Department of Mathematics

Department of Mathematics
Department
ofMathematics

Summary
Mathematics, originally centered in the concepts ofnumber, magnitude, and form, has long been growing
since ancient Egyptian times to the 21st century.
Through the use of abstraction and logical reasoning, it
became an indispensable tool not only in naturalsciences, but also in engineering and social sciences.
Recently, the remarkable development of computers is
now making an epoch in the history of mathematics.
Department of Mathematics is one of the six
departments of Graduate School of Science, Osaka
University. It consists of 6 research groups, all ofwhich are actively involved in the latest developmentsof mathematics. Our mathematics department has
ranked among the top seven in the country.
The department offers a program with 32 newstudents enrolled annually leading to post-graduate
degrees of Masters of Science. The department alsooffers a Ph.D. program with possibly 16 new students
enrolled annually.
Graduate courses are prepared so as to meet various
demands of students. Besides introductory courses forfirst year students, a number of topic courses are given
for advanced students. Students learn more specialized
topics from seminars under the guidance of thesis
advisors.
Our department has our own library equipped with
about 500 academic journals and 50,000 books inmathematics, both of which graduate students can use
freely. In addition, by an online system, students aswell as faculty members can look up references
through Internet.
Research Groups
Algebra, Geometry, Analysis, Global Geometry &Analysis, Experimental Mathematics, Mathematical
Science.

Theory, Dynamical Systems, Fractals, MathematicalEngineering, Information Geometry, Mathematical
Physics.
Faculty Members
Professors (16)Shin-ichi DOI, Akio FUJIWARA, Ryushi GOTO,
Masashi ISHIDA, Seiichi KAMADA, Soichiro
KATAYAMA, Takehiko MORITA, Hiroaki
NAKAMURA, Shin-ichi OHTA, Atsushi
TAKAHASHI, Naohito TOMITA, Takao
WATANABE, Katsutoshi YAMANOI, Kouji YANO,
Takehiko YASUDA, Masahiko YOSHINAGA.
Associate Professors (14)Shinpei BABA, Kento FUJITA, Takahisa INUI, Hisashi
KASUYA, Yoshihiko MATSUMOTO, Haruya
MIZUTANI, Tomonori MORIYAMA, Takahiro OBA,
Mamoru OKAMOTO, Shinnosuke OKAWA, Kazuto
OTA, Yuichi SHIOZAWA, Shuichiro TAKEDA,
Motoo UCHIDA.
Lecturer (1)Kazunori KIKUCHI.
Assistant Professors (9)Yasuhiro HARA, Takao IOHARA, Masataka IWAI,
Kohei KIKUTA, Erika KUNO, Shohei NAKAMURA,
Hiroyuki OGAWA, Koji OHNO, Toru SERA.
Cooperative Members in Osaka University
Professors (5)Daisuke FURIHATA, Eiko KIN, Katsuhisa MIMACHI,
Makoto NAKAMURA, Yoshie SUGIYAMA.
Associate Professors (6)Tsuyoshi CHAWANYA, Akihiro HIGASHITANI,
Yuto MIYATAKE, Norio NAWATA, Yasuhiro
WAKABAYASHI, Kouichi YASUI.
Home Page
http://www.math.sci.osaka‐u.ac.jp/eng/

Areas of Research
Number Theory, Ring Theory, Algebraic Geometry,
Algebraic Analysis, Partial Differential Equations,
Real Analysis, Differential Geometry, Complex
Differential Geometry, Topology, Knot Theory,
Discrete Subgroups, Transformation Groups, Complex
Analysis, Complex Functions of Several Variables,
Complex Manifolds, Discrete Mathematics, Probability

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Shinpei BABA
Low‐dimensional Geometry, Topology

My research is centered on surfaces (i.e. 2dimensional
manifolds),
which arefundamental object in geometry andtopology.
In particular, I am interested in therelations between geometric structures(locally homogeneous structures)onsurfaces, and representations of thefundamental groups of surfaces S (surface
groups) into Lie groups G.
In the case that the Lie groups G = PSL(2,R) or PSL(2,C) and that representations
have discrete images, beautiful theories
have been developed extensively, inparticular, in relation to the classificationand the deformation theorem of 2- and 3dimensional manifolds.

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ofMathematics

Shin‐ichi DOI
Partial Differential Equations

Partial differential equations have their
origins in various fields such as mathematicalphysics,
differential
geometry,andtechnology. Among them I am particularlyinterested in the partial differential equations
that describe wave propagation phenomena:
hyperbolic
equationsanddispersive
equations. A typical example of the former isthe wave equation, and that of the latter is the
Schroedinger equation. For many years I havestudied basic problems for these equations:existence and uniqueness of solutions,structure of singularities of solutions,
asymptotic behavior of solutions, and spectral
properties. Recently I make efforts tounderstand how the singularities of solutionsfor Schroedinger equations or, moregenerally, dispersive equations propagate.
The center of this problem is to determine
when and how the singularities of solutions
for the dispersive equations can be describedby the asymptotic behavior of solutions forthe associated canonical equations.

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Kento FUJITA
Algebraic Geometry

My research interest is algebraic geometry.
Especially, I am interested in birational
behavior of Fano varieties, a special class ofalgebraic varieties. I have researched the
Mukai conjecture, the existence of certain
good models over some reducible varieties,
and an algorithm to classify log del Pezzo
surfaces (joint work with Kazunori
Yasutake), etc. Recently, I am interested in
K-stability of Fano varieties; I (andindependently Chi Li) gave a birational
interpretation of K-stability of Fano
varieties.

