: “Information Spectrum Approach toSecond-Order Coding Rate in Channel Coding” IEEE Transactions on Information Theory,
Vol. 55, No. 11, 4947 - 4966 (2009). This prize is the most distinguished paper award in theinformation theory community.
• Japan IBM prize in the computer science section 2010: “Universal protocol in quantum
information and its application to quantum key distribution” This prize is one of the mostdistinguished prizes in Japan among information science for researchers across Japan under45.
• Funai Foundation for Information Technology Award in the computer science category 2010:“Universal quantum information protocol and its application to quantum cryptography”• 2001 SITA Encouragement Award (by The Society of Information Theory and its Applications(SITA)): “Variable length universal entanglement concentration by local operations”

Education and Appointments:
1998
2000
2003
2007
2012

JSPS Research Fellow at Kyoto University
Researcher, Brain Science Institute, RIKEN
Research Manager, ERATO Quantum Computation and Information Project, Japan
Science and Technology Agency
Associate Professor, Tohoku University
Professor, Nagoya University

Message to Prospective Students:
The following are candidates of topic for Master course:
Quantum information theory, Quantum cryptography, Quantum statistical inference, Information theory.
Since these topics are linked to each other, it is possible to tackle one topic based on knowledge obtained
from another topic. The following texts are useful for these topics.
• M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University
Press (2000).
• M. Hayashi,Quantum Information: An Introduction, Springer (2006).
• M. Hayashi, S. Ishizaka, A. Kawachi, G. Kimura, and T. Ogawa, Introduction to Quantum Information
Science, Graduate Texts in Physics, Springer (2014).
Since quantum information is a new area, students can relatively easily publish their own research article
while the possibility depends on their eﬀorts and their ability. Required preliminary knowledge is level 1,
which corresponds to the contents for ﬁrst, second, and third year courses. In particular, I ask students tomaster linear algebra, calculus, and elementary probability, which contains the central limit theorem but
does not necessarily contain measure theory, because they are very important for these topics. Since these
topics are opened to other research area, students are required to study topics outside of mathematics.
Since the above research topics are applied to practical problems, it is required not only to understandmathematically formulated problems but also to grasp the target problem itself. For doctoral course, I can
supervise group-theoretical approaches to the above topics as well as the above topics themselves.

15

HAYASHI, Takahiro

Associate Professor

Oﬃce:

Rm 443 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2416

Email:

hayashi@math.nagoya-u.ac.jp(ext. 2416)

Membership of Academic Societies:
Mathematical Society of Japan

Research Interest:• Hopf algebras and their generalizations• Reprentations of quantum groups• Tensor categories

Research Summary:
I am currently interested in Hopf algebras and their generalizations, such as face algebras,
weak Hopf algebras and Hopf algebebroids. These structures relate to many area of mathematics
and mathemacical physics, including, low dimensional topology, operator algebras, Yang-Baxter
equations and conformal ﬁeld theory.

Major Publications:[1] T. Hayashi, Sugawara operators and Kac-Kazhdan conjecture, Invent. Math., 94 (1988), no. 1,
13-52.
[2] T. Hayashi, Quantum group symmetry of partition functions of IRF models and its application
to Jones’ index theory, Commun. Math. Phys., 157 (1993), 331-345.
[3] T. Hayashi, Coribbon Hopf (face) algebras generated by lattice models. J. Algebra, 233 (2000),
614–641.
[4] T. Hayashi, A brief introduction to face algebras, in New trends in Hopf algebra theory, La
Falda 1999, Contemp. Math. 267, Amer. Math. Soc., 2000, pp. 161-176.

Education and Appointments:
1988
1995

Assistant, Nagoya University
Assistant Professor, Nagoya University

16Message to Prospective Students:
The following is a list of possible text books for Small Group Class (Seminar).
[1] C. Kassel, Quantum Groups, Graduate texts in Mathematics 155, Springer-Verlag, 1995.
[2] D. E. Radford, Hopf Algebras, World Scientiﬁc, 2012.
[3] J. Hong and S.-J. Kang, Introduction to Quantum Groups and Crystal Bases, Amer. Math.
Soc., 2002.
[4] J. C. Jantzen, Lectures on Quantum Groups, American Mathematical Society, 1996.
[5] V. G. Kac, Inﬁnite-Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, 1990.

17

HESSELHOLT, Lars

Professor

Oﬃce:

Rm 449 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2547

Email:

larsh@math.nagoya-u.ac.jp

Web-site:(ext. 2547)

http://www.math.nagoya-u.ac.jp/~larsh/

Membership of Academic Societies:
American Mathematical Society
European Mathematical Society
Danish Mathematical Society

Research Interest:• Algebraic K-Theory• Equivariant Homotopy Theory• p-Adic Arithmetic Geometry

Research Summary:My area of specialization is algebraic K-theory and algebraic topology, in general, and topological cyclic homology and equivariant stable homology theory, in particular. As homological algebra
came about because of the fact that not every module is projective, algebraic K-theory came aboutbecause of the fact that not every projective module is free. That is, homological algebra and algebraic K-theory are necessary in order to do linear algebra over a ring that is not a ﬁeld. Such rings
appear in all brances of mathematics, e.g. the coordinate ring of scheme in algebraic geometry andnumber theory and the integral and spherical group rings of a discrete group in geometric topology.
Despite the name, algebraic K-theory does not admit a purely algebraic deﬁnition. Instead,following Quillen, the algebraic K-groups of a ring R are deﬁned to be the homotopy groups
Kn (R) = πn (K(R))
of a (pointed) topological space K(R) built by gluing together simplices in a way that reﬂects
the structure of the category of ﬁnitely generated projective right R-modules. To understandthe structure of these groups is a very deep problem indeed. For instance, the statement that
K4m (Z) = 0 for all positive integers m is equivalent to the Kummer-Vandiver conjecture in number
theory which states that no prime number p divides the class number of the ﬁeld Q(ζp + ζp−1 ).
(The conjecture is known to hold for p < 163, 000, 000.) The cyclotomic trace map, which is a
generalization of the classical Chern character, is a natural map from algebraic K-theory to a
topological reﬁnement of Connes’ cyclic homology deﬁned by Bökstedt-Hsiang-Madsen. It is an
important tool for understanding algebraic K-theory. Indeed, for non-regular rings, it is currently
the only tool available. For instance, it is proved in the paper [1] that if OK a complete discrete
valuation ring with quotient ﬁeld K = OK [1/p] of characteristic 0 and algebraically closed residue
ﬁeld k = OK /mK of odd characteristic p, then there is the following canonical isomorphism.
K∗ (K, Z/pZ) → (W Ω∗OK (log mK ) ⊗ SZ/pZ (µp ))F =1
To formulate this result, it was necessarly to deﬁne the de Rham-Witt complex with log polesthat appear on the right-hand side. In this way, understanding the structure of algebraic K-theory18

often necessitates the creation of new mathematics. Motivic cohomology is another example of newmathematics that was created in this way.

Major Publications:[1] L. Hesselholt and I. Madsen, On the K-theory of local ﬁelds, Ann. of Math. 158 (2003), 1–113.
[2] T. Geisser and L. Hesselholt, The de Rham-Witt complex and p-adic vanishing cycles, J. Amer.
Math. Soc. 19 (2006), 1–36.
[3] T. Geisser and L. Hesselholt, Bi-relative algebraic K-theory and topological cyclic homology,
Invent. Math. 166 (2006), 359–395.
[4] L. Hesselholt, On the p-typical curves in Quillen’s K-theory, Acta Math. 177 (1996), 1–53.
[5] L. Hesselholt and I. Madsen, Cyclic polytopes and the algebraic K-theory of truncated polynomial algebras, Invent. Math. 130 (1997), 73–97.

Awards and Prizes:• Alfred P. Sloan Fellowship (1998)• Foreign Member of the Royal Danish Academy of Sciences and Letters (2012)• Niels Bohr Professor (2013–2018)• Clay Senior Scholar (2014)

Education and Appointments:
1994
1994
1997
2001
2008

Institut Mittag-Leﬄer Postdoctoral Fellow
C.L.E. Moore Instructor, M.I.T.
Assistant Professor, M.I.T.
Associate Professor, M.I.T.
Professor, Nagoya University

Message to Prospective Students:
The research area of algebraic K-theory is suitable for students with strong interest and background in algebra, number theory, and homotopy theory. It is an area that intersects numerous
mathematical ﬁelds and for that reason it is necessary to learn a lot of background material before
starting cutting-edge research. For that reason, it is important that students are capable of readingbooks and research papers independently. Here is a representative list of books and articles:[1] D. Quillen, Homotopical algebra, Lecture Notes in Math. 43, Springer, Berlin 1967.
[2] D. Quillen, Higher algebraic K-theory. I. Algebraic K-theory, I: Higher K-theories (Proc. Conf.,
Battelle Memorial Inst., Seattle, WA, 1972), pp. 85–147. Lecture Notes in Math., 341, Springer,
Berlin 1973.
[3] F. Waldhausen, Algebraic K-theory of spaces. Algebraic and geometric topology (New
Brunswick, N.J., 1983), 318?419, Lecture Notes in Math., 1126, Springer, Berlin, 1985.
[4] J. Milnor, Introduction to algebraic K-theory. Annals of Mathematics Studies, 72. Princeton
University Press, Princeton, N.J., 1971.

19

HIRAI, Hiroshi

Professor
Oﬃce:

Rm 501 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 2432

Email:

hirai.hiroshi@math.nagoya-u.ac.jp

Web-site:(ext. 2432)

https://www.math.nagoya-u.ac.jp/~hirai.hiroshi/index.html

Membership of Academic Societies:
The Japan Society for Industrial and Applied Mathematics (JSIAM) ,The Operations Research Society of Japan (ORSJ)

Research Interest:• Optimization• Algorithm• Discrete Mathematics

Research Summary:Discrete optimization (or combinatorial optimization) is a mathematical theory that aims at
designing algorithms to ﬁnd the most desirable one among a ﬁnite but huge number of combinations
in a realistic time. The theory of discrete optimization began in the mid-1960s with a series ofstudies by J. Edmonds, in which he proposed the thesis of “eﬃcient algorithm ≒ polynomial-timealgorithm,” introduced polyhedral methods, and showed the importance of matroids and submodular
functions in algorithm design. The polyhedral method is one of the basic paradigms of algorithm
design, which is ”to design an algorithm from the viewpoint of continuous optimization by embedding
a discrete optimization problem into a convex optimization problem in Euclidean space.” While
many discrete optimization problems are NP-hard, understanding the class P of problems withpolynomial-time algorithms remains an important challenge. Matroids and submodular functions,which are closely related to class P, are recognized as discrete convex sets and functions, and have
evolved into the theory of convex analysis on integer lattice points—discrete convex analysis—.
I worked on multi-commodity extensions of Ford-Fulkerson’s maximum ﬂow minimum-cut theorem in network ﬂow problem [1], and polynomial-time solvability classiﬁcation of facility location
problems that generalizes the minimum cut problem [2]. These studies deal with discrete convex
optimization on special graph structures, and I found that they can be embedded as geodesicallyconvex optimization on nonpositively curved metric spaces (CAT(0)-spaces) [3]. I then obtained
the idea of “developing convex optimization/algorithm theory for nonpositively curved spaces, andapplying it to discrete optimization”, which updates the above paradigm. Since then, I have been
conducting research based on this idea. Recently, I focus on the following two themes.
The ﬁrst is Edmonds problem, that is the symbolic rank compution of matrices containing variables. Although it is a fundamental problem with many applications, the existence of deterministic
polynomial-time algorithms is an important open problem in theoretical computer science. Also
it may be called algebraic combinatorial optimization, since it generalizes a class of combinatorial
optimization problems such as bipartite matching and linear matroid intersection. Recently, anoncommutative version of Edmonds problem was introduced, in which the variables are noncommutative. It was shown that the rank (noncommutative rank) in this setting can be computed in20polynomial time. The computation of the noncommutative rank becomes a new type of optimization problem over the family of vector subspaces. I have developed a polynomial-time algorithm
using convex optimization of a non-manifold CAT(0) space [8]. I further generalized it to a weighted
version (degree computation of noncommutative determinants) [5].
The second is study of convex optimization on Hadamard manifolds and symmetric spaces. The
noncommutative Edmonds problem over C, which can also be formulated as a norm minimization
problem over an orbit of group action, becomes convex optimization on symmetric spaces of nonpositive curvature. Various other applications have been found. While studying diﬀerential geometry,
I am trying to develop convex analysis and interior point methods on such manifolds [9, 10].
I have also studied discrete structures such as lattices, matroids, buildings, and discrete metricspaces due to my interest as stages for discrete optimization [4, 6, 7].
For more information on my research, please visit my web page.

Major Publications:[1] H. Hirai: The maximum multiﬂow problems with bounded fractionality, Mathematics of Operations Research 39 (2014), 60–104.
[2] H. Hirai, Discrete convexity and polynomial solvability in minimum 0-extension problems, Mathematical Programming, Series A 155, (2016) 1–55.
[3] H. Hirai: L-convexity on graph structures, Journal of the Operations Research Society of Japan61 (2018), 71–109.
[4] H. Hirai: Uniform semimodular lattices and valuated matroids, Journal of Combinatorial Theory, Series A 165 (2019), 325–359.
[5] H. Hirai: Computing the degree of determinants via discrete convex optimization on Euclidean
buildings, SIAM Journal on Applied Algebra and Geometry 3 (2019), 523–557.
[6] J. Chalopin, V. Chepoi, H. Hirai and D. Osajda: Weakly modular graphs and nonpositive
curvature, Memoirs of the AMS 268, no.1309, (2020).
[7] H. Hirai: A nonpositive curvature property of modular semilattices, Geometriae Dedicata 214(2021), 427–463.
[8] M. Hamada and H. Hirai: Computing the nc-rank via discrete convex optimization on CAT(0)spaces, SIAM Journal on Applied Geometry and Algebra 5 (2021), 455–478.
[9] H. Hirai: Convex analysis on Hadamard spaces and scaling problems, Foundations of Computational Mathematics, to appear.
[10] H. Hirai, H. Nieuwboer, and M. Walter: Interior-point methods on manifolds: theory andapplications, FOCS 2023, to appear.

Awards and Prizes:• 2014, Research Award, The Operations Research Society of Japan.
• 2018, The Young Scientists’ Award, The Commendation for Science and Technology by the
Minister of Education, Culture, Sports, Science and Technology.

21

Education and Appointments:
2004
2004
2007
2010
2014
2023

Master of Information Science and Technology, Graduate School of Information Science and Technology, The University of Tokyo
Research Associate, Research Institute for Mathematical Sciences, Kyoto University
Assistant Professor, Research Institute for Mathematical Sciences, Kyoto University
Lecturer, Graduate School of Information Sciences and Technology, The Universityof Tokyo
Associate Professor, Graduate School of Information Sciences and Technology, The
University of Tokyo
Professor, Graduate School of Mathematics, Nagoya University

Message to Prospective Students:
In the small group seminar, you will study the following basic textbooks on optimization andalgorithms, and tackle speciﬁc problems according to your interests.
[1] S. Boyd and L. Vandenberghe: Convex Optimization, Cambridge University Press, 2004.
[2] J. Kleinberg and E. Tardos: Algorithm Design, Pearson, 2006.
I expect that you will ﬁnd interesting research directions if you review various problems in mathematical science from the viewpoints of optimization, algorithms, and computational complexity.

22

HISHIDA, Toshiaki

Professor

Oﬃce:

Rm 507 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 4838

Email:

hishida@math.nagoya-u.ac.jp(ext. 4838)

Membership of Academic Societies:
Mathematical Society of Japan

Research Interest:• Nonlinear partial diﬀerential equations• Navier-Stokes system

Research Summary:My research interest is focused on nonlinear partial diﬀerential equations (PDEs) arising fromﬂuid mechanics. The motion of the ﬂuid is essentially nonlinear on the one hand and analysis is often
involved even at the level of linearized systems (around a basic state) on the other. Furthermore,some qualitative properties, such as asymptotic structure at spatial/temporal inﬁnity, of solutions
could depend on the ﬂuid region and on the motion of obstacles immersed there. Mathematicalanalysis of the Navier-Stokes system is traced back to a series of celebrated papers by Leray in
1930s. Later on, remarkable progress has been made by a lot mathematicians and it has always
had much inﬂuence on analysis of some other PDEs. In spite of their eﬀorts since the landmark by
Leray, the following problem still remains open: The unique existence of regular solution globallyin time without any smallness assumption on initial data no matter how smooth they are. Besides
this, however, we have many other challenging problems, in particular, it is of utmost importanceto ﬁnd mathematical properties (asymptotic structure, stability, attainability,...) of ﬂows arisingin physically relevant situations, such as the ﬂow past/around a rotating obstacle [1, 2, 4, 5, 9] andeven ﬂuid-structure interaction [3, 8].

