Low dimensional topology and number theory IX
March 15 - 18, 2017
AiRIMaQ Seminar Room, Innovation Plaza, Momochihama, Fukuoka,
JAPAN
ProgramMarch 15 (Wednesday)
9:30 〜 10:30
Toshie Takata (Kyushu University)
The slope conjecture and periodic construction
10:50 〜 11:50
Shunsuke Tsuji (The University of Tokyo)
Construction of an invariant for integral homology spheres
14:00〜 15:00
Shinya Harada (The University of Tokyo)
Deformation varieties of hyperbolic two-bridge link complements andtheir zeta functions
15:20 〜 16:20
Megumi Takata (Kyushu University)
The inﬁnite base change lifting associated to an APF extension of amixed characteristic local ﬁeld
March 16 (Thursday)
9:30 〜 10:30
Sakie Suzuki (Kyoto University, RIMS)The universal quantum invariant and colored ideal triangulation
10:50 〜 11:50
Seidai Yasuda (Osaka University)
Ihara bracket for group schemes
14:00 〜 15:00
Nariya Kawazumi (The University of Tokyo)
The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebrain genus zero1

15:20 〜 16:20
Jinsung Park (Korea Institute for Advanced Study)
Reidemeister torsion, complex volume, and Zograf inﬁnite productMarch 17 (Friday)
9:30 〜 10:30
Yasushi Mizusawa (Nagoya Institute of Technology)
On pro-p link groups of number ﬁelds
10:50 〜 11:50
Jun Ueki (The University of Tokyo)bp-adic Mahler measure, entropy, and Z-covers
14:00 〜15:00
Eiko Kin (Osaka University)
Small asymptotic translation lengths of pseudo-Anosov maps on the
curve complex
15:20 〜 15:40
Junhyeung Kim (Kyushu University)
On the foliation cohomology groups and dynamical zeta functions forsurface bundles over S 1
15:50 〜 16:40
Masanori Morishita (Kyushu University)
Local symbols and the reciprocity law on foliated 3-manifolds
March 18 (Saturday)
9:30 〜 10:30
Takefumi Nosaka (Kyushu University)
Massey products of free groups and Milnor-Orr link invariants.
10:50 〜 11:50
Kazuo Habiro (Kyoto University, RIMS)The Kontsevich integral for bottom tangles in handlebodies
14:00 〜 15:00
Tetsuya Ito (Osaka University)
On a structure of Dehn surgery along knots and LMO invariant

2

Abstract
Kazuo Habiro (Kyoto University, RIMS)The Kontsevich integral for bottom tangles in handlebodies
Using the Kontsevich integral, we deﬁne a functor from the category B
of bottom tangles in handlebodies to a category A of chord diagrams.
This functor can be thought of as a (partial) reﬁnement of the LMO
functor on Lagrangian cobordisms. I also plan to explain the algebraic
structure of the category A. This is joint work with Gwenael Massuyeau.
Shinya Harada (The University of Tokyo)
Deformation varieties of hyperbolic two-bridge link complements andtheir zeta functions
After a brief survey on SL2 -character varieties and their zeta functionsof hyperbolic 3-manifolds, I will talk on a work in progress about the
deformation varieties attached to the canonical decompositions of certain hyperbolic two-bridge link complements and their zeta functions.
Tetsuya Ito (Osaka University)
On a structure of Dehn surgery along knots and LMO invariant
In this talk we use the LMO invariant to study a structure of Dehn
surgery along a knot in S 3 . Through a computation of the LMO invariant, we give various constraints for a knot to admit cosmetic surgery(Dehn surgery along the same knot with diﬀerent slopes, yielding thesame 3-manifold), or the Lens space surgeries.
Nariya Kawazumi (The University of Tokyo)
The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebrain genus zero
In view of results of Goldman and Turaev, the free vector space over thefree loops on an (connected) oriented surface has a natural Lie bialgebra
structure. The Goldman bracket has a formal description by using a special (or symplectic) expansion of the fundamental group of the surface.
It is natural to ask for a formal description of the Turaev cobracket.
This Lie bialgebra is closely related to the mapping class group of thesurface. In this talk we will show how to obtain a formal description of
the Goldman-Turaev Lie bialgebra for genus 0 using a solution of theKashiwara-Vergne problem. A similar result was recently obtained by
Massuyeau using the Kontsevich integral. If time permits, I would liketo discuss positive genus analogues of the result. This talk is based on3