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Akio FUJIWARA
Mathematical Engineering"What is information?" Having this naive
yet profound question in mind, I have beenworking
mainlyon
noncommutative
statistics, information geometry, quantum
information
theory,andalgorithmic
randomness theory.
One can regard quantum theory as anoncommutative extension of the classicalprobability theory. Likewise, quantum
statistics is a noncommutative extension of
the classical statistics. It aims at finding thebest strategy for identifying an unknown
quantum object from a statistical point ofview, and is one of the most exciting research
field in quantum information science.
Probability theory is usually regarded as abranch of analysis. Yet it is also possible toinvestigate the space of probability measuresfrom a differential geometrical point of view.
Information geometry deals with a pair ofaffine connections that are mutually dual
(conjugate) with respect to a Riemannian
metric on a statistical manifold. It is known
that geometrical methods provide us with
useful guiding principle as well as insightful
intuition in classical statistics. I am interested
in extending information geometricalstructure to the quantum domain, admitting
an operational interpretation.
I am also delving into algorithmic andgame-theoretic
randomnessfromaninformation geometrical point of view.
SomedayI
wish
toreformulate
thermal/statistical physics in terms ofalgorithmic information theory.

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Ryushi GOTO
Geometry

My research interest is mostly in complex
and differential geometry, which are closely
related with algebraic geometry andtheoretical physics. My own research started
with special geometric structures such as
Calabi-Yau, hyperKaehler, G2 and Spin(7)structures. These four structures exactly
correspond to special holonomy groups whichgive rise to Ricci-flat Einstein metrics on
manifolds. It is intriguing that these moduli
spaces are smooth manifolds on which local
Torelli type theorem holds. In order tounderstand these phenomenon, I introduce a
notion of geometric structures defined by asystem of closed differential forms andestablish a criterion of unobstructed
deformations of structures.

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When we apply this approach to CalabiYau, hyperKaehler, G2 and Spin(7)structures, we obtain a unified construction ofthese moduli spaces. At present I also explore
other interesting geometric structures andtheir moduli spaces.

Yasuhiro HARA
Topology

The field of my study is topology and,especially, I study the theory oftransformation groups. The Borsuk-Ulamtheorem is one of famous theorems abouttransformation groups. This theorem is often
taken up as an application in elementarylectures about the homology theory. The
content of the theorem is as follows: for every
continuous map from the n-dimensional
sphere to the n-dimensional Euclidian space,
there exists a point such that the map takes
the same value at the point and at the
antipodal point. A famous application of thistheorem is the following. ''On the earth, there
is a point such that the temperature andhumidity at the point are the same as those at
the antipodal point.'' We consider a free

action of a group of order two on the ndimensional sphere to prove the Borsuk-Ulamtheorem. Then for any equivariant map (anycontinuous map which preserves the structureof the group action) from the sphere to itself,
the degree of the map is odd. By using thisfact, we obtain the Borsuk-Ulam theorem. In
the case of the Borsuk-Ulam theorem, weconsider spheres and free actions of a groupof order two. Actually, when we consider
other manifolds and actions of other groups,
there are some restrictions of homotopy typesof equivariant maps. I study such restrictionsof homotopy types of equivariant maps byusing the cohomology theory, and I study
relationships between homotopy types of
equivariant maps and topological invariants.

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Takahisa INUI
Nonlinear Partial Differential Equations

My research subject is nonlinear partial
differential equations. Especially, I am
interested in the global behavior of thesolutions to nonlinear dispersive equations or
nonlinear wave equations.
Dispersive equations describe the dispersion
phenomena of waves and are basic equationsin quantum physics. For example,
Schroedinger equation and Klein--Gordon
equation are typical dispersive equations,which appear in quantum physics orrelativistic quantum field theory. Wave
equations express the properties of motion inwaves. Considering nonlinear interactions
between particles, we can treat various
physical phenomena, for example, optics,
superconductivity,
andBose-Einstein

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condensate, by the nonlinear dispersive or
wave equations. Nonlinear dispersive or wave
equations have two properties.
One is dispersion and the other isnonlinearity. They conflict each other. Thus,
there are so many behaviors of the solutions. I
study this subject to find all behavior.

Takao IOHARA
Nonlinear Partial Differential Equations

My research interest is concerned withnonlinear partial differential equations
appearing in fluid mechanics. The current
research topic is the equations of the motionof viscous incompressible fluid which has
free moving surface. The motion of viscousincompressible fluid is governed by the
Navier-Stokes equations, which are not easy
to solve because of their nonlinearity. The
free moving surface adds anothernonlinearity to the problem and the study ofit needs more elaborate technique than the
problems on fixed domain.

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Masashi ISHIDA
Differential Geometry

My research interest is in geometry,particularly, interaction between topology anddifferential geometry. For instance, I am
studying the nonexistence problems ofEinstein metrics and Ricci flow solutions on
4-manifolds by using the Seiberg-Witten
equations. I am also interested in thegeometry of the Yamabe invariant. The
computation of the Yamabe invariant for agiven manifold is a difficult problem ingeneral. By using the Seiberg-Witten
equations, I determined the exact value of the
Yamabe invariant for a large class of 4manifolds which includes complex surfaces
as special cases. Furthermore, I am alsointerested in both the Ricci flow in higher
dimension and some generalized versions of
the Ricci flow like the Ricci Yang-Mills flow.

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The Ricci flow was first introduced by R.
Hamilton in 1981 and used as the main tool in
G. Perelman's solution of the Poincare
conjecture in 2002. Perelman introduced
many new and remarkable ideas to prove theconjecture. The theory developed by
Hamilton and Perelman is now called the
Hamilton-Perelman theory. One of my recent
interests is to investigate geometric analyticalproperties of the generalized versions of the
Ricci flow from the Hamilton-Perelman
theoretical point of view.

Masataka IWAI
Complex Geometry, Algebraic Geometry, Several Complex Variables

My main research object is a projective
manifold, which is a submanifold of thecomplex projective space. Especially, I study the
structure of projective manifolds whose tangent
bundles or cotangent bundles have semipositive
curvature. I have researched the structure of
projective manifolds whose tangent bundles are
semipositive by using the theory of singular
Hermitian metrics. Finally, I established thestructure theorem of semipositive foliations.
Now, I'm studying the structure of projective
manifolds whose cotangent bundles are
semipositive.
The interesting part of studying projective
manifolds is that we can use any techniques,
such as complex geometry, algebraic geometry,several complex variables, and so on. Because

projective manifolds are complex manifolds bydefinition, we can use the techniques of complex
geometry and differential geometry. On theother hand, since projective manifolds can be
represented by a zero set of polynomials by
Chow's theorem, we can use the techniques ofalgebraic geometry and birational geometry.
Furthermore, projective manifolds are built froma collection of unit balls in the complex Euclid
space, we can use the techniques of several
complex variables. In this way, I study
projective manifolds using methods in a wide
range of fields.