Major Publications:[1] T. Hishida, Spatial pointwise behavior of time-periodic Navier-Stokes ﬂow induced by oscillationof a moving obstacle, J. Math. Fluid Mech. 24 (2022), Paper No.102.
[2] T. Hishida and M. Kyed, On the asymptotic structure of steady Stokes and Navier-Stokes ﬂows
around a rotating two-dimensional body, Paciﬁc J. Math. 315 (2021), 89–109.
[3] T. Hishida, A. Silvestre and T. Takahashi, Optimal boundary control for steady motions of aself-propelled body in a Navier-Stokes liquid, ESAIM: Control, Optimisation and Calculus of
Variations 26 (2020), Paper No.92.
[4] T. Hishida, Decay estimates of gradient of a generalized Oseen evolution operator arising fromtime-dependent rigid motions in exterior domains, Arch. Rational Mech. Anal. 238 (2020),
215–254.
[5] T. Hishida, Large time behavior of a generalized Oseen evolution operator, with applications
to the Navier-Stokes ﬂow past a rotating obstacle, Math. Ann. 372 (2018), 915–949.
[6] T. Hishida, Stationary Navier-Stokes ﬂow in exterior domains and Landau solutions, Handbookof Mathematical Analysis in Mechanics of Viscous Fluids 299–339, Springer, 2018.
23

[7] T. Hishida and P. Maremonti, Navier-Stokes ﬂow past a rigid body: attainability of steadysolutions as limits of unsteady weak solutions, starting and landing cases, J. Math. Fluid Mech.
20 (2018), 771–800.
[8] T. Hishida, A. Silvestre and T. Takahashi, A boundary control problem for the steady selfpropelled motion of a rigid body in a Navier-Stokes ﬂuid, Annales de l’Institut Henri Poincare/ Analyse non lineaire 34 (2017), 1507–1541.
[9] T. Hishida, Mathematical analysis of the equations for incompressible viscous ﬂuid around arotating obstacle, Sugaku Expositions 26 (2013), 149–179.
[10] T. Hishida, The nonstationary Stokes and Navier-Stokes ﬂows through an aperture, Contributions to Current Challenges in Mathematical Fluid Mechanics, 79–123, Adv. Math. Fluid Mech.,
Birkhäuser, Basel, 2004.
[11] T. Hishida, An existence theorem for the Navier-Stokes ﬂow in the exterior of a rotating
obstacle, Arch. Rational Mech. Anal. 150 (1999), 307–348.

Awards and Prizes:• Analysis Prize (2007)

Education and Appointments:
1993
1993
1994
1997
2000
2008

Dr. Sci., Waseda University
Research Associate, Waseda University
Research Associate, Kumamoto University
Assistant Professor, Niigata University
Associate Professor, Niigata University
Professor, Nagoya University

Message to Prospective Students:
As the subject in the seminar of master course, I can propose, for instance, (1) elliptic PDEs ofsecond order; (2) the method of functional analysis such as semigroup theory; (3) large time behavior of solutions to evolution equations by means of spectral analysis; (4) mathematical analysis of
the Navier-Stokes system, which are related with each other. As the textbook, I would recommend
1. L. C. Evans, Partial Diﬀerential Equations, Amer. Math. Soc., 1998.
2. D. Gilbarg and N. S. Trudinger, Elliptic Partial Diﬀerential Equations of Second Order,
Springer, 1977.
3. H. Sohr, The Navier-Stokes Equations, An Elementary Functional Analytic Approach, Birkhäuser,
2001.
4. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations,
Steady Problems, Second Edition, Springer, 2011.
5. T.-P. Tsai, Lectures on Navier-Stokes Equations, Amer. Math. Soc., 2018.
For those who wish to proceed to the doctor course, they are asked to read some related
papers. In the doctor course, what is important is to ﬁnd a problem and to develop analysis byhimself/herself. It might be better to work on a bit diﬀerent subject from mine.

24

ITO, Kentaro

Associate Professor
Oﬃce:

Rm 425 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 5594

Email:

itoken@math.nagoya-u.ac.jp

Web-site:(ext. 5594)

http://www.math.nagoya-u.ac.jp/~itoken/

Membership of Academic Societies:
Mathematical Society of Japan

Research Interest:• Hyperbolic geometry, conformal Geometry• Kleinian groups, Riemann surfaces, Teichmüller theory• Low-dimensional topology

Research Summary:
My major research interest is hyperbolic geometry. Especially I am interested in hyperbolic 3manifolds and their deformation spaces. A hyperbolic 3-manifold is obtained as the quotient manifold of the hyperbolic 3-space by the action of a Kleinian group, a discrete subgroup of PSL(2, C).
The boundary of the deformation space of a Kleinian group has fractal structure, and is very
complicated. I am eager to understand the ‘complexity’ of the boundaries of these spaces.
I also interested in higher dimensional Kleinian groups, especially 4-dimensional Kleinian groups
acting on the hyperbolic 4-space. In this case, limit sets of Kleinian groups are fractal objects in
3-sphere. Figures below are computer-generated limit sets of 4-dimensional Kleinian groups with
two generators.

Major Publications:[1] K. Ito, Convergence and divergence of Kleinian punctured torus groups, Amer. J. Math. 134(2012), 861–889.
[2] Y. Araki and K. Ito, An extension of the Maskit slice for 4-dimensional Kleinian groups, Conform. Geom. Dyn. 12 (2008), 199–226.
[3] K. Ito, On continuous extensions of grafting maps, Trans. Amer. Math. Soc. 360 (2008), 3731–
3749.
[4] K. Ito, Exotic projective structures and quasi-Fuchsian space, II, Duke Math. J. 140 (2007),85–109.
25

[5] K. Ito, Schottky groups and Bers boundary of Teichmuller space, Osaka J. Math. 40 (2003),
639–657.
[6] K. Ito, Exotic projective structures and quasi-Fuchsian space, Duke Math. J. 105 (2000), 185–
209.

Education and Appointments:
2000
2000
2007

Ph.D. at Tokyo Institute of Technology
Assistant Professor, Nagoya University
Associate Professor, Nagoya University

Message to Prospective Students:
Some basic references of this area are [1], [2] and [3]. More advanced topics can be found in[4], [5], [6] and [7]. Especially [5] is the best reference to get an impression of this area. Some
master course students used [3], [8], [9] and [10] as textbooks of seminar. [11] will be used in theforthcoming master course seminar. [12] is a good guidebook for drawing computer graphics as in
the previous page. Sutudentswho are interested in such computer graphics are also welcome.
[1] S. Katok, Fuchsian Groups, The University of Chicago Press, 1992.
[2] F. Bonahon, Low-dimensional geometry: from euclidean surfaces to hyperbolic knots, AMS,
2009.
[3] A. F. Beardon, The geometry of Discrete Groups, Springer GTM 91, 1983.
[4] A. Marden, Outer Circles, Cambridge University Press, 2007.
[5] K. Matsuzaki and M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford Science
Publications, 1998.
[6] Y. Imayoshi and M. Taniguchi, An Introduction to Teichmüller Spaces, Springer, 1992.
[7] A. Fathi, F. Laudenbach and V. Poenaru, (translated by D. M. Kim and D. Margalit) Thurston’s
work on surfaces, Princeton University Press 2012.
[8] K. Stephenson, Introduction to Circle Packing, Cambridge University Press, 2005.
[9] P. J. Nocholls, The Ergodic Theory of Discrete Groups, Cambridge University Press, 1989.
[10] F. Dal’Bo, Geodesic and Horocyclic Trajectories, Springer, 2011.
[11] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton University Press,
2012.
[12] D. Mumford, C. Series, D. Wright, Indra’s Pearls, Cambridge University Press, 2002.

26

IZUMI, Keisuke

Associate Professor

Oﬃce:

Rm 502 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 5599

Email:

izumi@math.nagoya-u.ac.jp(ext. 5599)

Membership of Academic Societies:
The Physical Society of Japan (JPS)

Research Interest:• Gravitational theory• Cosmology

Research Summary:
I am working on gravitational theory. My ﬁnal purpose is the construction of the quantum
theory of gravity.
General relativity, proposed by A. Einstein, is the fundamental theory of classical gravity. This
theory has mathematically interesting aspects, and thus many researchers are investigating its
mathematical structure. Moreover, general relativity is an excellent theory in physics since noconﬂicts with experimental results and observational facts have been found.
Although general relativity is an excellent theory of classical gravity, there is a problem; “itsstraightforward quantization does not work well.” Since the fundamental physics except gravity is
described as quantum theory, the quantization of gravity theory is required to unify all theories offundamental physics. Because the quantization of general relativity doesn’t go well, by modifying
or extending general relativity (or also modifying the quantum theory) many researchers try toconstruct the uniﬁed theory describing all physical phenomenon. The strong candidate is the superstring theory.
Toward the construction of quantum gravity theory, we have the following three topics to workon;
• Investigating the mathematical structures of fundamental classical theory of gravity, generalrelativity• Approaching the quantum gravity theory by the investigation of the possible extension fromgeneral relativity.
• Investigating the mathematical structures of candidates of quantum gravity theory, such as
the string theory.
I am mainly working on the second topic, e.g. the property of the general relativity in higher
dimension and the causal structures in extended theory from the general relativity.

Major Publications:[1] K. Izumi, K. Koyama, O. Pujolas and T. Tanaka, “Bubbles in the Self-Accelerating Universe,”
Phys. Rev. D 76, 104041 (2007)[2] K. Izumi, “Orthogonal black di-ring solution,” Prog. Theor. Phys. 119, 757 (2008)[3] K. Izumi, “Causal Structures in Gauss-Bonnet gravity,” Phys. Rev. D 90, no. 4, 044037 (2014)[4] R. Emparan, K. Izumi, R. Luna, R. Suzuki and K. Tanabe, “Hydro-elastic Complementarity
in Black Branes at large D,” JHEP 1606, 117 (2016)27

2 gravity,”[5] Y. Abe, T. Inami and K. Izumi, “Perturbative S-matrix unitarity (S † S = 1) in RµνMod. Phys. Lett. A 36, no.16, 2150105 (2021)[6] K. Izumi, Y. Tomikawa, T. Shiromizu and H. Yoshino, “Area bound for surfaces in genericgravitational ﬁeld,” PTEP 2021, no. 8, 083E02 (2021)

Education and Appointments:
2009
2009
2011
2011
2015
2016
2021

D. Sc., Department of Physics, Kyoto University, Japan
Postdoctoral Fellow, Institute for the Physics and Mathematics of the Universe, The
University of Tokyo
Postdoctoral Fellow, Yukawa Institute for Theoretical Physics, Kyoto University
Distinguished Junior Postdoctoral Fellow, Leung Center for Cosmology and Particle
Astrophysics, National Taiwan University, Taiwan(R.O.C)
Postdoctoral Fellow, Department of Physics, University of Barcelona, Spain,
Assistant Professor, Department of Mathematics, Nagoya University
Lecturer, Department of Mathematics, Nagoya University

Message to Prospective Students:
General relativity is constructed with a simple equation, but has the various nontrivial properties. Meanwhile, the construction of quantum gravity theory is a challenging research topic. If you
want to study one of them (or both), don’t hesitate to ask me.
If you want to study these topics, it is better to start from the basics of general relativity. I
recommend the following book;• R. M. Wald, General Relativity, Chicago University Press
After mastering the basics of general relativity, you can go to the next step, reading the other books
and papers that you are interested in.

28

KANNO, Hiroaki

Professor

Oﬃce:

Rm 447 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2417

Email:

kanno@math.nagoya-u.ac.jp(ext. 2417)

Membership of Academic Societies:
JMS (The Mathematical Society of Japan),
JPS (The Physical Society of Japan)

Research Interest:• Topological gauge/string theory and enumerative geometry• Supersymmetric gauge theories and integrable system

Research Summary:
I am working on mathematical physics. In particular my recent interest is in topologicalgauge/string theory and supersymmetric gauge theories. Gauge theory, or the Yang-Mills theoryis the most fundamental mathematical framework underlying the Standard Model of Elementary
Particles, which is being conﬁrmed by the recent discovery of “Higgs particle” by LHC at CERN.
However, the Standard Model still has a few puzzles and does not incorporate gravity. The idea ofsupersymmetry is expected to resolve some of the puzzles and string theory is the most promising
candidate for the uniﬁed theory including gravity.
In the study of these theories the quantum dynamics at strong coupling plays a crucial role.
This is a diﬃcult problem in general, since we cannot rely on the perturbation theory, which has
been successful in the Yang-Mills theory at weak coupling. In such a circumstance exactly solvablemodels, even if they are toy models, are very valuable, since they will tell us some aspects of quantum dynamics beyond the perturbation theory. They are also called (quantum) integrable system.
Exact solvability in quantum ﬁled theory usually follows from inﬁnite dimensional symmetry and/or
the idea of dualities which exchanges the weak coupling and the strong coupling regions. Topological gauge/string theory is a typical example. The representation theory of inﬁnite dimensional
symmetry and the combinatorics are main mathematical tools for exact solvability. Combined with
the idea of the moduli space, exactly solvable models in supersymmetic gauge/string theories sometimes “solve” hard enumerative problems in symplectic/complex geometry, for example through themirror symmetry.

Major Publications:[1] H. Kanno, Weil Algebra Structure and Geometrical Meaning of BRST Transformation in Topological Quantum Field Theory, Z. Phys. C43 (1989) 477-484.
[2] T. Inami and H. Kanno, Lie Superalgebraic Approach to Super Toda Lattice and Generalized
Super KdV Equations, Commun. Math. Phys. 136 (1991) 519-542.
[3] L. Baulieu, H. Kanno and I.M. Singer, Special Quantum Field Theories in Eight and Other
Dimensions, Commun. Math. Phys. 194 (1998) 149-175.
[4] T. Eguchi and H. Kanno, Topological Strings and Nekrasov’s Formulas, JHEP 0312 (2003)006.
[5] H. Awata and H. Kanno, Instanton counting, Macdonald function and the moduli space of
D-branes, JHEP 0505 (2005) 039.
29

Education and Appointments:
1989
1991
1992
1993
1995
1998
2001
2004
2010

JSPS post-doctoral fellow at Yukawa Institute, Kyoto
Post-doctoral fellow at ICTP, Trieste
Research fellow of the Nishina memorial foundation at DAMTP, Cambridge Univ.
Assistant professor, Department of Mathematics, Hiroshima university
Lecturer, Department of Mathematics, Hiroshima university
Associate professor, Department of Mathematics, Hiroshima university
Associate professor, Graduate school of Mathematics, Nagoya university
Professor, Graduate school of Mathematics, Nagoya university
Joint appointment at KMI (Kobayashi-Maskawa Institute), Nagoya university

Message to Prospective Students:
The recent topics of Small Group Class (the tutorial seminar) in the master course include thetheory of solitons and instantons, the geometry of generalized complex structure and the quantum
theory of gauge ﬁelds. Though basic knowledge of classical and quantum mechanics is preferable,
it is not absolutely required. More important point is that you are full of curiosity.
The theory of integrable systems is one of the main subjects in mathematical physics. A typicalmethod in mathematical physics is to construct the models of physical system of interest andanalyze them by making a good use of mathematics. In this sense the exactly solvable models arequite remarkable both in mathematical and physical points of view. They give us valuable lessons
on physical phenomena which are hard to access by approximations, while deep mathematicalstructures, such as symmetry and duality, underlie the integrability of the models. In the study ofintegrable systems the computation by hand and/or computers is also an important business.
To give you some impression on the theory of integrable systems, let me mention my favorite
examples; the ﬁrst one is the theory of solitons which is closely related to conformal ﬁeld theory
on the Riemann surface. A good reference is the following textbook:• T. Miwa, M. Jimbo and E. Date ; translated by Miles Reid, Solitons : diﬀerential equations,symmetries and inﬁnite dimensional algebras, Cambridge University Press , 2000.
The second example is the Seiberg-Witten theory of N = 2 supersymmetric Yang-Mills theory infour dimensions. You may ﬁnd several good review articles at “arXiv” : http://arxiv.org/. The
followings are examples in early days:• Adel Bilal, Duality in N=2 SUSY SU(2) Yang-Mills Theory, http://arxiv.org/abs/hep-th/9601007.
• L. Alvarez-Gaume and S.F. Hassan, Introduction to S-Duality in N=2 Supersymmetric Gauge
Theory, http://arxiv.org/abs/hep-th/9701069.

30

KATO, Jun

Associate Professor
Oﬃce:

Rm 503 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 2410

Email:

jkato@math.nagoya-u.ac.jp(ext. 2410)

Membership of Academic Societies:
The Mathematical Society of Japan

Research Interest:• Partial Diﬀerential Equations• Fourier Analysis

Research Summary:My research area is the theory of partial diﬀerential equations. I am mainly interested in nonlinear partial diﬀerential equations arising in mathematical physics, especially nonlinear dispersive
equations and nonlinear wave equations. Equations in this category includes nonlinear Schrödinger
equations, KdV equations, nonlinear Klein-Gordon equations, Maxwell-Schrödinger equations, etc.
I am studying such type of equations in the frame work of functional analysis. Generally, it isdiﬃcult to give a explicit solution to nonlinear partial diﬀerential equations for a given data, e.g.
initial data, boundary values, we ﬁrst consider the existence of a solution (and uniqueness of solutions, continuous dependence of solutions to initial data) by using functional analytic techniques.
After that, we study qualitative behavior of solutions, e.g. regularity, asymptotic behavior. Fourier
analysis is crucial to catch properties of solutions to dispersive equations and wave equations.
I am also interested in equations which have a geometric background, such as wave maps andthe Schrödinger maps. These are considered as the generalization of the wave equation and theSchrödinger equation to the evolution of the maps between manifolds, and give interesting problems
in the ﬁeld of analysis and geometry, e.g. properties of the target manifold would change the global
behavior of solutions.

Major Publications:[1] J. Kato, The uniqueness of nondecaying solutions for the Navier-Stokes equations, Arch. Rational Mech. Anal. 169 (2003), 159–175.
[2] J. Kato, Existence and uniqueness of the solution to the modiﬁed Schrödinger map,
Math. Res. Lett. 12 (2005), 171–186.
[3] J. Kato, M. Nakamura, T. Ozawa, A generalization of the weighted Strichartz estimates forwave equations and an application to self-similar solutions, Comm. Pure Appl. Math. 60 (2007),
164–186.
[4] J. Kato, T. Ozawa, Endpoint Strichartz estimates for the Klein-Gordon equation in two spacedimensions and some applications, J. Math. Pures Appl. 95 (2011), 48–71.