a joint work by A. Alekseev, N. Kawazumi, Y. Kuno and F. Naef.
Junhyeung Kim (Kyushu University)
On the foliation cohomology groups and dynamical zeta functions forsurface bundles over S 1
C. Deninger initiated the cohomological study of the dynamical zetafunction for a manifold which is equipped with a 1-codimensional foliation and the transversal ﬂow. In this talk, we give concrete descriptionsof the foliation cohomology and the dynamical zeta function for thesimplest example, namely, a surface bundle over S 1 . We also discuss
some future problems.
Eiko Kin (Osaka University)
Small asymptotic translation lengths of pseudo-Anosov maps on the
curve complex
We consider the mapping class groups on the closed surface Sg of genus
g. We are interested in two invariants of pseudo-Anosov mapping classes.
The one is the entropy (the logarithm of the stretch factor of the pseudoAnosov map). The other is the asymptotic translation distance on thecomplex of curves. It is known that ﬁxing the genus g, both invariants
have minima. Furthermore the minimal entropies for Sg behaves like
1/g as g goes to inﬁnity. On the other hand, the minimal asymptotic
translation distance on the complex of curves for Sg behaves like 1/g 2
as g goes to inﬁnity. We describe a source of generating a sequence ofpseudo-Anosov maps on Sg whose both invariants are small, i.e, the twoinvariants of the sequence behave like 1/g and 1/g 2 respectively. Thisis a joint work with Hyunshik Shin (KAIST).
Yasushi Mizusawa (Nagoya Institute of Technology)
On pro-p link groups of number ﬁelds
As an analogue of a link group, we consider the Galois group of the
maximal pro-p-extension of a number ﬁeld with restricted ramiﬁcation
which is cyclotomically ramiﬁed at p, i.e, tamely ramiﬁed over the intermediate cyclotomic Zp -extension of the number ﬁeld. In some basic
cases, such a pro-p Galois group also has a Koch type presentation described by linking numbers and mod 2 Milnor numbers (Rédei symbols)of primes. Then the pro-2 Fox derivative yields a calculation of Iwasawa
polynomials analogous to Alexander polynomials.
Masanori Morishita (Kyushu University)4Local symbols and the reciprocity law on foliated 3-manifolds
The theory of local symbols (Hilbert symbols, tame symbols) is a beautiful subject in number theory and algebraic geometry, and plays an important role in class ﬁeld theory. In this talk, we introduce a local symbol on a foliated 3-manifold and show the reciprocity law. Our idea is toextend Deligne’s interpretation of a tame symbol on a Riemann surface,
by using Gomi-Terashima’s higher dimensional holonomy of a smooth
Deligne cocycle. Joint with Junhyeung Kim and Yuji Terashima.
Takefumi Nosaka (Kyushu University)
Massey products of free groups and Milnor-Orr link invariants.
TBA
Jinsung Park (KIAS)
Reidemeister torsion, complex volume, and Zograf inﬁnite productIn this talk, I will explain a formula which expresses the Reidemeister torsion in terms of complex volume and Zograf inﬁnite product for
closed hyperbolic 3-manifold. This can be understood as an analogueof the corresponding formula of Zograf and McIntyre-Takhtajan for the
regularized determinant of the hyperbolic Laplacian of compact Riemann surface.
Sakie Suzuki (Kyoto University, RIMS)The universal quantum invariant and colored ideal triangulation
The Drinfeld double of a ﬁnite dimensional Hopf algebra is a quasitriangular Hopf algebra with the canonical element as the universal
R-matrix, and one can obtain a ribbon Hopf algebra by adding theribbon element. The universal quantum invariant of framed links isconstructed using a ribbon Hopf algebra. In that construction, a copyof the universal R-matrix is attached to each crossing, and invarianceunder the Reidemeister III move is shown by he quantum Yang-Baxter
equation of the universal R-matrix. On the other hand, R. Kashaev
showed that the Heisenberg double of a ﬁnite dimensional Hopf algebra has the canonical element (the S-tensor) satisfying the pentagon
relation. In this talk we reconstruct the universal quantum invariant
using the Heisenberg double, and extend it to an invariant for coloredsingular triangulations of topological spaces, especially for colored ideal
triangulations of tangle complements. In this construction, a copy of
the S-tensor is attached to each tetrahedron, and invariance under thecolored Pachner (2; 3) moves is shown by the pentagon relation of the5S-tensor.
Megumi Takata (Kyushu University)
The inﬁnite base change lifting associated to an APF extension of amixed characteristic local ﬁeld
Langlands conjectured that there is a nice correspondence between automorphic forms/representations and Galois representations of number
ﬁelds. In the Langlands conjecture, the base change lifting is the counterpart on the automorphic side of the restriction functor on the Galois
side. For a ﬁnite cyclic extension of a number ﬁeld or a mixed characteristic local ﬁeld, the lifting was constructed by Langlands for GL(2)and by Arthur-Clozel for GL(n). In this talk, we give such a lifting fora totally ramiﬁed Zp -extension of a mixed characteristic local ﬁeld. By
Kazhdan’s theory of close ﬁelds, we can interpret this as an operation
which maps an automorphic representation of GL(n) over a local ﬁeldof mixed characteristic to that of positive characteristic.
Toshie Takata (Kyushu University)
The slope conjecture and periodic construction
The slope conjecture proposed by Garoufalidis asserts that the Jones
slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for some nonadequate, periodic knots. This is a joint work with Kimihiko Motegi.
Shunsuke Tsuji (The University of Tokyo)
Construction of an invariant for integral homology spheres via Kauﬀman bracket skein algebras
Using an explicit formula for the action of the Dehn twist along a simple
closed curve on the completed Kauﬀman bracket skein modules of thesurface, we introduce an embedding of the Torelli group into the completed skein algebra. This embedding and a Heegaard splitting enable
us to construct an invariant for an integral homology sphere which isan element of Q[[A + 1]]. This invariant induces a ﬁnite type invariantof order n + 1 which is an element of Q[[A + 1]]/((A + 1)n ).
Jun Ueki (The University of Tokyo)bp-adic Mahler measure, entropy, and Z-covers
My talk consists of two parts related to homology growth in Z-coversof 3-manifolds. Let p be a ﬁxed prime number.
(1) p-adic Mahler measure and Z-covers of links.