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Seiichi KAMADA
Topology

My research area is topology (geometric
topology-- knot theory, low dimensional
topology). I have been working on surfacelinks in 4-space, including 2-knots and 2links, 2-dimensional braids, braided surfaces,and 4-manifolds.
Surface-links are closed surfaces smoothly
embedded in the 4-space, and they are
considered equivalent if they are ambiently
isotopic. It is a quite difficult problem todecide whether two given surface-links areequivalent or not. Invariants are often used toshow that two surface-links are notequivalent. However, few invariants are
known for surface-links. I am interested instudying surface-links and their invariants
using "quandles" and graphics so-called"charts".

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A quandle is an algebra with a binary
operation which satisfies three axioms
corresponding to Reidemeister moves in knot
theory. A homology theory of quandles wasestablished and we can use it for constructionof invariants for surface-links. 2-dimensional
braids, or surface braids, are a generalizationof classical braids. They are in one-to-one
correspondence to oriented surface-links in 4space modulo certain basics moves. As amethod of describing 2-dimensional braids, achart description was introduced. Thismethod has been extended in a general theoryof charts. For example, charts can be used forstudies of Lefschets fibrations of 4-manifolds.

Hisashi KASUYA
Geometry

Until now, I have tried to extend thegeometry of nilpotent groups to the geometryof solvable groups. More precisely, I havestudiedthecohomology
theoryofhomogeneous spaces of solvable Lie groups
and complex geometry of non-Kähler
manifolds. It seems that the gap between
nilpotent groups and solvable groups is small.
But this gap contains a potential forgeometry. By the growing out of leftinvariance and non-triviality of local systemcohomology, I succeeded in giving a great
surprise.
Recently, I am interested in the geometry
which relates to reductive or semi-simple
groups in contrast to nilpotent or solvablegroups. In particular, I study non-abelian

Hodge theory, variations of Hodge structures,
lattices in semi-simple Lie groups and locallyhomogeneous spaces.

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Soichiro KATAYAMA
Nonlinear Partial Differential Equations

My research interest is in nonlinear partial
differential equations. To be more specific, I
am working on the initial value problem for
nonlinear wave equations (in a narrow sense),
and also for partial differential equations
describing the nonlinear wave propagation ina wider sense, such as Klein-Gordon
equations and Schroedinger equations.
The initial value problem is to find asolution to a given partial differential
equation with a given state at the initial time(a given initial value). However, in general, it
is almost impossible to give explicit
expression of solutions to nonlinear
equations. Therefore, in the mathematicaltheory, it is important to investigate theexistence of solutions and also their behavior

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when they exist.
If we consider the initial value problem forthe equations mentioned above and if theinitial value is sufficiently small, theexistence of solutions up to arbitrary time (theexistence of global solutions) is mainly
determined by the power of the nonlinearity.
Especially, when the nonlinearity has thecritical power, the existence and nonexistence of global solutions depend also on
the detailed structure of the nonlinear terms. I
am interested in this kind of critical case, andstudying sufficient conditions for the
existence of global solutions and their
asymptotic behavior.

Kazunori KIKUCHI
Differential Topology

I have been studying topology of smooth fourdimensional manifolds, in particular interested inhomology
genera,representationsofdiffeomorphism groups to intersection forms, andbranched coverings. Let me give a simple
explanation of what interests me the most, or
homology genera. The homology genus of a smooth
four-dimensional manifold M is a map associating toeach two-dimensional integral homology class [x] of
M the minimal genus g of smooth surfaces in M that
represent [x]. For simplicity, reducing thedimensions of M and [x] to the halves of them
respectively, consider as a two-dimensional
manifold the surface of a doughnut, or torus T, and a
one-dimensional integral homology class [y] of T.
Draw a meridian and a longitude on T as on theterrestrial sphere, and let [m] and [l] denote thehomology classes of T represented by the meridian
and the longitude respectively. It turns out that [y] =a[m] + b[l] for some integers a and b, and that [y] isrepresented by a circle immersed on T with only
double points. Naturally interesting then is thefollowing question: what is the minimal number n of
the double points of such immersions representing[y]? Easy experiments would tell you that, forexample, n = 0 when (a,b) = (1,0) or (0,1) and n = 1
when (a,b) = (2,0) or (0,2). In fact, it is proved with
topological methods that n = d−1, where d is thegreatest common divisor of a and b. It is the minimal
number n for T and [y] that corresponds to theminimal genus g for M and [x]. The study on theminimal genus g does not seem to proceed with only
topological methods; it sometimes requires methods
from differential geometry, in particular methods
with gauge theory from physics; though moredifficult, it is more interesting to me. I have beentackling the problem on the minimal genus g withsuch a topological way of thinking as to see things
as if they were visible even though invisible.

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Kohei KIKUTA
Algebraic Geometry, Group Theory

I'm interested in autoequivalence groups ofderived categories of algebraic varieties,
especially K3 surfaces. The objects of derived
categories are complexes of coherent sheaves
on algebraic varieties, and the morphisms are
given by using morphisms between
complexes. Derived categories contain muchinformation of algebraic varieties. Autoequivalence groups are just like automorphism groups of algebraic varieties. The
pull-back of automorphisms gives the naturalembedding of auto-morphism groups into
auto-equivalence groups. Autoequivalencegroups of derived categories are interesting
objects in also group-theory.
To understand autoequivalence groups, it is
not enough to study only the action on the

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cohomology groups. Then, via homologicalmirror symmetry originating the string theory
in physics, we compare autoequivalencegroups and (symplectic) mapping classgroups of symplectic manifolds. Rich
research history of mapping class groups
gives rich ideas to study autoequivalencegroups. Actually, motivating the analogue
with mapping class groups, I have been
studying autoequivalence groups from grouptheoretic/ metric geometric/ dynamical
viewpoints. But there's still a lot to work on
this topic. My main project is to study
autoequivalence groups of K3 surfaces by any
method in mathematics.