Awards and Prizes:• The MSJ Takebe Katahiro Prizes (2008)31

Education and Appointments:
2003
2003
2004
2006
2007
2009

Ph. D., Hokkaido University
COE Fellow, Tohoku University
JSPS Research Fellow, Kyoto University
Lecturer, Nagoya University
Associate Professor, Nagoya University∼ 2011, JSPS Postdoctoral Fellow for Research Abroad, New York University

Message to Prospective Students:
For the study of dispersive equations and wave equations, basic knowledge of functional analysis
and the Fourier analysis is required. So, in my small class I am planing to begin to read basic books
to make students be capable of reading recent research articles. The following books are examplesof the textbooks.
[1] H. Bahouri, J.-Y. Chemin, R. Danchin, “Fourier Analysis and Nonlinear Partial Diﬀerential
Equations,” Grundlehren der Mathematischen Wissenschaften 343, Springer (2011).
[2] C. D. Sogge, “Lectures on Nonlinear Wave Equations,” International Press (2008).
[3] S. Alinhac, “Hyperbolic Partial Diﬀerential Equations,” Universitext, Springer (2009).
[4] L. Grafakos, “Classical Fourier Analysis,” 2nd Ed., Graduate Text in Math. 249, Springer,
2008.
[5] F. Lin, C. Wang, “The Analysis of Harmonic Maps and Their Heat Flows,” World Scientiﬁc(2008).

32

KAWAMURA, Tomomi

Associate Professor

Oﬃce:

Rm 357 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 4534

Email:

tomomi@math.nagoya-u.ac.jp(ext. 4534)

Membership of Academic Societies:
The Mathematical Society of Japan

Research Interest:• Knot theory• Low dimensional topology

Research Summary:
Knot theory is a ﬁeld of geometry and topology, where we are interested in the complexity ofknots or links, i.e. simple closed curves in the space. Such complexity is expressed as numbers,
polynomials, etc. We call these values by invariants of knots or links. I mainly research the between
link diagrams and link invariants, for example, the number of crossing changes needed to unknot
the given link. It is amazing that a lot of seemingly easy formulas are proved using advanced
theories as singularity theory, contact geometry, gauge theory, or Khovanov homology theory.
The results in [1] of Major Publications are obtained from works due to Rudolph with related
to ‘Milnor’s conjecture’, solved by Kronheimer and Mrowka as researches on gauge theory. Theirworks relate to contact geometry and singularity theory. The articles [2, 3] are written on divide
knots deﬁned by A’Campo, a famous singularity theorist. Milnor’s conjecture has been reproved
combinatorially, using Khovanov homology theory established about 2000. The main result in [4]is partially reconsideration of the works in [1] after Khovanov homology theory. These results are
improved in [5]. In the paper [6], as a relation to crossing changes on knots and links, we obtained
a ”winning method” for an advanced version of the game ”Region Select” proposed in the text [9]mentioned below. We currently research how these games relate to other theories.
Recently, many researchers are interested in the relations between knot theory and the othertheory: representation theory, number theory, chemistry, biology, etc. It is hard to catch up withsuch researches, though I am locking forward to further evolution.

Major Publications:[1] T. Kawamura, On unknotting numbers and four-dimensional clasp numbers of links, Proc.
Amer. Math. Soc. 130 (2002), no. 1, 243–252.
[2] T. Kawamura, Quasipositivity of links of divides and free divides, Topology Appl. 125 (2002),
no. 1, 111–123.
[3] T. Kawamura, Links associated with generic immersions of graphs, Algebr. Geom. Topol. 4(2004), 571–594.
[4] T. Kawamura, The Rasmussen invariants and the sharper slice-Bennequin inequality on knots,
Topology 46 (2007), no. 1, 29–38.
[5] T. Kawamura, An estimate of the Rasmussen invariant for links and the determination forcertain links, Topology Appl. 196 (2015), 558–574.
33

[6] T. Kawamura, Integral region choice problem on link diagrams, Osaka J. Math. 60 (2023),
835–872.

Awards and Prizes:• 2003, MSJ Takebe Katahiro Prize for Encouragement of Young Researchers, Research on
divide knots and four-dimensional estimates of unknotting numbers

Education and Appointments:
2000
2000
2002
2007

Ph.D. in Mathematical Sciences, the University of Tokyo
JSPS Research FellowshipResearch Associate, Aoyama Gakuin University
Associate Professor, Nagoya University

Message to Prospective Students:
The students of the small group class I take charge of, usually read textbooks on knot theory
and low dimensional topology. Here are examples:[1] J. S. Birman, Braids, links and mapping class groups, Ann. of Math. Studies, 82, Princeton
University Press, 1974.
[2] J. M. Lee, Introduction to topological manifolds, Springer, 2000.
[3] L. H. Kauﬀman, Formal knot theory, Dover Publications, 2006.
[4] V. V. Prasolov and A.B.Sossinsky, Knots, links, braids and 3-manifolds, AMS, 1997.
[5] D. Rolfsen, Knots and links, Corrected reprint of the 1976 original. Math. Lect. Ser., 7. Publish
or Perish, Inc., Houston, TX, 1990.
[6] N. Ito, Category theory of knot theory (Japanese), Nippon hyoron sha, 2018[7] T. Ohtsuki, Knot invariants (Japanese), Kyoritsu Shuppan, 2015.
[8] A. Kawauchi, Knot theory (Japanese), Kyoritsu Shuppan, 2015.
[9] A. Kawauchi, K. Kishimoto, A. Shimizu, Knot theory and game : Mathematical world viewing
from the game ”Region Select” (Japanese) , Asakura Publishing, 2013.
[10] A. Hattori, Topology (Japanese), Iwanami Shoten, 1991.
They use not only main textbooks but also research papers and other books. They are expected
to decide the theme of master’s thesis by themselves, and read many articles where some errors oromissions exist occasionally. If you would like to start the study of knot theory quickly, I recommend
you to learn fundamental group and homology from textbooks on beginning of topology in advance.
I will also support your careful study on these.
I believe that you, a graduate student of Nagoya University, can manage to solve diﬃcult
problems and be glad to go a step further.

34

KITA, Nanao

Associate Professor
Oﬃce:

Rm 331 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 5612

Email:

kita at math.nagoya-u.ac.jp

Web-site:(ext. 5612)

https://sites.google.com/view/nanaokita

Research Interest:• Graph Theory• Discrete Optimization (Combinatorial Optimization)• Discrete Algorithm

Research Summary:Graphs are a mathematical abstraction of network structures. They are a major object of study
in discrete mathematics and theoretical computer science. Since network structures are ubiquitous,
unraveling the structures of graphs and designing algorithms for various problems on graphs attract
interest from many ﬁelds. There are numerous problems and questions that we need to solve. This
is why developing powerful theoretical tools eﬀective in diverse settings is crucial. Such tools will
enable us to build systematic theories and thus accelerate the growth of the ﬁeld. Decomposition
theorems for graphs are particularly eﬀective as this type of theoretical tool.
My research has focused on various decomposition theorems for graphs, which can serve asa foundation for studying them. In particular, I have engaged in canonical decomposition theory, including deriving new canonical decompositions and exploring their theoretical applications.
Canonical decompositions comprise a series of structure theorems with a common trait, providing a uniquely determined decomposition for each graph and enabling an observation of its entire
structure. Therefore, they function as powerful and versatile tools for studying graphs. Canonicaldecompositions were discovered around 1960 in the context of matching theory, a classical branchof graph theory and discrete optimization, and have been eﬀectively employed since then. However,each known canonical decomposition has its deﬁciencies, such as being applicable only to speciﬁcclasses of graphs. Consequently, canonical decomposition theory has long remained incomplete.
In my study, I have derived a new canonical decomposition that is applicable to all graphs andincludes the generalizations or reﬁnements of known canonical decompositions. (The generalized
cathedral and nonbipartite Dulmage–Mendelsohn decomposition [1, 3].) Furthermore, I have applied this new canonical decomposition to obtain an improved proof of some classical theorems,
such as the tight cut lemma by Edmonds et al. [2], and to characterize maximal barriers (dual
optimizers of the maximum matching problem) in terms of lattices [3].
These results have further opened new opportunities for various subjects of graph studies beyond
matching theory. That is, the theoretical foundations for these subjects can be innovated byproviding structure theorems similar to canonical decompositions. In light of this, my current focus
is particularly on parity factors (also known as T -joins) of graphs; see Kita [4] or other similar works.
Although parity factor theory is a classical topic in discrete optimization, encompassing various
routing problems as special cases, the theory still exhibits some immature aspects and needs a new
development in its foundational part. Parity factor theory is also well known for its connection
35

to a fundamental problem in statistical physics. Therefore, applying outcomes of my studies tostatistical physics is also one of my research agendas.
Discrete optimization (combinatorial optimization) contributes to the foundation of the computational complexity theory and therefore makes up a part of the foundation of theoretical computerscience. Grasping the true essence of discrete optimization problems solvable in polynomial time
has been a longstanding theme in theoretical computer science. One of my long-term objectives is
to approach this theme by exploring graph structures that represent polynomial time solvability insigniﬁcant discrete optimization problems.

Major Publications:[1] N. Kita, A partially ordered structure and a generalization of the canonical partition for generalgraphs with perfect matchings, Lecture Notes in Computer Science, vol. 7676 , 2012, pp. 85–94.
[2] N. Kita, Structure of towers and a new proof of the Tight Cut Lemma. Lecture Notes in
Computer Science, Vol. 10627, 2017, pp. 225-239.
[3] N. Kita, Nonbipartite Dulmage-Mendelsohn decomposition for Berge duality. Lecture Notes in
Computer Science, Vol. 10976, 2018, pp. 293–304.
[4] N. Kita, Graft Analogue of general Kotzig–Lovász decomposition, Discrete Applied Mathematics vol. 322, 2022, pp. 355–364.

Awards and Prizes:• 2012, Best student paper award, 23rd International Symposium on Algorithm and Computation

Education and Appointments:
2014
2015
2018
2022
2023

Postdoctoral Research Fellow, Graduate School of Frontier Science, University ofTokyo
Research Fellow of the Japan Society of the Promotion of Science (PD), National
Institute of Informatics
Assistant Professor, Faculty of Science and Technology, Tokyo University of Science
Specially Appointed Assistant Professor, Graduate School of Information Science,
Tohoku University
Associate Professor, Graduate School of Mathematical Science, Nagoya University

Message to Prospective Students:
Students with a strong interest in discrete mathematics and theoretical computer science areespecially welcome. For an idea of the textbooks typically used in our seminar, please refer to thefollowing books.
[1] W. Cook, et al., “Combinatorial optimization”, Wiley, 1997.
[2] A. Bondy, and U. S. R. Murty, “Graph theory”, Springer, 2008.
[3] B. Mohar, and C. Thomassen, “Graphs on surfaces”, Johns Hopkins University Press, 2001.
[4] M. Gröschel, et al., “Geometric algorithms and combinatorial optimization”, Springer, 2012.

36

LE GALL, François

Professor

Oﬃce:

Rm 404 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 2412

Email:

legall@math.nagoya-u.ac.jp

Web-site:(ext. 2412)

http://www.francoislegall.com/
https://www.math.nagoya-u.ac.jp/~legall/index.html (Group)

Membership of Academic Societies:
Information Processing Society of Japan, The Mathematical Society ofJapan, Association for Computing Machinery

Research Interest:• Algorithms• Computational complexity• Quantum computing

Research Summary:My research area is theoretical computer science, the ﬁeld of study that investigates computationfrom a mathematical perspective. Our group works mainly on quantum computing, and more
precisely quantum algorithms and quantum complexity. I give below a short description of these
two topics. Please have a look at the web page of our group (see 2nd URL above) for more details.
Quantum algorithms. While Shor’s algorithm for integer factoring and Grover’s algorithm forsearch are the most celebrated quantum algorithms, many other quantum algorithms have been
discovered in the past 25 years. One of the main goals of our group is to ﬁnd novel quantum
algorithms and discover “killer applications” of quantum computers. We also aim at showing
the superiority of quantum algorithms via rigorous mathematical analysis of their performance.
Recently we have been working on quantum algorithms for string problems [1], quantum algorithms
for chemistry and other problems from physics [2] and quantum distributed algorithms [3, 5].
Quantum complexity. The central question in quantum complexity theory is to understandin which situations quantum computation is more powerful than classical computation, and thenquantify the quantum advantage. In our group we investigate this question using mathematical
tools, in several setting such as time complexity, space complexity, query complexity, communication complexity or complexity classes. Recently we have been working for instance on establishing
quantum advantage for shallow quantum circuits [4] and making progress on the quantum PCP
conjecture [2].

Major Publications:[1] F. Le Gall and S. Seddighin. Quantum Meets Fine-grained Complexity: Sublinear Time Quantum Algorithms for String Problems. Algorithmica, Vol. 85(5), pp. 1251-1286, 2023.
[2] S. Gharibian and F. Le Gall. Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture. Proceedingsof the 54th ACM Symposium on Theory of Computing, 19-32, 2022.
[3] K. Censor-Hillel, O. Fischer, F. Le Gall, D. Leitersdorf and R. Oshman. Quantum Distributed
Algorithms for Detection of Cliques. Proceedings of the 13th Innovations in Theoretical Computer Science conference (ITCS 2022), pp. 35:1-35:25, 2022.
37

[4] F. Le Gall. Average-Case Quantum Advantage with Shallow Circuits. Proceedings of the 34th
Computational Complexity Conference, 21:1-21:20, 2019.
[5] F. Le Gall and F. Magniez. Sublinear-Time Quantum Computation of the Diameter in CONGEST Networks. Proceedings of the 37th ACM Symposium on Principles of Distributed Computing (PODC 2018), pp. 337-346, 2018.
[6] H. Kobayashi, F. Le Gall and H. Nishimura. Stronger Methods of Making Quantum Interactive
Proofs Perfectly Complete. SIAM Journal on Computing, Vol. 44(2), pp. 243-289, 2015.
[7] F. Le Gall: Powers of Tensors and Fast Matrix Multiplication. Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, pp. 296-303, 2014.
[8] F. Le Gall: Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments.
Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science, pp.
216-225, 2014.

Awards and Prizes:• NISTEP Award (2017)• ISSAC 2014 Distinguished Paper Award (2014)

Education and Appointments:
2006
2006
2009
2012
2016
2019
2022

PhD, Graduate School of Information Science and Technology, The University ofTokyo
Researcher, ERATO-SORST Quantum Computation and Information Project,
Japan Science and Technology Agency
Project Lecturer, Graduate School of Information Science and Technology, The University of Tokyo
Project Associate Professor, Graduate School of Information Science and Technology, The University of Tokyo
Program-Speciﬁc Associate Professor, Graduate School of Informatics, Kyoto University
Associate Professor, Graduate School of Mathematics, Nagoya University
Professor, Graduate School of Mathematics, Nagoya University

Message to Prospective Students:• My seminar for graduate research mainly focuses on quantum computation and quantum
information, but other topics in theoretical computer science (for instance, algorithms orcomplexity theory) are also possible. Students can basically choose any topic related to thosesubjects.
• The ﬁrst months of research are devoted to studying textbooks. The choice of the textbook depends on the subject and the students’ background. The following textbook is often
selected.
[1] M. Nielsen and I. Chuang. Quantum Computation and Quantum Information. Cambridge
University Press, 2010.
• Students then start reading technical research papers and getting familiar with recent research. After choosing an appropriate open problem (by discussing with me), they are encouraged to conduct original research.

38

MINAMI, Kazuhiko

Associate Professor

Oﬃce:

Rm 347 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 5578

Email:

minami@math.nagoya-u.ac.jp

Web-site:(ext. 5578)

http://www.math.nagoya-u.ac.jp/~minami/

Membership of Academic Societies:
Physical Society of Japan

Research Interest:• Statistical Mechanics• Mathematical Physics

Research Summary:
Statistical mechanics concerns basic raws which become valid in the limit of inﬁnite degrees of
freedom. It has been investigated in connection with many-body systems and condensed matterproblems. Statistical mechanics is the theoretical bases to investigate various physical phenomena
with large degrees of freedom, i.e. the bases of statistical physics. Statistical mechanics for nonequilibrium systems does not yet fully formulated except for the cases which lie in the neighborhoodof equilibrium. It becomes diﬃcult to investigate but becomes much more rich when interactionsin the systems are assumed to be quantum mechanical.
Statistical mechanics is basically an area in traditional physics, but can also be viewed as arepresentation of some mathematical structure. Statistical mechanics is a subject for mathematicalphysics, when one concentrate especially on its mathematical structures.
Statistical mechanics relate pure mathematics through integrable systems especially solvablelattice models, e.g. Onsager’s exact solution for the two-dimensional Ising model, some of thequantum spin models and vertex models. Quantum group was introduced in connection with the
Yang-Baxter equation which is a key concept in this area.
Statistical mechanics has not yet been fully applied for systems with quite large but ﬁnite
degrees of freedom, and still cannot explain non-equilibrium phenomena. However, in these areas,
one can ﬁnd interesting examples such as social and ecological phenomena, pattern formulations in
network systems etc. In addition to it, quantum mechanics itself are now again investigated fromthe viewpoint of the new information theory that is based on purely quantum mechanical eﬀects.
I will concern these new aspects of the statistical and quantum mechanics, in addition to the
problems which I have been investigated, i.e. the problems of phase transitions in two-dimensionallattice models, exact solutions for classical and quantum lattice models.

Major Publications:[1] K. Minami and M. Suzuki, Non-universal critical behaviour of two-dimensional Ising systems,
J. Phys. A27 (1994) 7301-7311.
[2] K. Minami, The zero-ﬁeld susceptibility of the transverse Ising chain with arbitrary spin , J.
Phys. A29 (1996) 6395-6405.
[3] K. Minami, The susceptibility in arbitrary directions and the speciﬁc heat of general Ising-type
chains with uniform, periodic and random structures, J.Phys.Soc.Jpn. 67 (1998) 2255-2269.
39

[4] K. Minami, An equivalence relation of boundary/initial conditions and the inﬁnite limit properties, J.Phys.Soc.Jpn. 74 (2005) 1640.
[5] K. Minami, The free energies of six-vertex models and the n-equivalence relation, J. Math.
Phys. 49 (2008) 033514.

Education and Appointments:
1993
1995
1998

Ph D, School of Science, the University of Tokyo
Assistant Professor, Graduate School of Mathematics, Nagoya University
Associate Professor, Graduate School of Mathematics, Nagoya University

Message to Prospective Students:
Let me list the textbooks which I have used in my seminar. It is recommended to move tooriginal topics after ﬁnishing reading your textbooks of the statistical mechanics and the quantum
mechanics.