6An asymptotic formula of homology torsion growth with use of Mahler
measure is well-known, and its p-adic reﬁnement called the Iwasawa type
formula is studied by Morishita and others. We introduce a p-adic analogue of Mahler measure imitating the Shnirel’man integral, and provea p-adic analogue of the asymptotic formula. In addition, we give abalance formula among p-adic Mahler measure, Iwasawa mu-invariant,and Bowen’s p-adic entropy.
(2) Proﬁnite completions of knot groups determine Alexander polynomials.
It is interesting to ask what topological properties of knots are determined by the proﬁnite completions of their knot groups; in other words,
what the systems of ﬁnite quotients of knot groups know. By resultsof Hempel and Perelman, knot groups inject into their proﬁnite completion. Grothendieck conjectured that such ﬁnitely generated groups
would be determined by their proﬁnite completions. However, counterexamples were given by Bridson–Grunewald. We generalize results ofBoileau–Friedl and Bridson–Reid by proving that the proﬁnite completions of knot groups determine their Alexander polynomials. In thecourse of proof, we use Fried’s theorem on cyclic resultant related to
the Artin–Mazur zeta function of dynamical systems, and study modules over the completed group ring Ẑ[[tẐ ]]. Our method is suggested bya viewpoint of Arithmetic Topology.
Seidai Yasuda (Osaka University)
Ihara bracket for group schemes
TBA.

7

Low dimensional topology and number theory X
March 26 - 29, 2018
AiRIMaQ Seminar Rm, Innovation Plaza, Momochihama, Fukuoka, JAPAN