Erika KUNO
Topology

My research theme is mainly mapping classgroups of surfaces (including non-orientablesurfaces) and 3-dimensional handlebodies. In
particular, I am interested in exploring themfrom the viewpoints of geometric grouptheory. Geometric group theory is a new fieldamong a lot of areas of mathematics and it isprogressing significantly. One of the mostimportant problems in geometric grouptheory is classifying finitely generated groups
by “quasi-isometries”. Two finitely generated
groups are quasi-isometric if roughly
speaking, their word metrics are the same up
to linear functions. An interesting part of thegeometric group theory is that the propertiesof the infinite groups are revealed one by one
by measuring with a coarse scale of quasi-

isometries, but not isometries. Currently,
groups which are quasi-isometric to mappingclass groups have hardly been found. Then
what I am wondering is the question "Whichgroups are quasi-isometric to the mappingclass group?". Based on this big theme, I
would like to elucidate properties of mappingclass groups, and deepen their understanding.

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Yoshihiko MATSUMOTO
Differential Geometry, Several Complex Variables

Working on differential geometry, partly with some flavor offunction theory of several complex variables. I’ve been mainly
studying geometry of “asymptotically complex hyperbolic
spaces,” with emphasis on a partial differential equation called
Einstein’s equation on them. While being a generalization ofgeometry of bounded strictly pseudoconvex domains infunction theory of several complex variables, it’s beyond thescope of the field of complex geometry. Based on thisexperience, I’m now aiming toward some more general theory
that applies to other “spaces that converge to ones with muchsymmetry,” which are called “asymptotically symmetricspaces.”Asymptotically hyperbolic spaces, which are the most basic
examples of asymptotically symmetric spaces, are lacking
symmetries such as the homogeneity or the isotropicity of thegenuine hyperbolic spaces in the strict sense. However, as apoint in the space moves more away from a fixed one, its
neighborhood looks more like an open set of the hyperbolic
space. Recall the “natural” conformal structure on the sphere at
infinity of the hyperbolic space—the boundary at infinity of an

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asymptotically hyperbolic space is equipped with a conformal
structure in the same way.
In the case of asymptotically complex hyperbolic spaces,
whose model has little less isotropicity than that of thehyperbolic space, the associated geometric structure on theboundary at infinity is the CR (Cauchy–Riemann) structure.
The fundamental question of geometry of asymptoticallysymmetric spaces is the following: what property of the spacereflects the geometric structure at infinity and the topology ofthe space, and in what way? This includes the question whether
or not there is a space with some certain property under a given
condition on the structure at infinity and the topology. The
existence problem of Einstein metrics is a typical example.
Although asymptotically hyperbolic spaces have been studied
for several decades, many fundamental questions remain
unsolved. And, if we think of understandings from the general
viewpoint of geometry of asymptotically symmetric spaces, ourfield still seems to be kind of a wilderness. I’d cultivate it bygoing back and forth between analyses on special cases andabstract considerations.

Haruya MIZUTANIPartial Differential Equations

The Schrödinger equation is thefundamental equation of physics fordescribing quantum mechanical behavior.
I am working on the mathematical theoryof the Schrödinger equation and myresearch interest includes scattering
theory, semiclassical analysis, spectral
theory, geometric microlocal analysis andso on. My current research has focused on
various estimates such as decay orStrichartz inequalities, which describe
dispersive or smoothing properties ofsolutions and are fundamental forstudying linear and nonlinear dispersive
equations. In particular, I am interested inunderstanding quantitatively the influenceof the geometry of associated classicalmechanics on the behavior of quantum
mechanics, via such inequalities.

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Takehiko MORITA
Ergodic Theory, Probability Theory, Dynamical System

I specialize in ergodic theory. To be more
precise, I am studying statistical behavior ofdynamical systems via thermodynamic
formalism and its applications.
Ergodic theory is a branch of mathematics
that studies dynamical systems withmeasurable structure and related problems. Its
origins can be found in the work ofBoltzmann in the 1880s which is concernedwith the so called Ergodic Hypothesis.
Roughly speaking the hypothesis wasintroduced in order to guarantee that thesystem considered is ergodic i.e. the space
averages and the long time averages of thephysical observables coincide. Unfortunately,
it turns out that dynamical systems are notalways ergodic in general. Because of such a

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background, the ergodic problem (= theproblem to determine a given dynamicalsystem is ergodic or not) has been one of theimportant subjects since the theory came into
existence. In nowadays ergodic theory has
grown to be a huge branch and has
applications not only to statistical mechanics,
probability, and dynamical systems but also
to number theory, differential geometry,
functional analysis, and so on.

Tomonori MORIYAMA
Number Theory

I am interested in automorphic forms ofseveral variables. A classical automorphic(modular) form of one variable is aholomorphic function on the upper half plane
having certain symmetry. Such functions
appear in various branches of mathematics,
say notably number theory, and have been
investigated by many mathematicians.
There is a family of manifolds called
Riemannian symmetric spaces, which is ahigher-dimensional generalization of theupper half plane. The set of isometries of a
Riemannian symmetric space forms a Liegroup G. Roughly speaking, an automorphic
form of several variables is a function on a
Riemannian symmetric space satisfying therelative
invariance
under
an

"arithmetic“ subgroup of G and certain
differential equations arising from the Liegroup G. Studies on automorphic forms ofseveral variables started from C. L. Siegel's
works in 1930s and have been developed
through interaction with mathematics of theday.
Currently I am working on two themes: (i)the zeta functions attached to automorphic
forms and (ii) explicit constructions ofautomorphic
forms,
byemployingrepresentation theory of reductive groups
over local fields. One of the joy in studying
this area is to discover a surprisingly simple
structure among seemingly complicatedobjects.