D. N. Zubarev, ” Non-equilibrium Statistical Thermodynamics”
L. D. Landau and E. F. Lifshitz, ”Quantum Mechanics”, ”Statistical Physics”
J. von Neumann, ”Mathematical Foundations of Quantum Mechanics”
A. Einstein, ”Selected Papers of Professor A. Einstein”
A. S. Kompaneyets ,”Quantum Mechanics”
A. Arai and H. Ezawa, ”Mathematical Structures of Quantum Mechanics”
M. A. Nielsen and I. L. Chuang, ”Quantum computation and quantum information”
N. Masuda and N. Konno, ”Complex Networks”
R. Kubo, ”Statistical Mechanics”
Y. Higuchi, ”Percolation”

40

MORIYOSHI, Hitoshi

Professor

Oﬃce:

Rm 504 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 4746

Email:

moriyosi@math.nagoya-u.ac.jp(ext. 4746)

Membership of Academic Societies:
MSJ (Mathematical Society of Japan)

Research Interest:• Diﬀerential Topology, Diﬀerential Geometry, Global Analysis• Noncommutative Geometry• The Atiyah-Singer Index Theorem

Research Summary:My research interest is in the study of diﬀerentiable manifolds, in particular a generalization of
the Atiyah-Singer index theorem [Ann. of Math. 87 (1968)], which reveals a profound relationshipbetween topological and analytic invariants on diﬀerentiable manifolds. Nowadays, it is considered
as one of the most beautiful and invaluable theorems in Mathematics. For instance, one of thebastonishing consequences of the Atiyah-Singer index theorem is that the L and A-genera
must beintegers, which is a priori just rational numbers by deﬁnition. Milnor exploited the integrality
theorem of L-genus to prove the existence of exotic spheres (manifolds that are homeomorphic
but not diﬀeomorphic to standard spheres). In recent years one exploits the index theorem in an
elaborated way so that many remarkable results are obtained in Low-dimensional Topology. Now
the index theorem is accepted as one of the central issues in modern Mathematics.
In the 80’s A. Connes, who was awarded the Fields medal in 1990, proposed a new frameworkin Mathematics called Noncommutative Geometry (NCG) [6]. Obviously the Atiyah-Singer index
theorem was a strong motivation for him to establish NCG. He also introduced new methods
into NCG like K-theory and cyclic cohomology and extended the Atiyah-Singer index theorem on
foliated manifolds, spaces with ergodic actions, noncompact homogeneous spaces and so on. Suchan index theorem can be developed on the Kronecker foliation in 2-dimension, which appears in
the next page (taken from Connes’ book [6]).
The inﬂuence of NCG is broad and profound in many areas of Mathematics. Thus students will
be required considerable knowledge on various subjects. However, the broader you are required,the deeper you understand. I am sure that you will be more attracted to Mathematics by studying
the index theorem.

Major Publications:[1] H. Moriyoshi and T. Natsume, The Godbillon-Vey cyclic cocycle and longitudinal Dirac operators, Paciﬁc J. Math., 172 (1996), no. 2, 483–539.
[2] H. Moriyoshi and P, Piazza, Eta cocycles, relative pairings and the Godbillon-Vey index theorem, Geom. Funct. Anal., 22 (2012), 1708–1813[3] H. Moriyoshi and T. Natsume, Operator algebras and geometry, Translations of Mathematical
Monographs 237, AMS, 2008.

41

Education and Appointments:
1986
1990
1990
1991
1995
1998
2009

M.S., Univ. of Tokyo, Japan
Ph.D., Pennsylvania State Univ., USA
Visiting Lecturer, SUNY at Buﬀalo, USA
Research Associate, Tokyo Institute of Technology, Japan
Associate Professor, Hokkaido University, Japan
Associate Professor, Keio University, Japan
Professor, Nagoya University, Japan

Message to Prospective Students:
Students in the Graduate Program for Master’s degree should attend Seminar (small groupclass). The following is a list of topics we dealt with in recent years:• Characteristic classed of vector bundles• de Rham cohomology and the Chern-Weil theory• Topological K-theory• Foliated manifolds and secondary characteristic classes• The Atiyah-Singer index theorem
The former three are subjects related to cohomology and characteristic classes, which lay thefoundation of Topology and Geometry. The latter are elaborated ones comparing to the former.
Suitable references are listed in the following. P. Shanahan [4] and Roe [5] are recommended to
study the Atiyah-Singer index theorem.
[1] J. Milnor, Characteristic classes, Princeton University Press,[2] R. Bott and L. Tu, Diﬀerential Forms in Algebraic Topology, GTM 82, Springer-Verlag,[3] J. Dupont, Curvature and characteristic classes, LNM, Vol. 640, Springer-Verlag.
[4] P. Shanahan, The Atiyah-Singer Index Theorem, LNM, Vol. 638, Springer-Verlag.
[5] J. Roe, Elliptic operators, topology and asymptotic methods, Longman.
[6] A. Connes, Noncommutative Geometry, Academic, 1994.
Prerequisites for attending Seminar (small group class) are subjects of level 1; see our web
cite. They include Calculus, Linear Algebra, Complex analysis. Students are also expected tohave backgrounds on Diﬀerentiable manifolds, Diﬀerential Geometry and Homology and Homotopy
theory. However, what is most important is enthusiasm for Mathematics. I am looking forward
to meeting such students who want to study every subject they got interested in. It is just like
Terentius, a Roman playwright, Humani nil a me alienum puto.
In Ph.D. course, students will be supervised who are interested in Diﬀerential Topology, Diﬀerential Geometry and Global Analysis.
4. GEOMETRIC EXAMPLES OF VON NEUMANN ALGEBRAS

R

U

55

dim F

R

codim F’

Figure 6. Foliation

A. Connes, Noncommutative Geometry, Academic Press.

42

NAGAO, Taro

Professor
Oﬃce:

Rm 508 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 5392

Email:

nagao@math.nagoya-u.ac.jp

Web-site:(ext. 5392)

http://www.math.nagoya-u.ac.jp/~nagao/nagaoeng.html

Membership of Academic Societies:
The Physical Society of Japan

Research Interest:• Random matrix theory and its applications• Semiclassical quantum theory

Research Summary:
I study random matrices from the viewpoints of fundamental theory and various applications.
The theory of random matrices (matrices with random number elements) originated in the ﬁeld ofmathematical statistics at the beginning of the twentieth century. Since Wigner introduced randommatrices to the research of nuclear physics, the range of their applications has been expanded tomany areas, including analytic number theory, combinatorics, elementary particle physics, solidstate physics, statistical mechanics and ecology. In particular, the developments in the last two
decades can be described as explosive and new discoveries have been reported one after another.
One of the most important problems in random matrix theory is the universality of energy
level statistics. It is known that the energy levels of quantum systems are distributed according touniversal laws depending on the underlying classical dynamics. If the underlying classical dynamics
is chaotic, universal level correlations are observed in agreement with the prediction of randommatrix theory. I have investigated the relation between random matrices and semiclassical quantum
theory in order to clarify the cause of the universality.
Random matrices can also be applied to the network theory as mathematical models describing
the connection pattern of networks. The network theory is a focus of current interest due to the
popularization of mobile phones and internet. I wish to contribute to the study of such realistic problems, and as well to obtain inspirations from them to deepen the understanding of thefundamental theory.

Major Publications:[1] T. Nagao, Correlation functions for multi-matrix models and quaternion determinants, Nucl.
Phys. B602 (2001) 622-637.
[2] T. Nagao, Dynamical correlations for vicious random walk with a wall, Nucl. Phys. B658 (2003)
373-396.
[3] T. Nagao and T. Sasamoto, Asymmetric simple exclusion process and modiﬁed random matrix
ensembles, Nucl. Phys. B699 (2004) 487-502.
[4] T. Nagao, P. Braun, S. Müller, K. Saito, S. Heusler and F. Haake, Semiclassical theory forparametric correlation of energy levels, J. Phys. A: Math. Theor. 40 (2007) 47-63.
[5] P.J. Forrester and T. Nagao, Eigenvalue statistics of the real Ginibre ensemble, Phys. Rev. Lett.
99 (2007) 050603.
43[6] T. Nagao and G.J. Rodgers, Spectral density of complex networks with a ﬁnite mean degree,
J. Phys. A: Math. Theor. 41 (2008) 265002.
[7] G. Akemann and T. Nagao, Random matrix theory for the Hermitian Wilson Dirac operator
and the chGUE-GUE transition, J. High Energy Phys. 2011 (2011) 60.
[8] G.J. Rodgers and T. Nagao, Complex Networks, The Oxford Handbook of Random Matrix
Theory (ed. by G. Akemann, J. Baik and P.Di Francesco, 2011) Chapter 43.

Awards and Prizes:• Ryogo Kubo Memorial Prize (2011) “Random matrix theory and its applications to physics”

Education and Appointments:
1994
1994
2004
2009

Doctor (Science), University of Tokyo
Assistant Professor, Osaka University
Associate Professor, Nagoya University
Professor, Nagoya University

Message to Prospective Students:
Possible themes in the master course can be listed as, for example, probability theory, statisticalmechanics, chaotic dynamical systems and network theory. These themes are universal and fundamental subjects in mathematical sciences, and deeply related to the theory of random matrices. It
is also possible to treat other themes, depending on the wish of the participants.
In the doctor course, I recommend students to take part in the research of novel areas, in which
the number of researchers is still relatively small, and to create their own styles of research. I also
like to encourage them to ﬁnd inﬂuential and solvable problems by keeping eyes on the newest
developments.

44

NAITO, Hisashi

Professor
Oﬃce:

Rm 408 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 2415

Email:

naito@math.nagoya-u.ac.jp

Web-site:(ext. 2415)

http://www.math.nagoya-u.ac.jp/~naito/

Membership of Academic Societies:
MSJ (Mathematical Society of Japan)

Research Interest:• Geometric Variational Problems• Nonlinear Partial Diﬀerential Equations• Discrete Geometric Analysis

Research Summary:
I am interested in “Geometric Variational Problems”. In particular, I study “Nonlinear Differential Equations with respect to Geometric Variational Problems” and “Discrete GeometricAnalysis”.
Geometric Variational Problems are geometric objects deﬁned by variational principles. For
example, “Harmonic Maps” and “Minimal Surfaces” are well-known geometric objects deﬁned byvariational principles, and they are deﬁned by non-linear diﬀerential equations. I my research, I
study diﬀerential equations of above types by using their geometric properties (cf. [3]).
On the other hand, from mathematical view points. “Crystal Lattice” in Material Sciences isdeﬁned by graphs with symmetries, and they are also deﬁned by variational principles. I also study
Crystal Lattices by using geometric view points (cf. [1, 2]).

Major Publications:[1] M. Itoh, M. Kotani, H. Naito, T. Sunada, Y. Kawazoe, T. Adschiri, New metallic carbon crystal,
Physical Review Letters, 102, (2009) 055703.
[2] H. Naito, Visualzation of standard realized crystal lattices, Comptenporary Mathematics, 484,(2009) 153–164.
[3] H. Naito, Finite time blowing-up for the Yang-Mills gradient ﬂow in higher dimension, Hokkaido
Math. J., 23 (1994), 451–464.

Education and Appointments:
1988
1995

Assistant Professor, Nagoya University
Associate Professor, Nagoya University
45

Message to Prospective Students:
Main subject of my classroom is “mathematics by computer approach”. Examples of subjects
are “Numerical analysis of diﬀerential equations”, “Computer approach for discrete geometric analysis”. In my classroom, I recommend that students does not study only methematical theory of a
subject, but should also study computational experiments them.
For that reason, I require that stundents have studied basic mathematics (eg. calculas, linear
algebra), and have skill of standard programmings.

46

NAKANISHI, Tomoki

Professor

Oﬃce:

Rm 406 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 5575

Email:

nakanisi@math.nagoya-u.ac.jp

Web-site:(ext. 5575)

http://www.math.nagoya-u.ac.jp/~nakanisi/

Membership of Academic Societies:
Japanese Mathematical Society

Research Interest:• integrable systems• cluster algebras• quantum groups

Research Summary:Since the late 80’s I have been working on the mathematical methods and aspects of the integrable quantum systems, such as conformal ﬁeld theories, S-matrix models, and quantum spin
chains. Until 2008, my method was mainly based on the aﬃne Lie algebras and their quantizations(quantum groups). In that method we found various interesting interrelations between integrable
systems and representation theory of Lie algebras and quantum groups. To name few key words,
the level-rank duality, the Kirillov-Resehtikhin modules, the dilogarithm identities, Q-, T - and Y systems, q-characters, fermionic formulas, and so on. Many problems were solved (by myself andother researchers) as a result of the development of the representation theory of quantum groups
in these twenty years, but some problems were left almost untouched.
Since 2008, I drastically changed the strategy, and started to study these problems from thepoint of view of cluster algebras. The cluster algebras are a new class of commutative algebras
introduced by Sergey Fomin and Andrei Zelevinsky around 2000. (So, it is one of mathematics
in this century!.) The original motivation of the introduction of cluster algebras was to study the
structure of the coordinate ring of some algebraic varieties. However, it turns out that a cluster
algebra is a key concept to understand some of the unsolved problems mentioned above, and,amazingly and unexpectedly, some of the seemingly untouchable problems were suddenly solved
with this new method.
Here I list some of the main results which I obtained with several collaborates. See the next
section for the list of references.
• The dilogarithm identities for the quantum aﬃne algebras of simply laced type [1].
• The periodicity of Y -systems and the associated dilogarithm identities for the quantum aﬃne
algebras of nonsimply laced type [2].
• The tropicalization method in cluster algebras [2].
• The dilogarithm identities associated with any period of a cluster algebra [3].
• The duality of the c-vectors and the g-vectors [3].
• The relation between the classical and quantum dilogarithm identities [4].
• The determination of the c-vectors and d-vectors of the cluster algebras of ﬁnite type [5].
• The periodicity of the sine-Gordon Y -systems and the associated dilogarithm identities [6].

47

Major Publications:(Only after 2008. See http://www.math.nagoya-u.ac.jp/ nakanisi/research/publications.html
for the compete list)[1] T. Nakanishi, Dilogarithm identities for conformal ﬁeld theories and cluster algebras: simply
laced case, Nagoya Math. J. 202 (2011) 23–43.
[2] R. Inoue, O. Iyama, B. Keller, A. Kuniba, T. Nakanishi, Periodicities of T and Y-systems,
dilogarithm identities, and cluster algebras I: Type Br , arXiv:1101.1880, to appear in Publ.
RIMS.
[3] T. Nakanishi, Periodicities in cluster algebras and dilogarithm identities, in Representations ofalgebras and related topics (A. Skowronski and K. Yamagata, eds.), EMS Series of Congress
Reports, European Mathematical Society, 2011, pp.407-444.
[4] R. M. Kashaev, T. Nakanishi, Classical and Quantum Dilogarithm Identities, SIGMA 7 (2011),
102, 29 pages.
[5] T. Nakanishi, S. Stella, Diagrammatic description of c-vectors and d-vectors of cluster algebras
of ﬁnite type, arXiv:1210.6299.
[6] T. Nakanishi, S. Stella, Wonder of sine-Gordon Y-systems, arXiv:1212.6853.

Education and Appointments:
1985-1990
1990–1994
1992–1994
1994–
2012

Graduate School of Science, Tokyo University(Doctor in Science, 1990)
Assistant Professor, Nagoya University
JSPS Fellow for Research Abroad, Harvard University
Associate Professor, Nagoya Univerisy
Research Member, MSRI, Berkeley

Message to Prospective Students:
You do not have to follow my research interest. On the contrary, it is important to ﬁnd yourresearch interest for yourself, because in most cases the research in the graduate school will inﬂuence
decisively the lifetime direction of your research.

48

NAKAOKA, Hiroyuki

Associate Professor

Oﬃce:

Rm 345 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2545

Email:

nakaoka.hiroyuki@math.nagoya-u.ac.jp(ext. 2545)

Membership of Academic Societies:
MSJ (The Mathematical Society of Japan)

Research Interest:• homological algebra• representation theory of algebras• category theory

Research Summary:
By virtue of the abstraction, category theory is suitable for providing general theory applicable
to a wide range of areas. My research aims to capture categorical frameworks appearing in algebra.
Especially in recent years I have been interested mainly in categories and functors used in the
representation theory of algebras. The representation theory of algebras involves categories invarious ways, including major class of categories for homology algebra such as abelian categories,
exact categories, triangulated categories, and more advanced ones. Recently I am engaged inresearch on the structure of categories and functors related to this area.

Major Publications:[1] H. Nakaoka and Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model
structures, Cah. Topol. Géom. Diﬀér. Catég. 60 (2019) no. 2, 117–193.
[2] H. Nakaoka, A simultaneous generalization of mutation and recollement on a triangulated category, Appl. Categ. Structures, 26 (2018) no. 3, 491–544.
[3] H. Nakaoka, General heart construction on a triangulated category (I): unifying t-structuresand cluster tilting subcategories, Appl. Categ. Structures, 19 (2011) no.6, 879–899.

Awards and Prizes:• MSJ Takebe Katahiro Prize for Encouragement of Young Researchers (2010)

Education and Appointments:
2009
2009
2009
2010
2019

Ph.D. at The University of Tokyo
Project Researcher, The University of Tokyo
Project Research Associate, The University of Tokyo
Associate Professor, Kagoshima University
Associate Professor, Nagoya University49Message to Prospective Students:
Envisaged theme for the seminar is category theory appearing in the representation theory ofalgebras. Mainly supposed are category theory related to homological algebra in abelian categories
and triangulated categories. I hope you to have a home ground (a research area dealing withconcrete objects) which will become a source of your research, rather than entirely focused on
abstract category theory, so I recommend to the learn representation theory of algebras. The
following is a candidate of textbook in this direction.
• I. Assem; D. Simson; A. Skowroński, Elements of the representation theory of associative
algebras. Vol. 1. Techniques of representation theory. London Mathematical Society Student
Texts, 65. Cambridge University Press, 2006.
Basic knowledge of linear algebra, group theory and ring theory are desirable prior to entering themaster course. In particular, it is necessary to have knowledge of modules over rings dealt in theundergraduate course. Besides, it is desirable to have some familiarity with homological algebra inabelian categories or like that.