ProgramMarch 26 (Monday)
9:30 〜 10:30
Thang Le (Georgia Institute of Technology)
The skein algebra of surfaces and hyperbolic TQFT
10:50 〜 11:50
Tsukasa Ishibashi (The University of Tokyo)
Cluster Dehn twists in cluster modular groups
14:00〜 15:00
Makoto Sakuma (Hiroshima University)
On symmetries of knots
15:20 〜 16:20
Masanobu Kaneko (Kyushu University)
Genus character L-functions of quadratic orders and class numbers
March 27 (Tuesday)
9:30 〜 10:30
Hwajong Yoo (IBS, Center for Geometry and Physics, Postech)
Examples in arithmetic Chern-Simons theory
10:50 〜 11:50
Romyar Shariﬁ (University of California, Los Angeles)
Modular symbols and arithmetic
13:40 〜 14:40
Tomoki Mihara (Tokyo Institute of Technology)
Homotopy Theory for Metric Spaces
15:00 〜 16:00
Pavel Zalesskii (Universidade Brasilia)
The proﬁnite colmpetion of 3-manifold groups

8Banquet
March 28 (Wednesay)
9:30 〜 10:30
Takahiro Kitayama (The University of Tokyo)
Torsion polynomial functions and essential surfaces
10:50 〜 11:50
Jun Murakami (Waseda University)
On a q-deformation of PSL(2) representations of knot groups
14:00 〜15:00
Yoshitaka Hachimori (Tokyo University of Science)
Functional equations and positively ramiﬁed extensions
15:20 〜 16:40
Zdzislaw Wojtkowiak (Universié de Nice)
TBA
March 29 (Thursday)
9:30 〜 10:30
Koichiro Sawada (RIMS, Kyoto University)
Finiteness of isomorphism classes of hyperbolic polycurves with prescribed fundamental groups
10:50 〜 11:50
Benjamin Collas (Universitat Bayreuth)
Arithmetics of Moduli Spaces of Curves and Topological Approaches
14:00 〜 15:00
Katsumi Ishikawa (RIMS, Kyoto University)
A link-homotopy invariant for surface links
15:20 〜 16:40
Kent Orr (Indiana University)
Transﬁnite 3-manifold invariants and a problem of John Milnor

9Abstract
Benjamin Collas (Universitat Bayreuth)
Arithmetics of Moduli Spaces of Curves and Topological Approaches
The moduli stack of curves are endowed with two stratiﬁcations, a divisorial one induced by the Knudsen morphisms in the stable compactiﬁcation, and a stack one induced by the automorphism of curves. Theyboth encapture some remarkable arithmetical and combinatorial properties, which are the core of Grothendieck-Teichmüller theory, i.e. thestudy of the absolute Galois group of rational in terms of topology. The
goal of this talk is to present how Grothendieck-Teichmüller theory –combined with group properties of the mapping class groups, and withhomotopy properties of operads – leads to a better understanding ofthese arithmetics, as well as some promising research lines.
Yoshitaka Hachimori (Tokyo University of Science)
Functional equations and positively ramiﬁed extensions
We discuss functional equations of some elements in Iwasawa algebras,arising from the theory of positively ramiﬁed extensions which was developed by Alexander Schmidt.
Tsukasa Ishibashi (The University of Tokyo)
Cluster Dehn twists in cluster modular groups
A cluster modular group, which is introduced by Fock-Goncharov, is an
automorphism group of a cluster algebra. The cluster modular group
acts on a pair (A, X) of contractible manifolds called a cluster ensemble. These objects are associated with a combinatorial data called aseed. For the seed associated with an ideal triangulation of a punctured
surface, it is known that cluster modular group=the (tagged) mappingclass group, A = the decorated Teichmuller space, X = the enhanced
Teichmuller space. Taking a suitable seed, we can also describe various objects: double Bruhat cells of algebraic groups, canonical basesof quantum groups, higher Teichmuller spaces, and so on. In this talk,
we introduce the concept of ”cluster Dehn twists” for a general cluster
modular group, which is a generalization of Dehn twists and half-twists
in the mapping class group of a surface. We show that orbits of the action of a cluster Dehn twist on the A-space have the similar asymptotic
behavior as those of (half) Dehn twists. Moreover, for several seeds ofﬁnite mutation type, we show that the corresponding cluster modulargroup is generated by cluster Dehn twists. It is a generalization of theclassical fact that the mapping class group of a surface is generated by10