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Shohei NAKAMURA
Harmonic Analysis, Real Analysis

I am currently working in Harmonic analysis on
Euclidean spaces and real analysis. In particular,
my research purpose is to understand the“oscillation” phenomena quantitatively via theinequalities. Let us think about a sequence a_n =(-1)^n for instance. If one sums up the sequence
up to n=1000, then the result is 1 as +1 and -1
would be cancelled out. However, if one ignoresthe cancellation (oscillation) and sums up thesequence |a_n|, then the result is 1000 which
seems to have a big difference than before.
Although this is the simplest example ofoscillation phenomena, the problem would bemuch more involved once one has the oscillationin more complicated way. For example, one has
the oscillation e^{-ixy}, which is the linear and so
relatively simpler oscillation, in the definition of
the Fourier transform. However, in the definitionof the Schrodinger propagator, one has e^{-i (xy +t y^2)}, which is the quadratic oscillation. Hence
its treatment would be no longer trivial and oneneeds to invent a clever way to exploit thecomplicated oscillation.
Such oscillation phenomena appear in several
mathematical fields. For instance, I have alreadymentioned before, in the theory of partial
differential equations as well as in Fourier
analysis. Not only in analysis, but one can alsofind such phenomena in number theory; a problem
to count a number of integer solutions of certain
Diophantine equation can be boiled down to thequestion how to capture the “oscillation”. Amongthem, I am in particular interested in how tocapture the oscillation of the Fourier transform ona hypersurface. Related to this problem, I am alsointerested in geometric inequalities with the sharp
constant, like Young’s convolution inequality.

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Takahiro OBA
Topology

My research interests lie in the topology ofmanifolds. In particular, I am interested in thatof contact and symplectic manifolds. One of
the main problems in this area is how muchthese geometric structures impose constraints
on the topology of those manifolds. One canaddress this problem by various methods.
Among them, I have mainly used fibration-likestructures: Lefschetz fibrations and open bookdecompositions. One can extract information
about the topology of contact and symplectic
manifolds from fibration-like structures, andvice versa.
I have studied low-dimensional, namely 3-/4dimensional contact and symplectic manifolds.
I have recently started trying to understandhigher dimensions. Little is known about thiscase, because of lack of knowledge on

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mapping class groups of higher-dimensional
manifolds, so this is challenging. With various
techniques such as holomorphic curve
techniques, now I am diving into the higherdimensional world.

Hiroyuki OGAWA
Algebraic Number Theory

I have an interest in periodic objects.
Expanding rational numbers into decimal
numbers is delightful. The decimal number
expansion becomes the repeat of a sequenceof some integers. I have an appetite forcontinued fraction expansions, never get tired
to calculate it, and want to find continued
fractions with sufficiently long period. It is on
the way to Gauss' class number one
conjecture. Recently, I am studying iterationof rational functions. For a rational function
g(x) with rational coefficients, a complex
number z with g(g(...g(z)...))=z is called aperiodic point on g(x) and is an algebraic
number. I expect that number theoreticalproperties which such an algebraic number z
has is described by the rational function g(x).

This does not seem to work out anytime, but
one can find many rational functions g(x) that
describe the Galois group, the class number,the class group, and so on of a periodic pointof g(x). I think that this should be surely
useful, and calculate like these every day.

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Koji OHNO
Algebraic Geometry

When I was a student, I thought I knew number
theory, geometry, but algebraic geometry was
unfamiliar for me. One may say algebra andgeometry are different fields, but you know thetheory of quadratic curves and are aware ofefficiency of algebraic methods for solving
geometric problems. The field called algebraic
geometry lies on such a line. When I was studying
the theory of quadratic curves, I wondered, ''Why dothey only treat special equations like quadratics?
There are many other equations. But how can they betreated?". When I discovered the answer might lie on
this field, I decided to enter this field. The easiest
non-trivial equation has the form such as "the second
power of y = an equation of x of degree three",which defines the so called "an elliptic curve". The
theory of elliptic curve was one of the greatest
achievements of nineteenth century and keeps
developing today. Recently, the famous Fermat's

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conjecture has been solved using this theory. The
theory of quadratic and elliptic curves involve only
two variables x, y. It is natural to think of theequations with many variables. In fact the algebraic
geometers are expanding the theory, curves tosurfaces and higher dimensional cases these days.
Two dimensional version of elliptic curves are called
K3 surfaces, which can be treated only using thetheory of linear algebra(!) thanks to the Torelli's
theorem. These days, the 3-dimensional versions,which is called Calabi-Yau threefolds are fascinating
for algebraic geometers like me. Somehow
theoretical physicists are also interested in this field.
To study Calabi-Yaus by specializing these to ones
with a fiber structure (on which field, I'm now
working) might be one method, but I have been
thinking that a new theory is needed. These days,
many intriguing new theories have appeared and one
may find more!

Shin‐ichi OHTA
Differential Geometry, Geometric Analysis

My research subject is geometry,especially differential geometry
andgeometric analysis related to analysis andprobability theory. A keyword of myresearch is “curvature” which represents
how the space is curved. As seen in the
difference between the sums of interior
angles of triangles in a plane and a sphere,
the behavior of the curvature influences
various properties of the space (the shapesof triangles, the volume growth ofconcentric balls, how heat propagates, thebehavior of entropy, etc.). This powerfuland versatile conception has been applied to
Riemannian manifolds, metric spaces,
Finsler manifolds, Banach spaces, as well as
discrete objects such as graphs.

Graduate School of Science
2023‐2024

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Mamoru OKAMOTO
Nonlinear Partial Differential Equations

My research interest is in nonlinear partial
differential equations. Especially, I am working on
the Cauchy problem for nonlinear dispersive
equations, such as nonlinear Schrödinger andnonlinear wave equations. The Cauchy problem isa problem to find a solution to a differential
equation satisfying what are known as initialconditions (initial data) at a certain (initial) time.
For nonlinear problems, it is usually not possible
to obtain explicit solutions, so we need to show an
existence result of a solution. It then may happen
that the solution changes significantly when theinitial data change small. However, thisphenomenon causes some difficulties on analysisof physical models and differential equations.
Hence, it is often required the well-posedness of
the Cauchy problem, which ensures that thesolution continuously depends on initial data. Oneof the distinguishing features of linear dispersive

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equations is that the speed of the spread ofsolutions depends on its frequency. Therefore,
some kind of smoothing effect can be obtained byincorporating the oscillation caused by dispersion
due to time evolution into the analysis.
Furthermore, by grasping what kind of oscillationis generated from each nonlinear term, we can use
the smoothing effect in the analysis of nonlinear
dispersive equations.
I am studying the well-posedness of the Cauchy
problem for dispersive equations by using thesmoothing effect due to such a dispersive nature.
Recently, I am also interested in asymptotic
behavior of solutions to the Cauchy problem andstochastic dispersive equations.