50

NAKASHIMA, Makoto

Associate Professor

Oﬃce:

Rm 453 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2421

Email:

nakamako@math.nagoya-u.ac.jp

Web-site:(ext. 2421)

http://www.math.nagoya-u.ac.jp/~nakamako/

Membership of Academic Societies:
MSJ (The Mathematical Society of Japan)

Research Interest:• Probability• Large scale interacting systems

Research Summary:
Probability theory means “measure theoretic probability theory” which is completely diﬀerent
from combinatorial probability and statistics. Lectures in measure theory are usually given to
junior students majoring in mathematics. In measure theory, we deﬁne “areas” of measurable sets
in a set S and then we can construct Lebesgue integral of a measurable function. In measure
theoretical probability, measurable sets in sample space Ω is called events and their “areas” areidentiﬁed as probabilities. Measure theoretic probability allows us to consider “inﬁnitely many
trials” mathematically and analyze them.
The large scale interacting systems are closely related to physics. We shall explain it throughpolymers which is one of my research interests.
Polymers are large molecules created via polymerization of small molecules, monomers, e.g.
(−CH2 −). Each bond between −C−s has a freedom of rotation, i.e. is random. We will simplify
the models by ignoring exclusivity between monomers. We consider a polymer chain with lengthn + 1 and we set the position of one endpoint (monomer 0) of chain as 0 and denote by Si theposition of ith-monomers. Moreover, we assume that the displacements {Xi := Si − Si−1 }ni=1 are
independent and identically distributed. Then, this is a random walk that is well-studied. Now,
we study the shape of polymer chains. Investigating polymers macroscopically can be regarded as
the scaling limit of random walk. When {Xi }ni=0 satisﬁes a proper assumption, the scaling limit of{Si }ni=0 is the Brownian motion (invariance principle). Thus, the shape of polymers in an ideal
medium can be regarded as the trace of Brownian motion.
How about the case for the polymers in a medium with impurities? We may believe that
there exists some interaction between impurities and monomers. This interaction is described via
a new probability measure called Gibbs measure. This measure is generated by assigning a weight
representing an interaction to each path {Si }ni=0 . Changing a parameter (e.g. concentration of
impurities), the shape of polymers under Gibbs measure is changed completely(phase transition). Thus, interaction between impurities and monomers (large scale interaction) yields a newphenomenon.
There are other physical models classiﬁed as large scale interacting systems and many branches.
Usually, the phase transition is characterized in terms of the quantity “free energy”. Recently,
I have studied the asymptotics of the free energy and found the universality structure behind thephysical models.
51

Major Publications:[1] C. Cosco, S. Nakajima, M. Nakashima: Law of large numbers and ﬂuctuations in the sub-criticaland L2 regions for SHE and KPZ equation in dimension d ≥ 3. Stochastic Process. Appl. 151(2022), 127–173.
[2] M. Nakashima: Free energy of directed polymers in random environment in 1+1-dimension athigh temperature. Electron. J. Probab. 24 (2019), No. 50.
[3] M. Nakashima: Branching random walks in random environment and super-Brownian motion
in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 4, 1251–1289.

Awards and Prizes:• 2014, MSJ Takebe Katahiro Prize for Encouragement of Young Researchers 「Study of branching random walks in random environment」

Education and Appointments:
2012
2015

Assistant Professor, University of Tsukuba
Associate Professor, Nagoya University

Message to Prospective Students:
When you learn measure theoretic probability, calculus and linear algebra are needed. Moreover,
it is better to master the measure theory. Before entering graduate school, you have to learn themeasure theoretic probability (σ-algebra, independence, law of large numbers, central limit theorem,
conditional expectation). The following is a standard textbook of measure theoretic probability.
[1] Williams「Probability with martingales」Cambridge Mathematical Textbooks,

52

1991.

NAYATANI, Shin

Professor

Oﬃce:

Rm 429 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2814

Email:

nayatani@math.nagoya-u.ac.jp(ext. 2814)

Web-site:
Membership of Academic Societies:
Mathematical Society of Japan

Research Interest:• Rigidity of discrete groups• Conformal geometry

Research Summary:
My subject of study is geometry. I learned diﬀerential geometry when I was a student, andsince then I have studied topics in diﬀerential geometry and those related to geometry. In past, I
studied the instability index of minimal surfaces, scalar curvature equations, geometric structuresassociated with real and complex Kleinian groups, self-dual metrics, CR geometry and quaternionic
CR geometry. The key words for them will be surface geometry, geometric analysis and conformal
geometry.
My recent research is on the actions of discrete groups (that is, inﬁnite countable groups) on
spaces. It is supposed that the spaces have metrics and the actions preserve them. The prototype
of such situation is the action of the fundamental group of a Riemannian manifold on its universal
covering space. In such a case, the action is good in the sense that it is properly discontinuous.
We consider actions of more general discrete groups on more general metric spaces, but the metricspaces are not completely arbitrary; they will be assumed to be similar to Riemannian manifolds
of nonpositive sectinal curvature. In this setting, according to the choices of discrete groups andmetric spaces, it can happen that good actions are extremely rare or never exist. Such discretegroups are known to have very special algebraic property, but it has not been understood very wellwhether such groups (in particular, having also the property of so-called hyperbolicity) exist inabundance.
We could prove that such groups do exist in abundance by using methods with origin in differential geometry and geometric analysis (in particular, maps minimizing certain energy) andcombining them with the theory of random groups. An interesting future problem is to construct
such groups explicitly, instead of appealing to the random group theory.
The original motivation for this research was the following problem: Prove geometrically the
Margulis superrigidity theorem, a rigidity theorem for lattices of certain algebraic groups, when thealgebraic groups are the matrix groups with coeﬃcients in p-adic numbers. The Margulis theorem
can be formulated in the above setting, but a geometric proof is not yet completed. I’m working
on this problem again, though I abandoned it for some time.

Major Publications:[1] H. Izeki and S. Nayatani, Combinatorial harmonic maps and discrete-group actions on
Hadamard spaces, Geom. Dedicata 114 (2005), 147–188.
53

[2] S. Nayatani, Patterson-Sullivan measure and conformally ﬂat metrics, Math. Z. 225 (1997), no.
1, 115–131.

Awards and Prizes:• 2004, Geometry Prize.

Education and Appointments:
1990
1991
1994
1998
2005

JSPS Post-doctoral Fellow, Osaka University
Research Associate, Tohoku University
Associate Professor, Tohoku University
Associate Professor, Nagoya University
Professor, Nagoya University

Message to Prospective Students:
The themes of my seminar for master course students have been Riemannian goemetry andgeometric analysis, geometry and analysis of groups, hyperbolic geometry, and so one. For details,
please look at the past course designs (in Japanese) available at
http://www.math.nagoya-u.ac.jp/ja/education/archive/
In each year I propose a theme of the seminar, but the details are decided by discussing with thestudents. (They depend on the students’ preliminary knowledge, interest and the future career.)
The provisional purpose of the seminar will be to master the assigned theme through working with
it for one or two years. The students, however, may shift to other topics in geometry if they getinterested in them in the course of their study, as I consider it desireable to ﬁnd the object of yourinterest by yourself.

54

OHIRA, Toru

Professor
Oﬃce:

Rm 341 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2824

Email:

ohira@math.nagoya-u.ac.jp(ext. 2824)

Membership of Academic Societies:
Physical Society of Japan,
Japanese Society for Mathematical Biology

Research Interest:• Mathematical Biology and Physiology• Delayed Stochastic Systems• Chases and Escapes

Research Summary:
Many natural and artiﬁcial systems are associated with noise and feedback or interaction
delay . Examples includes stochastic diﬀerential equations with delay, bio-physiological controls,neural networks, traﬃc ﬂows, electrical circuits, and so on. These systems can show surprisingly
rich behaviors to otherwise simple systems.
The main theme of my research has been investigating and seeking applications of suchdelayed stochastic systems . For example, I proposed the concept of ”delayed random walks”
as a mathematical framework for studying such systems. A delayed random walk is a random
walk in which the transition probability depends on the position of the walker at a ﬁxed timeinterval in the past. It has been used to model human postural controls and neural activities incomparison to experimental data. Typically, oscillatory autocorrelation function is associated withdelayed random walks of suﬃciently long delay. To study such oscillatory behavior in stationary
and transient states, we have studied analytically tractable models. On the basis of this theory, we
have also devised a method of estimating delay from noisy time series coming out of linear delayed
feedback systems.
I have also proposed the concept of ”delayed stochastic control”. The main motivation of sucha hypothesis is the fact that humans can often handle situations or objects whose time constantis much faster than their reaction time. Compared to artiﬁcial systems, humans are ”very slow”
with a reaction time of a few hundred mili-seconds. Of course, one cannot relay only on feedbacks,
predictive controls are also important. However, the key question is whether they are enough ornot. For example, by combining these traditional control schemes, can we create a robot with areaction time of that of a human (approximately 200 mili-seconds), and which can ride a unicycle?
Recent experiments, for example, a human balancing a stick on the ﬁngertips began to pose thesequestions. Most of the ﬂuctuating movement of the stick is much faster than 200 mili-seconds.
Delayed stochastic control is a new scheme, which takes advantage of resonant phenomena with anappropriate amount of noise level and feedback delay time. We analyzed this delayed stochastic
resonant phenomena by considering the stability of repulsive delayed random walks. We also
discovered a new eﬀect: someone can better balance the stick on the ﬁngertips, if they move an
object with the other hand in a ﬂuctuating manner. This is likely to be a piece of supporting
evidence for delayed stochastic controls.
55

These theoretical works on delayed stochastic systems have been used by other researchers tomodel or analyze their experimental results. Examples include feedback resonant phenomena ofsolid-state laser experiments, analysis of human eye saccade movements, and so on. As related
topics, I have also worked on problems such as high-frequency currency exchange market behaviors
and simple computer network traﬃc models. I authored a book in Japan compiling these research
activities in 2006.
I have also recently proposed a new theme of
Escapeor

Pursuit and Evasion

Group Chase and Escape .

Chase andis mainly mathematical research topic with long history. Its

main applications have been that of military issues. The topic has also found a connection with thegame theory, developed under the name of adiﬀerential game theory . The collective motionsof self-driven particles, on the other hand, have been studied actively in recent years. They include
models of school of ﬁsh, ﬂock of birds, traﬃc ﬂow, and so on. My motivation to introduce thetheme ofGroup Chase and Escapeis to provide a platform of research to combine above two

ﬁelds. The models we proposed are simple. Yet, it showed rather unexpected and complex group
behaviors. Our paper is covered in aNews and Views

section of Nature (by Tamas Vicsek, 1

July, 2010, vol. 466, pp.43-44; attached), noting its originality. At the same time, however, there
are much to be done from this point on this topic.
First, we need to develop theoretical method to analyze the result of computer simulations
on our models. We have observed escaping and chasing in groups with their size varying in time.
These spatio-temporal patterns are yet to be understood mathematically. Eﬀects of delay and noise
are also to be studied. Secondly, I hope to extend models into various directions. An example isto consider chases and escapes in the context of traﬃc models such as optimal velocity models.
Combination with various models of swarms and swarm intelligence will be another direction ofinterest.
Applications and connections to natural and artiﬁcial systems are also of importance. Actual
data from animal or ﬁsh hunting and evading in groups could be of high value in this context. There
are recent developments in this direction such as study of a ﬂock of pigeons using GPS devices.
How to reﬂect these data sets to modeling or how to suggest experimental paradigm or chasing
strategies from model simulations will be an important research direction. The same story can beenvisioned with a large number of small interacting robots. Also, eﬀective virus tracking both inphysical and cyber spaces would be an important issue socially.

Major Publications:[1] T. Ohira and Y. Sato: Resonance with Noise and Delay, Physical Review Letters, 82 (1999),
2811–2815.
[2] T. Ohira and T. Yamane: Delayed Stochastic Systems, Physical Review E, 61 (2000), 1247–
1257.
[3] A. Kamimura and T. Ohira: Group Chase and Escape, New Journal of Physics, 12 (2010),
053013

Education and Appointments:
1993
1993
2012

Ph.D. in Physics, The University of Chicago
Reseacher, Sony Computer Science Laboratories
Professor, Nagoya University

56Message to Prospective Students:
I have worked with researchers from various ﬁelds. Though it can be challenging to step outsideof mathematics, there are many phenomena which still need to be formulated in the language ofmathematics. This side of mathematical endeavor is as interesting as solving diﬃcult known mathematical problems. Some representative textbooks, which reﬂect my interests, are the following:[1] B. Balachandran, T. Kalmar-Nagy and D.E. Gilsinn, Delay Diﬀerential Equations: Recent
Addvances and New Directions, Springer, 2009.
[2] L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton Univ.
Press, 2007.
[3] P. J. Nahin, Chases and Escapes: The Mathematics of Pursuit and Evasion, Princeton Univ.
Press, 200757

OHKUBO, Shun

LecturerOﬃce:

Rm 351 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2431

Email:

shuno@math.nagoya-u.ac.jp(ext. 2431)

Membership of Academic Societies:
MSJ（The Mathematical Society of Japan）

Research Interest:• p-adic representations of a complete discrete valuation ﬁeld
• p-adic diﬀerential equation

Research Summary:
I study p-adic aspects of algebraic number theory. In modern number theory, as in the proof
of Fermat’s last theorem due to Taylor and Wiles, it is important to study algebraic varieties andits p-adic étale cohomology V . The absolute Galois group GQ of Q acts naturally on V and westudy the Galois action by restricting GQ to its decomposition groups D. The most critical case is
D = GQp , that is, the absolute Galois group of the p-adic number ﬁeld Qp . Fontaine establisheda fundamental theory of p-adic representations of GQp arising in the above situation: He classiﬁed
p-adic representations V by using rings of p-adic periods, then he associated linear algebraic objects(p-adic Hodge structure) to V . Berger related a p-adic Hodge structure of V to the solution spaceof a certain p-adic diﬀerential equation, and he proved Fontaine’s p-adic monodromy conjecture.
Brinon generalized Fontaine’s theory where GQp is replaced by the absolute Galois group of acomplete discrete valuation ﬁeld with imperfect residue ﬁeld. In my early study ([1,2]), I proved apartial generalization of Berger’s theory in Brinon’s setup.
I am also studying the asymptotic behavior of solutions of p-adic diﬀerential equations. On
the p-adic number ﬁeld Qp , a naı̈ve analogue of analytic continuation fails because of its totally
discontinuity. So, when we study p-adic diﬀerential equations, beside the existence of solutions,
it is important to study the asymptotic behavior of the solutions around the boundary, that is,
the edge of its convergent disc. Around the 1970s, Dwork established the fundamental theory
on the asymptotic behavior of solutions of p-adic diﬀerential equations. He also (vaguely) statedsome basic conjectures, but there had been little progress until recently. In late 2000s, André andChiarellotto-Tsuzuki re-considered Dwork’s theory, and they obtained some important results on
Dwork’s conjecture. In [3], I gave a negative answer to a problem of André on the logarithmic
growth Newton polygon of p-adic diﬀerential equations by constructing a certain p-adic diﬀerential
equation of rank 2.

Major Publications:[1] S. Ohkubo，The p-adic monodromy theorem in the imperfect residue ﬁeld case, Algebra andNumber Theory 7 (2013), No. 8, 1977–2037.
[2] S. Ohkubo，On diﬀerential modules associated to de Rham representations in the imperfectresidue ﬁeld case, arXiv:1307.8110.
[3] S. Ohkubo，A note on logarithmic growth Newton polygons of p-adic diﬀerential equations, toappear in International Mathematics Research Notices 2014; doi: 10.1093/imrn/rnu017.

58

Awards and Prizes:• MSJ Takebe Katahiro Prize for Encouragement of Young Researchers (2014)

Education and Appointments:
2012-2015

JSPS post-doc at The University of Tokyo

Message to Prospective Students:• Background knowledge
When I was an undergraduate student, I learned basics of algebraic number theory by reading
J.-P. Serre, ”A Course in Arithmetic”
J. W. S. Cassels，A．Frohlich, ”Algebraic number theory”.
To study number theory, I think that it is better to have some knowledge about elliptic curves. I
learned it by reading
J. H. Silverman, ”The arithmetic of elliptic curves”.
(A) Basic texts on Fontaine’s theory on p-adic representations
1. J.-M. Fontaine, Y. Ouyang，”Theory of p-adic Galois representations”
2. O. Brinon, B. Conrad，”Notes on p-adic Hodge theory”
3. L Berger，”An introduction to the theory of p-adic representations”
4. J. Tate，”p-divisible groups”(B) Basic texts on p-adic diﬀerential equations
5. K S. Kedlaya，”p-adic diﬀerential equations”
6. B. Dwork, ”On p-Adic Diﬀerential Equations II: The p-Adic Asymptotic Behavior of Solutionsof Ordinary Linear Diﬀerential Equations with Rational Function Coeﬃcients”
Potential students who want to study Fontaine’s theory on p-adic representations or p-adic
diﬀerential equations under my supervision should look at [3] or § 0 of [5] respectively.