Dehn twists and half-twists.
Katsumi Ishikawa (RIMS, Kyoto University)
A link-homotopy invariant for surface links
A surface link is an oriented closed (not necessarily connected) surface
smoothly embedded in the 4-sphere. Any 2-link, i.e. a surface link withall components being 2-spheres, is known to admit a link homotopy
which pulls the 2-spheres apart, but we know only a little about homotopy classiﬁcation of other surface links: for example, we have only a
few link-homotopy invariants for them. In this talk, we introduce a linkhomotopy invariant for surface links as a reﬁnement of the asymmetriclinking number. This is a surface-link version of Milnor’s link-homotopy
invariant for 1-links, and the triple linking number is calculated from it.
This invariant works well for homotopy classiﬁcation and the speaker
expects it to detect the null-homotopy surface links.
Masanobu Kaneko (Kyushu University)
Genus character L-functions of quadratic orders and class numbers
For general quadratic orders, the genus character L-functions are explicitly computed. As an application, we generalize a formula due toHirzebruch and Zagier which expresses the class number of imaginary
quadratic ﬁelds in terms of continued fraction expansion. This is a joint
work with Yoshinori Mizuno.
Takahiro Kitayama (The University of Tokyo)
Torsion polynomial functions and essential surfaces
We will discuss an application of torsion invariants to an extension of
the Culler-Shalen construction of essential surfaces in a 3-manifold. The
coeﬃcients of twisted Alexander polynomials of a 3-manifold induce
functions on its SLn -character variety. We will describe how the homology class of an essential surface given by an ideal point is restricted byregularity of the function of the highest degree.
Thang Le (Georgia Institute of Technology)
The skein algebra of surfaces and hyperbolic TQFT
The skein algebra of surface has close relations to the character variety and the quantum Teichmüller space; it serves as a bridge between
quantum topology and classical topology. We will discuss representations of the skein algebra and show how they can be used to potentiallyconstruct a hyperbolic TQFT.

11

Tomoki Mihara (Tokyo Institute of Technology)
Homotopy Theory for Metric Spaces
This is a joint work with Frederic Paugum. By a technique by Nikolai
Durov, metric spaces are regarded as normed inﬁnity groupoids. Using the identiﬁcation, I formulate homotopy theory for metric spaces interms of normed inﬁnity category theory.
Jun Murakami (Waseda University)
On a q-deformation of PSL(2) representations of knot groups
In this talk, I would like to explain an idea to construct a q-deformationof PSL(2) representations of knot groups. The main tool is the braided
quantum group BSL(2) constructed by S. Majid. The knot group is
given by Wirtinger presentation, which is interpreted as a conjugate
quandle. Here we construct a braided version of the conjugate quandle, and then associate the braided quantum group to the dual of the
braided conjugate quandle., and then explain how the braided conjugate quandle relate to a knot diagram. This is joint with Roland van
der Veen.
Kent Orr (Indiana University)
Transﬁnite 3-manifold invariants and a problem of John Milnor
In the mid-1950s, Milnor introduced his link invariants, a vast and profound extension of the classical the linking number. His examples, results, and a seminal list of problems have driven decades of research.
One of Milnor’s original questions remains unresolved. How can one extract a version of Milnor’s invariants from the transﬁnite lower central
series. We present a solution to this problem for 3-manifolds, developing
transﬁnite invariants and realizing non-trival values.
Makoto Sakuma (Hiroshima University)
On symmetries of knots
After giving a quick survey on the study of symmetries of knots, I would
like to talk about my joint work with Luisa Paoluzzi concerning amphicheirality and free periodicity of knots.
Koichiro Sawada (RIMS, Kyoto University)
Finiteness of isomorphism classes of hyperbolic polycurves with prescribed fundamental groups
A hyperbolic polycurve is a successive extension of families of hyperbolic

12

curves, which have been regarded as a typical example of ”an anabelianvariety”. In other words, roughly speaking, a hyperbolic polycurve overa certain type of a ﬁeld may be completely determined by its arithmetic
fundamental group. In this talk, we show that a hyperbolic polycurveis determined by its arithmetic fundamental group up to ﬁnitely many
possibilities.
Romyar Shariﬁ (University of California, Los Angeles)
Modular symbols and arithmetic
In this talk, I will explain a conjectural connection between modular
symbols modulo an Eisenstein ideal and values of Steinberg symbols of
cyclotomic units. I will then discuss recent developments in a program
which has arisen from this.
Zdzislaw Wojtkowiak (Universié de Nice)
TBA
Hwajong Yoo (Universie de Nice)
Examples in arithmetic Chern-Simons theory
We introduce a basic idea behind of arithmetic Chern-Simons theory.
Then, we compute the arithmetic Chern-Simons invariants in some cases
using decomposition formula. As an application, we give a short proof
of non-existence of certain quaternion extensions of the rational number
ﬁeld.
Pavel Zalesskii (Universidade Brasilia)
The proﬁnite colmpetion of 3-manifold groups
b of a 3We shall present structural results of the proﬁnite completion G
manifold group G and its interrelation with the structure of G. Residual
properties of G also will be discussed.