Shinnosuke OKAWA
Algebraic Geometry

Geometric objects which are described as thecollection of solutions of algebraic equation(s), suchas ellipses, parabolas, and hyperbolas, are called
algebraic varieties. These are the subject of study inthe field of algebraic geometry, and I have beenworking on various problems in this area.
In the early days, I was studying topics about
Geometric Invariant theory (GIT) and birational
geometry. GIT is a theory about quotients ofalgebraic varieties by algebraic group actions, and inbirational
geometrycertain"transformation(modification)" of algebraic varieties is studied. The
definition of a GIT quotient depends on a choice of aparameter, which is called a stability condition, andone obtains different quotients by changing thestability conditions. Typically these quotients are all
birational to each other, and in good situations it turns
out that the birational geometry of the quotient
variety is complete described in this way.
I have proved several properties of this class ofvarieties. Recently I am mainly investigating
algebraic varieties from categorical points of view.
One of my interests is the "irreducible
decomposition" of the derived category of coherent
sheaves. This in fact is motivated by birational
geometry, and important techniques of birational
geometry, e.g. the canonical bundle formula, areused. I am also studying non-commutative
deformations of algebraic varieties and their moduli
spaces. Derived categories again plays a central role,
but other interesting topics such as GIT, geometry of
elliptic curves, and birational transformation of noncommutative algebraic varieties also show up.
Through the study, I found an interesting andunexpected relationship with an old invariant theory
which goes back to the end of the 19th century. I am
also trying to understand to what extent the derived
category of coherent sheaves keeps the geometric
information of the original variety.

Graduate School of Science
2023‐2024

Department of Mathematics

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Kazuto OTA
Number Theory

My research area is number theory. Moreprecisely, I am interested in mysterious
relations
betweenL-functions
andarithmetic invariants which are attached toalgebraic varieties defined over therationals. Roughly speaking, L-functions are
regarded as a vast generalization of the
Riemann zeta function, and arithmetic
invariants here are rational solutions ofalgebraic equations, Selmer groups etc.
Although there are conjectural formula forgeneral algebraic varieties of several forms
nowadays, almost all of them are still
unsolved. For example, the BSD conjecture
is such a conjecture for elliptic curves(algebraic varieties of dimension one), andDepartment
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it is one of The Millennium Prize Problemsby the Clay Mathematics Institute. By thisexample, we may see difficulty andimportance of such conjectures connecting
L-functions and arithmetic invariants. I
would like to contribute to them. So far, I
have studied relations between L-functions
and Selmer groups of elliptic curves andelliptic modular forms by p-adic methods,
where p is a rational prime number. I will
next study to understand higher-dimensional
objects as well.

Toru SERA
Ergodic Theory, Probability Theory, Dynamical Systems

My research interest is in ergodic
transformations with infinite invariant
measures. Typical examples are intermittent
maps, i.e., non-uniformly expanding maps
with indifferent fixed points. Intermittent
maps have been studied as toy models ofintermittent phenomena and are importantresearch subjects in mathematics andstatistical physics.
In the case of probability-preserving
ergodic transformations, Birkhoff's ergodic
theorem states that the long-time average ofan appropriate observable converges to thespace average. In the case of infinite ergodic
transformations, Birkhoff's ergodic theorem
is not very useful. Nevertheless, many
researchershavestudiedvariousprobabilistic limit theorems. For example,
given an appropriate initial distribution for
an intermittent map, the long-time averageof a suitable observable converges to a
generalized arcsine distribution.
My research aim is to deepen and develop
probabilistic limit theorems for intermittent
maps. For this purpose, I have been focusing
on the probabilistic similarities between
one-dimensional diffusion processes andintermittent maps. Although they are
different from each other in origin, we canuse the analysis methods of the former tostudy the latter under some modifications.

Graduate School of Science
2023‐2024

Department of Mathematics

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Yuichi SHIOZAWA
Probability Theory

My research area is probability theory. In
particular, I am working on the sample pathanalysis for symmetric Markov processes
generated by Dirichlet forms. Dirichlet form
is defined as a closed Markovian bilinear
form on the space of square integrable
functions. The theory of Dirichlet forms
plays important roles in order to construct
and analyze symmetric Markov processes.
I am interested in the relation between theanalytic information on Dirichlet forms andthe sample path properties of symmetricMarkov processes. I am also interested in
the global properties of branching Markov
processes, which are a mathematical modelfor the population growth of particles bybranching.

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Atsushi TAKAHASHI
Complex Geometry, Algebra, Mathematical Physics

My current interests are mathematical aspects ofthe superstring theory, in particular, algebraic
geometry related to the mirror symmetry.
More precisely, I am studying homologicalalgebras and moduli problems for categories of"D-branes" that extend derived categories ofcoherent sheaves on algebraic varieties.
Indeed, I am trying to construct Kyoji Saito'sprimitive forms and their associated Frobenius
structures from triangulated categories defined via
matrix factorizations attached to weighted
homogeneous polynomials.
For example, I proved that the triangulated
category for a polynomial of type ADE isequivalent to the derived category of finitely
generated modules over the path algebra of the
Dynkin quiver of the same type.
Now, I extend this result to the case when thepolynomial corresponds to one of Arnold's 14

exceptional singularities and then showed the"mirrorsymmetry"
betweenweightedhomogeneous singularities and finite dimensional
algebras, where a natural interpretation of the"Arnold's strange duality" is given.