59

OHTA, Hiroshi

Professor

Oﬃce:

Rm 325 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2543

Email:

ohtaZmath.nagoya-u.ac.jp (Please replace Z by @)(ext. 2543)

Membership of Academic Societies:
Mathematical Society of Japan

Research Interest:• symplectic geometry• Floer theory• gauge theory

Research Summary:
I am working on symplectic geometry, whose origin goes back to classical mechanics. Symplectic
manifold is by deﬁnition a smooth manifold admitting a non-degenerate closed 2-form. Typicalexamples are cotangent bundle on which classical mechanics is described, and submanifolds incomplex projective spaces.
My recent interest is Floer theory and relationship between singularities and symplectic/contact
geometry. I am now working on Floer theory from the point of view of certain homotopical algebra,
so called A∞ algebra. Such a homotopical algebra is a classical object originally arising fromtopology but it is now making new progress, partially motivated from physics. In particular,
collaborating with K. Fukaya, Y-G. Oh and K. Ono, I constructed a ﬁltered A∞ algebra associatedto a Lagrangian submanifold of a symplectic manifold and developed Lagrangian intersection Floer
theory based on the ﬁltered A∞ algebra. (See Reference 1-[1] below.) This A∞ algebra plays
an important and fundamental role in mirror symmetry, which claims correspondence between
symplectic geometry on a symplectic manifold X and complex geometry on the mirror complex
manifold X̌. As a consequence, for example, certain symplectic invariant of X deﬁned by using
solutions to some non linear partial diﬀerential equation will be surprisingly derived from certain
complex geometric invariants of X̌ deﬁned by some linear diﬀerential equation. Our theory gives
not only mathematical foundation in mirror symmetry but also provides some new applications toconcrete problems in symplectic geometry. (See Reference 1-[2][3], for example.)

Major Publications:
1. Floer theory and mirror symmetry:[1] K. Fukaya, Y-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory –Anomaly
and Obstruction–. vol 46-1, vol 46-2. AMS/IP Studies in Advanced Mathematics. American
Mathematical Society/International Press (2009).
[2] K. Fukaya, Y-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds
I. Duke Math. J. 151, 23–175. (2010).
[3] K. Fukaya, Y-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds
II: Bulk deformations. Selecta Math. New Series, 17, 609-711. (2011).
[4] K. Fukaya, Y-G Oh, H. Ohta and K. Ono, Lagrangian Floer theory and mirror symmetry on
compact toric manifolds. Astérisque, 376, Société Mathématique de France (2016).
60

2. Singularity and symplectic/contact geometry:[1] H. Ohta and K. Ono, Simple singularities and topology of symplectically ﬁlling 4-manifold.
Comment. Math. Helv. 74. 575–590. (1999).
[2] H. Ohta and K. Ono, Simple singularities and symplectic ﬁllings. J. Diﬀerential Geom. 69, 1–42.
(2005).
[3] H. Ohta and K. Ono, Examples of isolated surface singularities whose links have inﬁnitely many
symplectic ﬁllings. J. Fixed Point Theory and Applications. 3, (V.I. Arnold Festschrift Volume)
51–56. (2008).
3. Gauge theory:[1] M. Furuta and H. Ohta, Diﬀerentiable structures on punctured 4-manifolds. Topology and its
Appl. 51. 291–301 (1993).
[2] H. Ohta and K. Ono, Notes on symplectic 4-manifolds with b+
2 = 1, II. Internat. J. of Math. 7.
755–770. (1996).
[3] H. Ohta, Brieskorn manifolds and metrics of positive scalar curvature. Advance Studies Pure
Math. 34. 231–236. (2002).

Message to Prospective Students:
Here are some examples of texts which I used in my seminar (for the ﬁrst year of master course).
1. M. Audin, Torus actions on symplectic manifolds, 2nd revised edition, Birkhäuser (2004).
2. M. Audin and M. Damian, Morse theory and Floer homology, Springer. (2014).
3. D. McDuﬀ and D. Salamon, Introduction to symplectic topology, Oxford Univ. Press (1995).
4. N. Hitchin, The self-dual equations on a Riemann surface, Proc. London Math. Soc 55(1987) 59-126.
5. H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser.
(1994).
It is expected to already master manifold theory, (co)homology theory, elementary diﬀerential
geometry, topology but the most important is to study by yourself what you don’t know. Of
course, I will give some advice and suggestion, if necessary. To get an impression on the basic
literature, please look at the following books:
1. K. Fukaya, Symplectic geometry, Iwanami, (1999) (in Japanese).
2. D. McDuﬀ and D. Salamon, J-holomorphic curves and symplectic topology, American Math.
Soc. (2004).
3. P. Seidel, Fukaya categoryies and Picard-Lefscetz theory, Zurich Lectures in Advanced Math.,Eurp. Math. Soc. (2008).
28/Nov/2017

61

OKADA, Soichi

Professor
Oﬃce:

Rm 427 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 5596

Email:

okada@math.nagoya-u.ac.jp(ext. 5596)

Web-site:
Membership of Academic Societies:
Mathematical Society of Japan, American Mathematical Society

Research Interest:• Enumerative and Algebraic Combinatorics• Combinatorial Representation Theory

Research Summary:My research interests are in combinatorics and its connection with algebra, representation
theory, and integrable systems. I am working in the area where combinatorics meets with other ﬁelds
of mathematics (and science). More speciﬁcally, I am interested in combinatorial problems arisingfrom the representation theory of classical groups and related algebras, and also in enumeration
problems of plane partitions, alternating sign matrices, and so on.
Combinatorial objects and techniques play an important role in representation theory. For example, Young diagrams (see the left ﬁgure below) are used to parameterize the irreducible representations of symmetric groups and general linear groups. Many concrete problems in representation
theory (e.g., construction of irreducible representations, irreducible decomposition of a given representation) can be solved or attacked by using combinatorial methods. On the other hand, somecombinatorial algorithms, e.g., the Robinson–Schensted correspondence, are now interpreted interms of the crystal basis for quantum groups from the view point of representation theory. With
these interactions between combinatorics and representation theory in mind, I study symmetricfunctions (characters and their generalizations) and determinant/Pfaﬃan identities ([2]).
Plane partitions are certain arrays of non-negative integers, which can be visualized as a stackof unit cubes (see the middle ﬁgure below). And alternating sign matrices are certain matrices
with entries 1, 0 and −1 (see the right ﬁgure below), which are a generalization of permutation
matrices. These objects were deﬁned with purely combinatorial motivation, but they turned out tohave relations with representation theory, statistical physics and so on. In my research, I enumerate
certain classes of plane partitions, alternating sign matrices and related combinatorial objects byrevealing hidden algebraic structures ([3]). In particular, I am interested in mysterious relationshipbetween alternating sign matrices and totally symmetric self-complementary plane partitions.

0
0

0

0

1
0

62


0 1
0 0 0
1 −1 1 0 0
0 1 −1 0 1
0 0
0 1 0
0 0
0 0 0
0 0
1 0 0

Major Publications:[1] S. Okada, Algebras associated to the Young–Fibonacci lattice, Trans. Amer. Math. Soc., 346(1994), 549 – 568.
[2] S. Okada, Applications of minor summation formulas to rectangular-shaped representations ofclassical groups, J. Algebra 205 (1998), 337 – 367.
[3] S. Okada, Enumeration of symmetry classes of alternating sign matrices and characters ofclassical groups, J. Algebraic Combin. 23 (2006), 43 – 69.
[4] S. Okada, “Representation Theory of Classical Groups and Combinatorics”, Baifukan, 2006 (inJapanese).
[5] M. Ishikawa and S. Okada, Identities for determinants and Pfaﬃans, and their applications,Sugaku 62 (2010), 85–114 (in Japanese).

Education and Appointments:
1990
1990
1995
2006

Doctor of Science, University of Tokyo
Research Associate, Nagoya University
Associate Professor, Nagoya University
Professor, Nagoya University

Message to Prospective Students:
I think that combinatorics is one of the most active and interesting areas in mathematics.
Combinatorial problems or structures can be found in many branches of mathematics (and science),
and the methods of algebraic combinatorics are applicable to these problems. So it is important totake interest not only in combinatorics but also in other related ﬁelds.
The interested students can start their study in combinatorics and its connection to algebra
and representation theory with one of the following books.
1. R. P. Stanley, “Enumerative Combinatorics”, Vol. 1 (2nd ed.), Vol. 2, Cambridge Univ. Press,
2012, 1999.
2. W. Fulton, “Young Tableaux : With Applications to Representation Theory and Geometry”,
Cambridge Univ. Press, 1996.
3. I. G. Macdonald, “Symmetric Functions and Hall Polynomials”, Oxford Univ. Press, 1995.
4. J. Hong and S.-J. Kang, “Introduction to Quantum Groups and Crystal Bases”, Amer. Math.
Soc., 2002
5. A. Björner and F. Brenti, “Combinatorics of Coxeter groups”, Springer, 2005.
Basic knowledge of linear algebra and abstract algebra is a necessary prerequisite for most of thesebooks. Also the students can start their research activities in combinatorics at a relatively earlystage.

63

SHIROMIZU, Tetsuya

Professor

Oﬃce:

Rm 445 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 5577

Email:

shiromizu@math.nagoya-u.ac.jp

Web-site:(ext. 5577)

http://www.math.nagoya-u.ac.jp/~shiromizu/

Membership of Academic Societies:
The Physical Society of Japan
The Astronomical Society of Japan

Research Interest:• General relativity• Cosmology• Gravity

Research Summary:
I am studying general relativity and cosmology. Due to the recent development in the observations/experiments for the universe, we could have the standard model for the universe now.
However, we encounter various new issues. To address such things, I often employed the diﬀerential
geometry to have unique consequences. The keywords for my research would be black hole spacetimes, asymptotic structure of spacetimes, string-inspired higher dimensional model for cosmology,
positive mass theorem and so on. For example, using the second variation formula for the minimal
surface, I could show the presence of the upper bound for black hole area in asymptotically deSitter spacetimes. And I showed the uniqueness of static black holes in higher dimensions. I prefer
non-trivial application of diﬀerential geometry or so into general relativity and cosmology.

Major Publications:[1] T. Shiromizu, K. Nakao, H. Kodama and K. -I. Maeda, “Can large black holes collide in deSitter space-time? An inﬂationary scenario of an inhomogeneous universe,” Phys. Rev. D 47,
3099 (1993).
[2] T. Shiromizu, K. -i. Maeda and M. Sasaki, “The Einstein equation on the 3-brane world,” Phys.
Rev. D 62, 024012 (2000).
[3] G. W. Gibbons, D. Ida and T. Shiromizu, “Uniqueness and nonuniqueness of static black holes
in higher dimensions,” Phys. Rev. Lett. 89, 041101 (2002).
[4] K. Tanabe, S. Kinoshita and T. Shiromizu, “Asymptotic ﬂatness at null inﬁnity in arbitrary
dimensions,” Phys. Rev. D 84, 044055 (2011).
[5] M. Nozawa and T. Shiromizu, “Modeling scalar ﬁelds consistent with positive mass” Phys. Rev.
D 89, 023011(2014).

Awards and Prizes:• 2005 20th Nishinomiya-Yukawa Memorial Awards• 2006 The Young Scientists’ Prize(The Commendation for Science and Technology by the
Minister of Education, Culture, Sports, Science and Technology)• 2010 Daiwa Adrian Prize (for UK-Japan teams(Japanese Team Leader:M.Sasaki))64

Education and Appointments:
1996
1996
2002
2008
2014

Department of Physics, Kyoto University PhD
Assistant Professor, Department of Physics, The University of Tokyo
Associate Professor, Department of Physics, Tokyo Institute of Technology
Associate Professor, Department of Physics, Kyoto University
Professor, Department of Mathematics, Nagoya University

Message to Prospective Students:
A typical application of diﬀerential geometry to physics is in general relativity(GR). GR predicts
the presence of black holes and expanding universe. Meanwhile the superstring theory, which is atheory of physics for everything, is formulated in higher dimensions. So the higher dimensional GR
is also interesting research ﬁeld.
In my seminar,
R. M. Wald, General Relativity, Chicago University Press.
will be used. Depending student’s interest, we will consider astrophysics and cosmology too. In
addition, I encourage graduate student report new/important papers to look for the topics for master/doctor thesis. I think that the minimum way to have the original work is good discussion/chat.
They make student’s understanding of study deep one.

65

SUGIMOTO, Mitsuru

Professor

Oﬃce:

Rm 303 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 2544

Email:

sugimoto@math.nagoya-u.ac.jp

Web-site:(ext. 2544)

http://www.math.nagoya-u.ac.jp/~sugimoto/index-e.html

Membership of Academic Societies:
The Mathematical Society of Japan

Research Interest:• partial diﬀerential equations• Fourier analysis

Research Summary:
Various phenomena of nature can be treated mathematically by describing them in the languageof partial diﬀerential equations (PDE). Through the analysis I aim to extract new principles which
comprehend concrete phenomena. As a methodology of PDE, many properties of the solutions to
PDE can be deduced from their characteristics, and I employ this idea to investigate quantitative
properties of solutions like size, regularity, and so on. Simultaneously I proceed with the study ofFourier analysis as an important tool for such analysis.
That is the summary of my research, and I explain it in detail below. The tool Fourier integral
operator (FIOp) was theorized by Hörmander et al. in the beginning of 1970’s, and has been
applied to the study of PDE in various situation. In particular, it enables us to discuss PDE
after transforming them to their normal forms. FIOp is also used to express the solutions to
Cauchy problems of hyperbolic and Schrödinger equations, and from the expression we can extract
information on the position of singularities and how they are propagated. In this way, peculiar
information on solutions governed by PDE is inherent in FIOp as algebraic or geometric structure.
On the other hand, non-linear analysis is one of the most active research ﬁelds in the modern
theory of PDE, and many complex phenomena of nature have been clariﬁed thorough it. Knowing
size or regularity of solutions to PDE is an important task because they are reﬂected very sensitively
in phenomena. But unexpectedly, it is not a straightforward task to extract all these informationfrom FIOp. So we need help of Fourier analysis, and sometimes we need to develop Fourier analysis
itself.
With the idea of pushing ahead with the quantitative analysis of PDE via FIOp for a background, I have studied so far the following subjects:• “Lp -estimates for hyperbolic equations”
To determine the relation between Lp -type estimates for hyperbolic equations and the geometrical structure of their characteristics.
• “Smoothing properties of dispersive equations”
To understand why dispersive equations have extra gain of regularity if we take integral mean
in time variable.
Recently I am trying to induce estimates for solutions to PDE by transforming them to theirnormal forms. As a great advantage of this method, we can understand the mechanism of estimates66

from a high position. Having prepared theories of FIOp and function spaces as the fundamental
tools, I have successfully applied them to induce smoothing estimates for dispersive equations.

Major Publications:[1] M. Sugimoto, A priori estimates for higher order hyperbolic equations, Math. Z. 215 (1994),
519–531.
[2] M. Ruzhansky and M. Sugimoto, A smoothing property of Schrödinger equations in the criticalcase, Math. Ann. 335 (2006), 645–673.
[3] N. Tomita and M. Sugimoto, The dilation property of modulation spaces and their inclusion
relation with Besov spaces, J. Funct. Anal. 248 (2007), 79–106.
[4] M. Ruzhansky and M. Sugimoto, Structural resolvent estimates and derivative nonlinear
Schrödinger equations, Comm. Math. Phys. 314 (2012), 281–304.
[5] M. Ruzhansky and M. Sugimoto, Smoothing properties of evolution equations via canonicaltransforms and comparison principle, Proc. London Math. Soc. 105 (2012), 393–423

Awards and Prizes:• 2010, Daiwa Adrian Prizes, “Phase space analysis of partial diﬀerential equations”

Education and Appointments:
1992
1987–1990
1990–1998
1998–2008
2008–

PhD (University of Tsukuba)
Research Associate (University of Tsukuba)
Assistant Professor (Osaka University)
Associate Professor (Osaka University)
Professor (Nagoya University)

Message to Prospective Students:
PDE and Fourier analysis are tightly connected to each other, and are still developing under
their mutual interaction. I advise students to pick one topic in either subject (or both subjects)
and study it keeping other subject within his insight. The aim of this procedure is to get at least
one specialty of their own. Some examples of possible textbooks are listed below:
1. G. B. Folland, Introduction to Partial Diﬀerential Equations, Princeton University Press 1995
2. G. Eskin, Lectures on Linear Partial Diﬀerential Equations, American Mathematical Soc.
2011
3. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press 1993
4. L. Grafakos, Classical Fourier Analysis, Springer 2008
5. K. Gröchenig, Foundation of Time-Frequency Analysis, Birkhäuser 2001
For further study, I often encourage students to read (and write if possible) research papers.

67

SUZUKI, Hiroshi

Associate Professor

Oﬃce:

Rm 459 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 4830

Email:

hiroshis@math.nagoya-u.ac.jp(ext. 4830)

Membership of Academic Societies:
MSJ (The Mathematical Society of Japan)

Research Interest:• Algebraic number theory• Ideal class group• Capitulation problem

Research Summary:An algebraic integer is a complex number which is a root of a monic polynomial
X n + a1 X n−1 + · · · + an−1 X + an

(a1 , . . ., an ∈ Z, n ≧ 1)with rational integer coeﬃcients. An algebraic number ﬁeld K is an extension of the rational
number ﬁeld Q of ﬁnite degree. We call the ring OK consisting of the algebraic integers containedin K the ring of integers of K. Furtwängler showed the Principal Ideal Theorem which statesthat every ideal of OK becomes principal in the ring of integers of the Hilbert class ﬁeld H(K)
(namely the maximal unramiﬁed abelian extension) of K. In capitulation problem, we study ideals
which become principal in the ring of integers of an extension. In [1], we obtained that for any
intermediate ﬁeld L of H(K)/K, the number of ideal classes of K which become principal in L
is divisible by the degree [L : K] of the extension L/K. The paper [2] is a generalization whichcontains Tannaka–Terada’s Principal Ideal Theorem.
In recent years, I am interested in real quadratic ﬁelds of class number 1.

Major Publications:[1] H. Suzuki, A generalization of Hilbert’s theorem 94, Nagoya Math. J., 121 (1991), 161 – 169.
[2] H. Suzuki, On the Capitulation Problem, Advanced Stud. in Pure Math., Class Field Theory –Its Centenary and Prospect, 30 (2001), 483 – 507.
[3] Y. Odai and H. Suzuki, The rank of the group of relative units of a Galois extension II, Tohoku
Math. J. 56 (2004), 367 – 370.

Education and Appointments:
1991
2007

Lecturer, Nagoya University
Associate Professor, Nagoya University

68Message to Prospective Students:
In algebraic number theory, calculations of examples by hand often need enormous time andeﬀorts, so I suggest in my small class using software packages KASH, PARI etc.

69

TAKAHASHI, Ryo

Professor

Oﬃce:

Rm 433 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2834

Email:

takahashi@math.nagoya-u.ac.jp

Web-site:(ext. 2834)

https://www.math.nagoya-u.ac.jp/~takahashi/

Membership of Academic Societies:
MSJ (The Mathematical Society of Japan)

Research Interest:• Commutative Algebra• Representation Theory of Algebras

Research Summary:Commutative algebra is the theory of commutative rings. This theory has close relationships
with many areas of mathematics, including algebraic geometry, number theory, representation theory, noncommutative algebra, algebraic topology, algebraic combinatorics, computational algebra,
and recently, even physics and algebraic statistics. I have mainly been working in the boundary
between commutative algebra and representation theory of algebras.
The subject of representation theory of algebras is to understand the structure of the module
category, i.e., the category of ﬁnitely generated modules, over a given noetherian algebra. The
structure of the module category will be clariﬁed if we can classify all the indecomposable ﬁnitely
generated modules, but this is regarded as “impossible” in general. (Most algebras have wild
representation type, and it is known that over such an algebra it is hopeless to classify the indecomposable ﬁnitely generated modules.) Thus, in modern representation theory of algebras, themain approach to try to understand the structure of the module category is to investigate subcategories of the module category having good properties and triangulated categories associated to themodule category, like derived categories, stable categories and singularity categories.
My research area is “representation theory of commutative rings.” Namely, the purpose of mystudy is to understand the structure of the module category of a given commutative noetherian
ring. Representation theory of Cohen–Macaulay rings was born in the 1970s as a higher dimensionalversion of representation theory of ﬁnite dimensional algebras, which explores the subcategory of
the module category of a Cohen–Macaulay ring consisting of (maximal) Cohen–Macaulay modules.
The meaning of Cohen–Macaulay rings has initially been found in the ideal theory as a local theoryof algebraic geometry. These rings are important from the viewpoints of both homological algebra
and algebraic combinatorics, and have been playing a crucial role in modern commutative algebra. I
have been studying module categories of commutative rings and their subcategories and associatedtriangulated categories, always having in mind Cohen–Macaulay rings, especially Gorenstein rings,
which possess plenty of dualities and symmetries. My current biggest interests are in classifying
resolving subcategories of module categories and thick subcategories of derived categories.

Major Publications:[1] R. Takahashi, Classifying thick subcategories of the stable category of Cohen–Macaulay modules, Adv. Math. 225 (2010), no. 4, 2076–2116.
70

[2] R. Takahashi, Contravariantly ﬁnite resolving subcategories over commutative rings, Amer. J.
Math. 133 (2011), no. 2, 417–436.
[3] H. Dao; R. Takahashi, Classiﬁcation of resolving subcategories and grade consistent functions,Int. Math. Res. Not. (2015), no. 1, 119–149.
[4] S. B. Iyengar; R. Takahashi, Annihilation of cohomology and strong generation of module
categories, Int. Math. Res. Not. (2016), no. 2, 499–535.
[5] H. Matsui; R. Takahashi, Thick tensor ideals of right bounded derived categories, Algebra
Number Theory 11 (2017), no. 7, 1677–1738.

Awards and Prizes:• 2004, MSJ Takebe Prize, “Homological studies of Cohen–Macaulay rings”• 2020, MSJ Algebra Prize, “Subcategories of module categories of commutative rings”

Education and Appointments:
2000
2004
2006
2009
2012
2022

BhSc at Kyoto University
Ph.D. at Okayama University
Assistant Professor, Shinshu University
Associate Professor, Shinshu University
Associate Professor, Nagoya University
Professor, Nagoya University

Message to Prospective Students:“Commutative algebra is a beautiful and deep theory in its own right” — This sentence appears at the beginning of the introduction of [3]. When I was a third-year undergraduate, I met amathematician, who became my Ph.D. advisor later, and he gave me motivations to study commutative algebra. As soon as I started studying commutative algebra, its systematic theory very much
attracted me. Commutative algebra is an area whose entry level is low; one can start studying it
only by basic knowledge on rings given in the undergraduate course. If you have not yet studied
commutative algebra itself, go to a bookstore or a library to get and study [3]. Then, even if you
face a place which you cannot understand (i.e., which you cannot explain in the case where someone
asks you), do not skip it. Consider each sentence until you understand it.
The book [3] has been thought of as the most excellent book in commutative algebra all over theworld, and probably all of those who deal with commutative algebra possess this book. Basically
one can understand it by undergraduate algebra, that is, linear algebra, group theory, ring theory
and general topology. If [3] turns out to be too diﬃcult for you, you can begin with [1] alternatively.
After learning [3], you should move to [2]. This book contains a lot of basic facts which
are usually assumed as preliminary knowledge in papers on commutative algebra. Thus classicalcommutative algebra can be enjoyed enough by studying [3], but to understand recent results incommutative algebra and to get your own results, you need knowledge given in [2]. The book [4]handles representation theory of Cohen–Macaulay rings, and contains classiﬁcation of Gorensteinrings of ﬁnite Cohen–Macaulay representation type and Cohen–Macaulay modules over them, which
has been completed in the 1980s. One can understand this book after learning [2]. Since this is
deeply interesting, I recommend you to study it.
[1] M. F. Atiyah; I. G. MacDonald, Introduction to commutative Algebra, Westview Press, 1994.
[2] W. Bruns; J. Herzog, Cohen-Macaulay rings, Cambridge University Press, 1998.
[3] H. Matsumura, Commutative ring theory, Cambridge University Press, 1989.
71

[4] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, Cambridge University Press,
1990.

72

TANIMOTO, Sho

Professor

Oﬃce:

Rm 506 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 2424

Email:

sho.tanimoto@math.nagoya-u.ac.jp

Web-site:https://shotanimoto.wordpress.com(ext. 2424)

Research Interest:• Algebraic Geometry• Arithmetic Geometry• Diophantine Geometry

Research Summary:
The study of rational solutions to a system of polynomial equations with rational coeﬃcients iscalled as Diophantine geometry whose origin dates back at least to the age of Greece. For example,
Fermat’s last theorem claims that a naive equation xn + y n = z n possesses only trivial solutions
when n ≥ 3, and this theorem has been proved by Wiles using highly advanced mathematics.
A modern approach in Diophantine geometry is to consider rational solutions as points on ageometric object called an algebraic variety which is deﬁned by a system of polynomial equations,and this is a reason why rational solutions are called as rational points. My research mainly involves
applying recent advances of higher dimensional algebraic geometry to problems in Diophantine
geometry and applying the perspective of arithmetic geometry to problems in algebraic geometry.
My research has been centered around Manin’s conjecture which is a conjectural asymptotic
formula for the counting function of rational points on a Fano variety and the asymptotic formula isexpressed in terms of birational invariants of the underlying variety. I have been studying birational
geometric aspects of Manin’s conjecture using higher dimensional algebraic geometry, and I also
have been applying techniques from analytic number theory to prove Manin’s conjecture for certain
homogeneous spaces. So far my research can be summarized in three categories:• We proposed a conjectural description of exceptional sets appearing in Manin’s conjecture
and proved that it is a thin set using the minimal model program. ([1], [7])• We applied the above study of birational geometry of Manin’s conjecture to problems on
moduli spaces of rational curves on Fano varieties. ([2], [5], [8], [9])• We proved Manin’s conjecture and its variants for certain homogeneous spaces using themethod of height zeta functions. ([4], [6])Recently I am interested in homological stability of moduli spaces of rational curves and the motivic
version of Manin’s conjecture.

Major Publications:[1] B. Lehmann and S. Tanimoto, On the geometry of thin exceptional sets in Manin’s conjecture,
Duke Math. J. 166 (2017), no. 15, 2815–2869,[2] B. Lehmann and S. Tanimoto, Geometric Manin’s conjecture and rational curves, Compos.
Math. 155 (2019), no. 5, 833–862,73

[3] S. Tanimoto, On upper bounds of Manin type, Algebra Number Theory 14 (2020), no. 3,
731–761,[4] D. Loughran, R. Takloo-Bighash, and S. Tanimoto, Zero-loci of Brauer group elements on
semi-simple algebraic groups, J. Inst. Math. Jussieu, 19 (2020), no. 5, 1467–1507,[5] B. Lehmann and S. Tanimoto, Rational curves on prime Fano threefolds of index 1, J. Algebraic
Geom., 30 (2021), no. 1, 151–188,[6] M. Pieropan, A. Smeets, S. Tanimoto, and A. Várilly-Alavarado, Campana points of boundedheight on vector group compacriﬁcations, Proc. Lond. Math. Soc, 123 (2021), no. 1, 57–101,[7] B. Lehmann, A. K. Sengupta, and S. Tanimoto, Geometric consistency of Manin’s conjecture,
Compos. Math. 158 (2022), no. 6, 1375–1427[8] B. Lehmann and S. Tanimoto, Classifying sections of del Pezzo ﬁbrations, I, to appear in J.
Eur. Math. Soc. (JEMS),[9] B. Lehmann, E. Riedl, and S. Tanimoto, Non-free sections of Fano ﬁbrations, submitted,

Education and Appointments:
2012
2012
2015

2018

2021

Courant Institute of Mathematical Sciences,
New York University, Ph.D.
G.C. Evans Instructor,
Department of Mathematics, Rice University
PostDoc,
Department of Mathematical Sciences,
the University of Copenhagen
Associate Professor,
Priority Organization for Innovation and Excellence,
Kumamoto University
Associate Professor,
Graduate School of Mathematics, Nagoya University

Message to Prospective Students:
Undergraduate students need to study the foundation of algebraic geometry, i.e., schemes andcohomology, and I have been using [2] for my seminar. To understand these, students need to be
familiar with commutative algebra, and this can be acquired by reading [1] or [3].
I want my MS students to study higher dimensional algebraic geometry. Possible topics are: (1)the minimal model program (2) Positivity of divisors (3) Theory of rational curves. To understand(1), [4] is a standard textbook, but this book does not contain recent advances in the MMP suchas BCHM. These advances can be studied by reading original papers after completing [4]. To learn(2), [5] are best textbooks. I have not read these recently and I am interested in reading themagain. For (3), [6] is a good textbook. [7] is a more serious book, but it is a bit challenging to read.
MS students in their second year and Doctor students should conduct their own research.
Topics of my recent students are: (1) Examples of exceptional sets in Manin’s conjecture, (2)Geometric Manin’s conjecture for Fano varieties, (3) Constructions of derived categories. Any
topics in algebraic geometry and arithmetic geometry are welcome, so let me know if you have any
problem you are interested in.
[1] M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra,[2] R. Hartshorne, Algebraic Geometry, Springer[3] S. Bosch, Algebraic Geometry and Commutative Algebra, Springer
74

[4] J. Kollár and Sh. Mori, Birational Geometry of Algebraic Varieties, Cambridge[5] R. Lazarsfeld, Positivity in Algebraic Geometry I, II, Springer[6] O. Debarre, Higher Dimensional Algebraic Geometry, Springer[7] J. Kollár, Rational curves on Algebraic Varieties, Springer

75

TERASAWA, Yutaka

Associate Professor

Oﬃce:

Rm 457 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 4533

Email:

yutaka@math.nagoya-u.ac.jp(ext. 4533)

Membership of Academic Societies:(MSJ) The Mathematical Society of Japan

Research Interest:• Mathematical Analysis of Fundamantal Equations for Fluid Mechanics• Fourier Analysis

Research Summary:
My main research interest is in mathematical analysis of fundamental equations for ﬂuid mechanics. Recently, I study motion of non-Newtonian ﬂuids (ﬂuids of macromolecule such as gel andketchup). Navier-Stokes equations which describes the motion of incompressible Newtonian ﬂuids
is the following:∂t u − div(νDu) + u · ∇u + ∇p = f , div u = 0Here, u is a velocity vector ﬁeld of the ﬂuid, Du = 12 ∇u + (∇u)T , p is a pressure scalar ﬁeldand ν is a viscosity of the ﬂuid which is a positive constant and f is an external force. u and p
are unknown functions and a problem of solvability of the system under suitable initial and boundary data is initial-boundary-value problem of Navier-Stokes equations. Main characteristics of theequations is non-linearity, having several unknowns and incompressibility condition divu = 0. The
analysis of the equations are diﬃcult because one couldn’t use a maximum principle for the solutions which holds for certain non-linear heat equations. In fact, global in time existence of theclassical solutions for the Navier-Stokes equations in 3-dim. space is a famous open problem andis one of seven millennium problems posed by Clay Mathematics Institute. There have been manystudies about local in time existence of classical solutions and global in time existence of classicalsolutions concerning the equation.
My recent research interest is in non-Newtonian ﬂuids which is described by the equations whose
diﬀerence with the Navier-Stokes equations is that the viscosity is a non-constant function which
depends on the largeness of Du. Among the equations which describes various non-Newtonian
ﬂuids is power-law ﬂuid equations in which the Laplacian part of the Navier-Stokes equations are
replaced by the p-Laplacian type operator. The solvability of the equations depends on the valueof p. Maximal monotone operator theory is needed to show the existence of global weak solution
and in 60’s Ladyzhenskaya and Lions show the unique existence of global weak solutions for large
p. To show the existence of weak solutions for smaller p, Fourier analytic method called ”Lipschitz
truncation method” is needed and it is studied well by a group around Ruzicka after ’00. For
an analysis of two-phase Newtonian ﬂuids, such as oil and water, Lp estimates (or, Lp maximal
regularity) of its linearized equation around initial data is needed. A similar method can be
employed for single power-law ﬂuids and the local existence of its classical solutions were shown byBothe-Pr”uss(’07).
I am studying problems related with global existence of weak solution for two-phase power-law
ﬂuid and local existence of its classical solution. I am also interested in a study of deeper property76

of a linearized equation of a single power-law ﬂuid equation and related Probability Theory andFourier Analysis.

Major Publications:[1] H. Abels and Y. Terasawa, On Stokes operators with variable viscosity in bounded and unbounded domains, Math. Ann. 344 (2009), 381–429.
[2] H. Abels and Y. Terasawa, Non-homogeneous Navier-Stokes systems with order-parameterdependent stresses, Math. Methods Appl. Sci. 33 (2010), 1532–1544.
[3] H. Abels, L. Diening and Y. Terasawa, Existence of weak solutions for a diﬀuse interface modelof non-Newtonian two-phase ﬂows, Nonlinear Anal. Real World Appl. 15 (2014), 149–157.

Education and Appointments:
2007
2009
2010
2011
2012
2014

Graduated from Gradate school of Mathematics at Hokkaido University
Research Assistant, Tohoku University
Project Researcher, The University of Tokyo
JSPS Fellow, The University of Tokyo
Project Research Associate, The University of Tokyo
Associate Professor, Nagoya University

Message to Prospective Students:
To investigate a motion of incompressible viscous ﬂuids, one needs to master basic theories on
Partial Diﬀerential Equations, Functional Analysis and Fourier Analysis etc. In the seminar, it ispreferable that students ﬁrst master one of these topics well and then later study students study
more advanced topics such as solvability theory of incompressible viscous ﬂuid equations based on
it. It is also preferable that students expand their knowledge of other ﬁelds in the course of theirstudy of advanced topics. If students hope, both basic course and advanced course could be held.
I list a candidate for texts used in the course, but students could choose other books.
1. S. Krantz, A Panorama of Harmonic Analysis, The Mathematical Association of America.
2. T. Hytönen, Weighted Norm Inequalities, Lecture Note available on Web.
3. H. Tanabe, Functional Analytic Methods for Partial Diﬀerential Eqautions, CRC Press.
4. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Diﬀerential Equations, Springer.
5. M. Giaquinta, L. Martinazzi, An introduction to the regularity theory for elliptic system,
harmonic maps and minimal graphs, Edizioni Della Normale.
6. A. McIntosh, Operator Theory - Spectra and Functional Calculi, Lecture Note available on
Web.
Students who would like to take our master course small class are required to have soundknowledge of basic facts in Calculus, Ordinary Diﬀerential Equation, Complex Analysis, Lebesgue
Integration Theory and Functional Analysis. If a student doesn’t have enough knowledge to study
in the course, he or she needs to study it in case it is needed. After these studies, one studies
more specialized topics such as analysis of motion of incompressible ﬂuids. One could also choose
another subject related to Fourier analysis or PDE.
In their Doctor Course, I advise students on more advanced topics. In Doctor Course, students’hope is valued concerning a choice of their research topics.

77

UEDA, Yoshimichi

Professor

Oﬃce:

Rm 301 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 2823

Email:

ueda@math.nagoya-u.ac.jp

Web-site:(ext. 2823)

http://www.math.nagoya-u.ac.jp/~ueda/

Research Interest:• Operator algebras• Non-commutative analysis

Research Summary:
I have studied various topics based on operator algebras. The highlights of operator algebras are:ﬁghts against “non-commutative phenomena” and overcoming obstacles from “inﬁnite dimensionality”. Finding suitable algebraic formulations is important in view of “non-commutativity”, and
it is interesting that we need skills of analysis to overcome obstacles from “inﬁnite dimensionality”.
Operator algebras are so cool and quite an interesting and broad subject in mathematics.
The keywords of my research include: von Neumann algebras, free product (with amalgamation), HNN extension, quantum group, subfactor, Jones index, ergodic equivalence relation, freeprobability theory, random matrix, non-commutation function space.