Graduate School of Science
2023‐2024

Department of Mathematics

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Shuichiro TAKEDA
Number Theory, Representation Theory

My research area is called the Langlands
program. The Langlands program is at thecrossroad of several important branches ofmodern mathematics such as number theory,
representation theory and harmonic analysis.
Because of this, it is considered as one of themost difficult areas of mathematics, but at thesame time it has been producing various
interesting research outcomes.

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The major object of study in the Langlands
programiscalledautomorphic
representations and their L-functions.
Automorphic
representationsarerepresentations of certain matrix groups
called reductive groups, and considered as avast generalization of classical modular
forms. An automorphic representation isfurther decomposed into local representations
such as representations of p-adic groups. I
have been studying these representations by
using
the
methodcalledtheta
correspondence, which allows one toconstruct a representation of one reductive
group out of a representation of another. Via
this method, I have obtained numerous
interesting results in the theory of
automorphic representations.

Naohito TOMITAReal Analysis

My research field is Fourier analysis, and I
am particularly interested in the theory offunction spaces. Fourier series wereintroduced by J. Fourier(1768-1830) for the
purpose of solving the heat equation. Fourier
considered as follows: "Trigonometric series
can represent arbitrary periodic functions".
However, in general, this is not true. Then,
we have the following problem: "When canwe write a periodic function as an infinite (orfinite) sum of sine and cosine functions?".
Lebesgue space which is one of function
spaces plays an important role in this classicalproblem. Here Lebesgue space consists offunctions whose p-th powers are integrable.
In this way, function spaces are useful forvarious mathematical problems. As anotherexample, modulation spaces were recently
applied to pseudodifferential operators which
are important tool for partial differential
equations, and my purpose is to clarify theirrelation.

Graduate School of Science
2023‐2024

Department of Mathematics

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Motoo UCHIDA
Algebraic Analysis, Microlocal Analysis

My research field is algebraic analysis andmicro-local analysis of partial differential
equations. The view point of micro-local
analysis (with cohomology) is a new important
point of view in analysis introduced by Mikio
Sato in the early 1970s. Thinking from a
micro-local point of view helps us to wellunderstand a number of mathematicalphenomena (at least for PDE) and to find asimple hidden principle behind them. Even forsome classical facts (scattered as well known
results) we can sometimes find a new unified
way of understanding from a micro-local or
algebro-analytic viewpoint.

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Takao WATANABE
Algebraic Number Theory

My current interest is the Geometry of Numbers. The
Geometry of Numbers was founded by Hermann
Minkowski in the beginning of the 20th century.
Minkowski proved a famous theorem known as"Minkowski's convex body theorem", which assertsthat "there exists a non-zero integer point in V if V isan o-symmetric convex body in the n-dimensional
Euclidean space whose volume is greater than 2^n".
When V is an ellipsoid, this theorem is refined asfollows. Let A be a non-singular 3 by 3 real matrix andK(c) the ellipsoid consisting of points x such that the
inner product (Ax, Ax) is less than or equal to c > 0.
For i = 1,2,3, we define the constant c_i as theminimum of c > 0 such that K(c) contains i linearly
independent integer points. Then c_1, c_2, c_3
satisfies the inequality c_1c_2c_3 <= 2|det A|^2. Thisis called "Minkowski's second theorem". A similarinequality holds for any n-dimensional ellipsoid.
Namely, if A is a non-singular n by n real matrix andK(c) is the n-dimensional ellipsoid defined by (Ax,Ax) <= c, we can define c_i for i = 1,2, ..., n as theminimum of c > 0 such that K(c) contains i linearly
independent integer points. Then the inequality
c_1c_2...c_n <= h(n)|det A|^2 holds for any A. The
optimal upper bound h(n) does not depend on A, andis called Hermite's constant. We know h(2) = 4/3, h(3)= 2, h(4)= 4, ..., h(8) = 256, but h(n) for a general n is
not known. A recent major topic of this research areais the determination of h(24). In 2003, Henry Cohn
and Abhinav Kumar proved that h(24) = 4^24.
(Incidentally, h(3) was essentially determined by
Gauss in 1831, and h(8) was determined by Blichfeldt
in 1953. If you would determine h(9), then your namewould be recorded in treatises on the Geometry of
Numbers.) Now I study (an analogue of) the Geometryof Numbers on algebraic homogeneous spaces. One of
my results is a generalization of Minkowski's second
theorem to a Severi-Berauer variety. In addition, I am
interested in the reduction theory of arithmetic
subgroups, automorphic forms, the algebraic theory ofquadratic forms and Diophantine approximation.

Graduate School of Science
2023‐2024

Department of Mathematics

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Katsutoshi YAMANOI
Complex Analysis, Complex Geometry

My research interest is Complex geometry andComplex analysis, both from the view point ofNevanlinna theory. In the geometric side, I am
interested in the conjectural second main
theorem in the higher dimensional Nevanlinna
theory for entire holomorphic curves into
projective manifolds. Also I am interested in
the behavior of Kobayashi pseudo-distance of
projective manifolds of general type. These
problems are related to an algebraic geometricproblem of bounding the canonical degree ofalgebraic curves in projective manifolds ofgeneral type by the geometric genus of thecurves. In the analytic side, I am interested inclassical problems of value distribution theory
for meromorphic functions in the complex
plane.

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Kouji YANO
Probability Theory, Ergodic Theory

Probability theory contains several limit
theorems such as the law of large numbers, thelaw of iterated logarithm, the central limit
theorem, the Poisson law of small numbers, andthe arcsine law. They are mathematical theories
about limit properties in long time forstochastic processes which change randomly astime passes.
I have studied extensions of the arcsine law to
one-dimensional diffusion processes and to
random dynamical systems with fixed points
which are indifferent-in-mean. The arcsine law
describes the rate where a stochastic process is
attracted to positive and negative infinity, and it
also describes the rate where a dynamicalsystem with two indifferent fixed points is
attracted to them. It is an interesting problem

where the theory of stochastic processes and theinfinite ergodic theory intersect.
I have also studied the penalisation problem, alimit theorem where the stochastic process
considered is conditioned to prevent from acertain behavior for a long time. Compared tothe original stochastic process, the limit process
is absolutely continuous in finite time but issingular in infinite time, and the RadonNikodym density is given by a martingale
converging to zero. The way of taking the limit
here is called a clock. If we adopt instead of aconstant clock a random clock such as an
exponential clock and a hitting time clock, we
encounter various martingales. Looking at astochastic process from different viewpoints,
we came to find its interesting structures.