Major Publications:[1] Yoshimichi Ueda, A minimal action of the compact quantum group SUq (n) on a full factor.
Journal of Mathematical Society of Japan, Vol. 51 (1999), 449 – 461.
[2] Dimitri Shlyakhtenko and Yoshimichi Ueda, Irreducible subfactors of L(F∞ ) of index λ > 4.
Journal für die reine und angewandte Mathematik, Vol. 548 (2002), 149 – 166.
[3] Fumio Hiai, Denis Petz and Yoshimichi Ueda, Free transportation cost inequalities via randommatrix approximation. Probability Theory and Related Fields, Vol. 130 (2004), 199 – 211[4] Fumio Hiai, Takuho Miyamoto and Yoshimichi Ueda, Orbital approach to microstate free entropy. International Journal of Mathematics, Vol. 20 (2009), 227–273.
[5] Yoshimichi Ueda, On peak phenomena for non-commutative H ∞ . Mathematische Annalen, Vol.
343 (2009), 421–429.
[6] Yoshimichi Ueda, Factoriality, type classiﬁcation and fullness for free product von Neumann
algebras. Advances in Mathematics, Vol. 228 (2011), 2647–2671.
[7] Yoshimichi Ueda, Discrete cores of type III free product factors. American Journal of Mathematics, Vol. 138 (2016), 367–394.
[8] Yoshimichi Ueda, Matrix liberation process I: Large deviation upper bound and almost sureconvergence. Journal of Theoretical Probability, to appear.

Education and Appointments:
Mar. 1999
Apr. 1999
Sep. 1999
Apr. 2002
Oct. 2017

Completed Graduate program, Kyushu University
Research Associate, Hiroshima University
Doctoral degree (Ph.D.) from Kyushu University
Associate Professor, Kyushu University
Professor, Nagoya University78Message to Prospective Students:
You should ﬁnish to learn calculus, linear algebra, general topology, complex analysis, measure
theory, and elementary functional analysis prior to your entrance to our graduate school for yourstudy on operator algebras. You can obtain enough knowledge on complex analysis, measure
theory, and elementary functional analysis from two very ﬁne books “Real and Complex Analysis”
by W. Rudin and “A Guide to Functional Analysis” by S.G. Krantz. These three subjects form an
important basis for the study of any branches of mathematical analysis. It is also very nice if you
have already learnt a little about abstract algebra and geometry as well as probability theory.
I usually advise my students to study topics, which are diﬀerent from my research topics aswell as other student’s ones under my guidance, within operator algebras in a broad sense. Thisis because I hope that any student studies hard as an independent and important member of ourgroup.

79

UZAWA, Tohru

Professor

Oﬃce:

Rm 305 in Math. Bldg.

Phone:+81 (0) 52 – 789 – 2461

Email:

uzawa@math.nagoya-u.ac.jp(ext. 2461)

Membership of Academic Societies:
BSP (The Biophysical Society of Japan),
SMF (Société Mathématique de France)

Research Interest:• Representation Theory of Groups• Compactiﬁcations• Applied Mathematics

Research Summary:Representation theory of groups is a relative new-comer to mathematics: it started with a paper
by Frobenius titled ”On the theory of group characters” in 1887. It was ﬁrst a theory of characters,
and it was Schur that formulated the results in terms of representations, namely homomorphisms
from a group G to the general linear group GL(V ).
Representation developed rapidly, and is crucial to many ﬁelds in mathematics, physics, andapplications.
The most natural way to produce examples of groups is to look at the group of symmetries
preserving a structure. For example, one can take a vector space V over a ﬁnite ﬁeld Fq and lookat the group of linear isomorphisms. This is a ﬁnite group which is almost a simple group. Simple
groups are the fundamental building blocks in group theory, and one interesting fact is that all
but ﬁnitely many simple groups arise as groups preserving a geometry, and thus can be classiﬁedusing Dynkin diagrams. Dynkin diagrams classify simple compact Lie groups; this is yet anotherexample of the continuity joining ﬁnite groups and simple groups over various ﬁelds.
I have worked on real groups, and introduced a geometric method to give Matsuki correspondencees ([1]). I am also working on the use of covariant compactiﬁcations of group varieties andits applications to representation theory [2], and symmetric varieties over ﬁelds of arbitrary characteristics [1].
I am now also working on problems that arise from industry and medical research. Recent
progress in stem cell research oﬀers various interesting mathematical problems: the ﬁeld is far frommature, and novel mathematical methods may arise from such research.

Major Publications:[1] T. Uzawa: Symmetric varieties over arbitrary ﬁelds , C. R. Acad. Sci. Paris Sér. I Math. 333(2001), no. 9, 833–838,[2] T. Uzawa: Compactiﬁcations of symmetric varieties and applications to representation theory,Sūrikaisekikenkyūsho Kōkyūroku 10826 (1999), 137–142.
[3] Inui, N. and Katori, M. and Uzawa, T.: Duality and universality in non-equilibrium latticemodels, J. Phys. A 28 (1995), no. 7, 1817–1830.
[4] Mirković, I. and Uzawa, T. and Vilonen, K.: Matsuki correspondence for sheaves, Invent. Math.
109(1992) no. 22, 231–245.
80

[5] T. Yamada, H.Akamatsu, S. Hasegawa, N. Yamamoto, T.Yoshimura, Y.Hasebe, Y. Inoue,
H. Mizutani, T.Uzawa, K. Matsunaga, S. Nakata: Age-related changes of p75 Neurotrophin
receptor-positive adipose-derived stem cells, J. of Dermatological Science 58 (2010), no. 1,
36–42.

Education and Appointments:
1990
1991
1992
1997
2002

Assistant Professor, Penn State
Assistant Professor, Tokyo University
Associate Professor, Tohoku University
Associate Professor, Rikkyo University
Professor, Nagoya University

Message to Prospective Students:
The interplay between theory of groups and representation theory is fascinating, and the connection with automorphic forms is one of the most tantalizing connections to be found in mathematics.
I prefer to work with geometric methods, a student should have strong background in diﬀerential
geometry and algebraic geometry. My work in applied ﬁelds are scattered, but work related tostem cells are starting to take form; this is a rapidly progressing ﬁeld, and it is hard to say what
are the basic mathematical tools.
[1] I.M.Gelfand, M.I. Graev, I. Piatetski-Shapiro, Representation theory and automorphic functions.
[2] R.Hartshorne: Algebraic geometry.
[3] D. Vogan: Unitary Representations of Reductive Lie Groups[4] D.

MacKay:Information

Theory,Inference,

http://www.inference.phy.cam.ac.uk/itprnn/book.html[5] W.Fulton: Intersection theory.

81

and

Learning

Algorithms.

YANAGIDA, Shintaro

Associate Professor

Oﬃce:

Rm 441 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 5595

Email:

yanagida@math.nagoya-u.ac.jp

Web-site:(ext. 5595)

http://www.math.nagoya-u.ac.jp/~yanagida/index.html

Membership of Academic Societies:
The Mathematical Society of Japan

Research Interest:• Representation Theory• Algebraic Geometry• Mathematical Physics

Research Summary:My research areas are representation theory and algebraic geometry, in particular the topics
related to mathematical physics and special functions.
My interest in algebraic geometry is mainly on the derived category of sheaves on algebraic varieties. Two keywords may be named: Fourier-Mukai transforms and Bridgeland stability conditions.
On these topics, I have co-authored papers [3] and [10].
My interest in representation theory is mainly on quantum algebras, in particular quantum
groups, Hall algebras and vertex algebras. In the collaboration [1], we investigated the quantum
integrable system associated to Macdonald symmetric functions using representation theory of gl1
quantum toroidal algebra (also called the Ding-Iohara-Miki algebra). Since then, I have been
studying the Macdonald polynomials and the (double) aﬃne Hecke algebras associated to aﬃneroot systems [8, 9].
As an intersection of algebraic geometry and representation theory, I have been studying geometric aspects of vertex algebras. In [6], I introduced the gluing construction of vertex algebras ofclass S in the derived setting, using the derived symplectic/Poisson geometry. In [7], I introduced
an analogue of the canonical Li ﬁltration of a vertex algebra for an arbitrary SUSY vertex algebra,
and relate the representation theory of superconformal vertex algebras to the Poisson geometry of
the associated superschemes.
Recently, I collaborated with the doctor student Yusuke Nishinaka-san [4, 5] to establish thealgebraic operad encoding the structure of SUSY vertex algebras, and with Masamune Hattori-san[2] to introduce the dynamical Ding-Iohara algebroids which unify the elliptic quantum groups andDing-Iohara quantum algebras.

Major Publications:[1] B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, S. Yanagida, A commutative algebra on
degenerate CP1 and Macdonald polynomials, J. Math. Phys. 50 (2009), no. 9, 095215, 42 pp.
[2] M. Hattori, S. Yanagida, A dynamical analogue of Ding-Iohara quantum algebras, preprint(2022), arXiv:2210.02777, 42pp.
[3] H. Minamide, S. Yanagida, K. Yoshioka, The wall-crossing behavior for Bridgeland’s stability
conditions on abelian and K3 surfaces, J. Reine Angew. Math. 735 (2018), 1–107.
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[4] Y. Nishinaka, S. Yanagida, Algebraic operad of SUSY vertex algebra, preprint (2022),
arXiv:2209.14617, 42pp.
[5] Y. Nishinaka, S. Yanagida, Algebraic operad of SUSY Poisson vertex algebra, preprint (2023),
arXiv:2305.00714, 42pp.
[6] S. Yanagida, Derived gluing construction of chiral algebras, Lett. Math. Phys. 111 (2021),
Article no. 51, 103pp.
[7] S. Yanagida, Li ﬁltrations of SUSY vertex algebras, Lett. Math. Phys., 112 (2022), Article no.
103, 77pp.
[8] K. Yamaguchi, S. Yanagida, Specializing Koornwinder polynomials to Macdonald polynomialsof type B, C, D and BC, J. Algebraic Combin. (2022), online published, 56pp.
[9] K. Yamaguchi, S. Yanagida, A review of rank one bispectral correspondence of quantum
aﬃne KZ equations and Macdonald-type eigenvalue problems, RIMS Kokyuroku (2023) 36pp.;
arXiv:2211.13671.
[10] S. Yanagida, K. Yoshioka, Semi-homogeneous sheaves, Fourier-Mukai transforms and moduli
of stable sheaves on abelian surfaces, J. Reine Angew. Math. 684 (2013), 31–86.

Education and Appointments:
2012
2012
2012
2016

Ph.D. Mathematics at Kobe University
JSPS PD at RIMS, Kyoto University
Assistant Professor, RIMS, Kyoto University
Associate Professor, Nagoya University

Message to Prospective Students:
Undergraduate students interested in algebraic geometry or (algebraic/geometric) representation theory will be welcomed. The reading seminar will be on standard texts such as the textbooks
1,2 and 3.
I also welcome graduate students who are willing to study Bridgeland stability conditions andrelated topics, or geometric representation theory of quantum algebras. For examples of particular
topics, please see the books 4, 5 and 6 below.
1. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer (1977).
2. T. Tanisaki, Lie algebras and quantum groups (in Japanese), Kyoritsu-syuppan (2002).
3. Y. Yamada, Introduction to conformal ﬁeld theory (in Japanese), Baifukan (2006).
4. D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Univ. Press (2006).
5. D. Huybrechts, M. Lehn, The geometry of moduli spaces of sheaves, Cambridge University
Press (2010).
6. E. Frenkel, D. Ben-Zvi, Vertex algebras and algebraic curves, 2nd edition, Mathematical
Surveys and Monographs 88, American Mathematical Society (2004).

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YOSHIDA, Nobuo

Professor

Oﬃce:

Rm 439 in Sci. Bldg. A

Phone:+81 (0) 52 – 789 – 2420

Email:

noby@math.nagoya-u.ac.jp

Web-site:(ext. 2420)

http://www.math.nagoya-u.ac.jp/~noby/

Membership of Academic Societies:
MSJ (Mathematical Society of Japan)

Research Interest:• Directed polymers in random environment• Interacting particle systems

Research Summary:Imagine a hydrophilic polymer chain wafting in water. Due to the thermal ﬂuctuation, the shapeof the polymer should be understood as a random object. We now suppose that the water containes
randomly placed hydrophobic molecules as impurities, which repell the hydrophilic monomers which
the polymer consists of. The question we address here is;
How does the impurities aﬀect the global shape of the polymer chain?
My recent research interest centers around the above question. The above question is mathematically formulated in the framework of ”directed polymers in random environment” (DPRE). Thiscan be thought of as a model of statistical mechanics in which paths of the random walk interact
with a quenched disorder (impurities). We study the phase transition of this model, which depends
on the dimension of the sapce and the thermodynamic parameters.
More recently, it was recognized that the DPRE is closely connected to a certain class ofinteracting particle systems, as well as branching random walks in random environment. This
provides us with new perspectives of the research.

Major Publications:[1] F. Comets, N. Yoshida: Localization Transition for Polymers in Poissonian Medium. Commun.
Math. Phys. (to appear)[2] R. Fukushima, N. Yoshida On the exponential growth for a certain class of linear systems.
ALEA Lat. Am. J. of Prob. Math. Stat. 9 (2012), 323–336.
[3] Y. Nagahata, N. Yoshida Localization for a Class of Linear Systems. Electron. J. Prob.16(2011), no. 3, 657–687[4] F. Comets, N. Yoshida Branching Random Walks in Time-Space Random Environment: Survival Probability, Global and Local Growth Rates. J. Theoret. Prob.24 (2010), no. 3, 657–687

Awards and Prizes:• The 4th MSJ Analysis Prize (2005)

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Education and Appointments:
1991
1998
2003
2013

Assistant Professor, Kyoto University
Lecturer, Kyoto University
Associate Professor, Kyoto University
Professor, Nagoya University

Message to Prospective Students:
The text book of the seminar for graduate students will be choosen, for example from thefollowing subjects: Brownian motion, stochastic calculus, interacting particle systems, percolation.

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BOURNE, Chris

Associate Professor
Oﬃce:!! Invalid Oﬃce (Humanities Building-419)

Phone:+81 (0) 52 – 789 – 5181

Email:

cbourne@nagoya-u.jp

Web-site:(ext. 5181)

https://sites.google.com/site/khomologyzone/

Membership of Academic Societies:
International Association of Mathematical Physics (IAMP),The Mathematical Society of Japan (MSJ),
Australian Mathematical Society (AustMS)

Research Interest:• Operator algebras• Index theory and noncommutative geometry• Mathematical physics

Research Summary:
Much of my research is about the application of operator algebras, noncommutative geometry
and index theory to systems in non-relativistic quantum physics.
Quantum mechanical observables are represented by operators on a complex Hilbert space. We
can study algebras of observables/operators to rigorously deﬁne and study various properties of theunderlying physical system. Noncommutative geometry and index theory can then potentially be
used to determine a system’s topological properties. In more detail, we can use ideas from index
theory and K-theory of operator algebras to assign a topological index to a system that serves asa mathematical phase label. Topological phase labels are invariant under deformations or small
perturbations and physical systems described by topological phases, such as the quantum Halleﬀect, posses many novel properties.
In previous research, I studied topological phases of a variety of systems such as symmetricfree-fermions [4, 5] and many-body ground states in quantum statistical mechanics with a spectral
gap condition [3].
The study of index theory in quantum mechanics, where anti-linear symmetries such as timereversal symmetry may play an important role, has also motivated new research on K-theory andindex theory on real Hilbert spaces. The paper [1] develops spectral ﬂow and index theory on real
Hilbert spaces with additional ‘Cliﬀord algebra symmetries’, which can be connected to quantum
mechanical symmetries.
Recently I have been exploring new applications of index theory techniques in physics andbeyond. An example is the paper [2], which applies methods in coarse geometry, a frameworkused to study the index theory of non-compact manifolds, to the ground states of superconductor
models. I am also currently studying the relevance of operator algebras, noncommutative geometry
and index theory in quantum walks and quantum information theory.

Major Publications:[1] C. Bourne, A. L. Carey, M. Lesch and A. Rennie. The KO–valued spectral ﬂow for skew-adjoint
Fredholm operators. J. Topol. Anal., 14(2):505–556, 2022.
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[2] C. Bourne. Locally equivalent quasifree states and index theory. J. Phys. A: Math. Theor.,55(10):104004 (38 pages), 2022.
[3] C. Bourne and Y. Ogata. The classiﬁcation of symmetry protected topological phases of onedimensional fermion systems. Forum Math. Sigma, 9, Article No. e25 (45 pages), 2021.
[4] C. Bourne and B. Mesland. Index theory and topological phases of aperiodic lattices. Ann.
Henri Poincaré, 20(6):1969–2038, 2019.
[5] C. Bourne and A. Rennie. Chern numbers, localisation and the bulk-edge correspondence forcontinuous models of topological phases. Math. Phys. Anal. Geom., 21(3):16 (62 pages), 2018.

Awards and Prizes:• Journal of Physics A (Mathematical and Theoretical) Best Paper Prize 2020

Education and Appointments:
2015
2016
2017
2018
2018
2023

PhD (Mathematics), The Australian National University
Postdoc, Friedrich-Alexander Universität Erlangen-Nürnberg
JSPS Postdoc, Tohoku Univerisity
Assistant Professor, Tohoku University
Visiting Scientist, RIKEN iTHEMS
Associate Professor, Nagoya University

Message to Prospective Students:
My research is in the ﬁeld of noncommutative geometry, operator algebras and their applications
in physics. Noncommutative geometry and operator algebras bring together many ideas from
diﬀerent areas of mathematics. This combination of mathematical ideas makes the ﬁeld interesting
to work in, but it also means that one needs a basic knowledge of many diﬀerent mathematicalﬁelds, which can be daunting for new students. I hope to be able to help you navigate this barrierof entry and assist you to ﬁnd an area of study that you ﬁnd interesting. Some reference books
that we use in the small-group seminar are below:[1] J. M. Gracia-Bondı́a, J. C. Várilly, H. Figueroa – Elements of Noncommutative Geometry[2] K. Davidson – C*-Algebras by Example[3] N. E. Wegge-Olsen – K-Theory and C*-Algebras: A Friendly Approach[4] O. Bratteli, D. Robinson – Operator Algebras and Quantum Statistical Mechanics[5] D. D. Bleecker, B. Booß-Bavnbek – Index Theory with Applications to Mathematics and Physics[6] N. Higson, J. Roe – Analytic K-homology

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