Graduate School of Science
2023‐2024

Department of Mathematics

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Takehiko YASUDA
Algebraic Geometry, Singularity Theory

My main research object is singularities ofalgebraic varieties. An algebraic variety is a
"figure" formed by solutions of algebraic
equations. Such a figure often has points
where the figure is sharp-pointed or intersects
itself. Singularities make the study of analgebraic variety difficult. However since
they often appear under various constructions,
it is important to study them. Also
singularities are interesting research object
themselves. More specifically, I am interested
in resolution of singularities, the birationalgeometric aspect of singularities, the McKay
correspondence.

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Although these are classical research areas,changing a viewpoint or the setting of aproblem, one can sometimes find new
phenomenon. Such a discovery is the greatest
pleasure in my mathematical research. To
pursue research, I use various tools likemotivic integration, Frobenius maps, modulitheoretic blowups, non-commutative rings,
and sometimes make ones by myself.
Recently I am fascinated by mysterious
behaviors of singularities in positive
characteristic (a world where summing up
several 1's gives 0.)

Masahiko YOSHINAGA
Algebra, Topology, Combinatorics

The central object of my research is thehyperplanearrangement.
Ahyperplanearrangement is a collection of (n-1)-dimensional
subspaces in a n-dimensional space. Such objects
appear many area of mathematics. Recently, I am
mainly working in the following topics.
(1) The freeness of the module of logarithmic
vector fields and the construction of the basis.
(2) Topology of the Milnor fiber and covering
spaces.
(3) Lattice points counting and the characteristic
quasi-polynomials.
(1) has connections with algebraic geometry andrepresentation theory, and recently, is influenced
by the study of quantum integrable systems. In (2),the icosidodecahedral arrangement (Figure) plays
crucial role in recent studies. (3) has very richconnections with many other topics such as thetheory of polytopes, arrangements of tori, andgeneralizations of the notion of Tutte polynomial
for graphs and matroids.
I am also interested in the notion of themagnitude of metric spaces introduced bycategorists, and more generally, categorification of
combinatorial phenomena. My interests also
include the notion of “periods” that are real
numbers with integral expressions.

Graduate School of Science
2023‐2024

Department of Mathematics

SEMINARS and COLLOQUIA●ALGEBRA
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ofMathematics

Number Theory
SeminarNumber theory seminar at Osaka University
is a seminar for faculty members andgraduate students of Osaka University or
researchers
studying
nearbyOsaka
University. The subject of the seminarcovers wide topics concerning Number
theory, especially, algebraic number theory
and analytic number theory, modular forms,
arithmetic geometry, representation theory
and algebraic combinatorics. In this seminar,
we have reports of new results on these
topics and we exchange ideas and technics
of our research.

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Algebraic Geometry
Seminar

The seminar is held two or three times amonth and each time one speaker gives a talkof 90 minutes. After a talk, we have time forquestions and discussion. The purpose of theseminar is to learn important results by active
researchers in Algebraic Geometry andrelated fields, providing new perspectives on
the areas through lectures and discussions.
We also have survey lectures by experts forgraduate students and young researchers. We
have guest speakers not only from domestic
universities but also from foreign countries,
reflecting various aspects of the research area.

●GEOMETRY
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Geometry Seminar

This seminar on Mondays is intended fortalks that will be of interest to a wide range
of geometers. Topics discussed include
Riemannian, complex, and symplectic
geometry; PDEs on manifolds; mathematicalphysics.

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Topology Seminar

This seminar focuses on various aspects of
low-dimensional topology. Our major topics
are 2-, 3-, and 4-dimensional manifold
theory, knot theory, geometric group theory,
hyperbolic geometry, Riemann surfaces, andtransformation groups. In addition to ourgroup members from the math department,
associated faculty members from otherdepartments and their graduate students arealso regular participants.

Graduate School of Science
2023‐2024

Department of Mathematics●ANALYSIS
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Seminar ofDifferential Equations

Our seminar is held on every Friday from 15:30
to 17:00. One of the features of the seminar is tocover a wide variety of topics on Qualitative
Analysis of Differential Equations. In fact, we areinterested in ordinary differential equations,partial differential equations, linear differential
equations, nonlinear differential equations and soon. Lecturers are invited from not only domestic
universities but also foreign countries and present
us their original results or survey of recent
development of their fields. Furthermore, thisseminar provides opportunities to give a talk forour colleagues and Ph.D. students majoring indifferential equations. Moreover, we should
mention that we are pleased to have participantsfrom other universities located closed to ours. In
this way, we communicate with each other and try
to contribute to the progress of the theory ofdifferential equations.

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Seminar on Probability

Probability theory group, the graduate
school of science and the graduate school ofengineering science, organizes "Seminar on
Probability" on Tuesday evening. The topics
on this seminar are the following:
1. Probability theory, stochastic analysis andinfinite dimensional analysis, problems
arising from other areas of mathematics
such as real analysis, differential
equations and differential geometry.
2. Research fields related to Probability
theory, Ergodic theory, dynamical system,
stochastic control and mathematicalfinance.
We welcome visits and talks by many
researchers from other universities, domestic
and abroad.

●COLLOQUIA
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Dynamics andFractals Seminar

Researchers and students working on
various fields related to dynamical systems
and fractals attend this seminar. We meetonce a month for approximately 90 minutes.
Each talk on his/her research is followed bydiscussions among all participants.

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Mathematics
Colloquium

Colloquia take place on Monday afternoon
at 17:00 in Room E404. They are directed
toward a general mathematical audience. In
particular, one of the functions of these
Colloquia is to inform non-specialists andgraduate students about recent trends, ideas
and results in some area of mathematics, orclosely related fields.

Graduate School of Science
2023‐2024