13

Low dimensional topology and number theory XIMarch 11 - 14, 2019
Conference Room E404, Science Building. E, Toyonaka Campus,
Osaka University, JAPAN

ProgramMarch 11 (Monday)
10:00 〜 11:00
Ken Ono (Emory University)
Polya’s Program for the Riemann Hypothesis and Related Problems
11:20 〜 12:20
Tomoyoshi Ibukiyama (Osaka University)
Graded rings of modular forms of rational weight
14:00 〜15:00
Toshiki Matsusaka (Kyushu University)
A Kronecker limit type formula for hyperbolic Eisenstein series
15:20〜16:20
Anastasiia Tsvietkova (OIST)
Representations of knot groups

March 12 (Tuesday)
10:00 〜 11:00
Akira Sarashina (RIMS, Kyoto University)
Reconstruction of one-punctured elliptic curves in positive characteristic
by their geometric fundamental groups
11:20 〜 12:20
Wataru Yuasa (Kyoto University)
Andrews-Gordon type identities for A2 with one-row Young diagrams
14:00 〜 15:00
Masanobu Kaneko (Kyushu University)
On a variant of multiple zeta values of level two

14

15:20 〜 16:20
Madeline Dawsey (Emory University)
Higher Width Moonshine
17:15 〜 Banquet

March 13 (Wednesay)
10:00 〜 11:00
Ian Wagner (Emory University)
Harmonic Hecke eigenlines and Mazur’s problem
11:20 〜 12:20
Hikaru Hirano (Kyushu University)
On arithmetic Chern-Simons-Kim invariants for any number rings
14:00 〜15:00
Hidekazu Furusho (Nagoya University)
Betti side of the double shuﬄe theory
15:20 〜 16:20
Jae Choon Cha (Postech)
Homotopical properties and freely slicing good boundary links

March 14 (Thursday)
10:00 〜 11:00
Jun Ueki (Tokyo Denki University)
Chebotarev link is stably generic
11:20 〜 12:20
Densuke Shiraishi (Osaka University)
Galois actions on fundamental groups of P1 \{0, 1, ∞} and triple ℓ-th
power residue symbols

15

Abstract
Jae Choon Cha (Pohang University of Science and Technology)
Homotopical properties and freely slicing good boundary links
A question of fundamental importance in 4-dimensional topology iswhether surgery works for arbitrary fundamental groups. It is known tobe equivalent to the free slicing problem for certain class of links which
are called good boundary links. We show that good boundary links
satisfying a homotopically trivial plus property are freely slice. This
generalizes all previously known methods for freely slicing good boundary links with two or more components, and presents new freely slice
good boundary links.
Madeline Locus Dawsey (Emory University)
Higher Width Moonshine
Weak moonshine for a ﬁnite group G is the phenomenon where an inﬁnite dimensional graded G-module
M
VG =VG (n)
n≫−∞has the property that its trace functions, known as McKay-Thompson
series, are modular functions. Recent work of Dehority, Gonzalez, Vafa,
and Van Peski established that weak moonshine holds for every ﬁnite
group. Since weak moonshine only relies on character tables, which
are not isomorphism class invariants, non-isomorphic groups can havethe same McKay-Thompson series. We address this problem by extending weak moonshine to arbitrary width s ∈ Z+ . Namely, for each1 ≤ r ≤ s and each irreducible character χi , we employ Frobenius’ r(r)character extension χi : G(r) → C to deﬁne McKay-Thompson series(r)of VG := VG × · · · × VG (r copies) for each r-tuple in G(r) := G × · · · × G(r copies). These series are modular functions. We ﬁnd that complete
width 3 weak moonshine always determines a group up to isomorphism.
Furthermore, we establish orthogonality relations for the Frobenius rcharacters, which dictate the compatibility of the extension of weak
moonshine for VG to width s weak moonshine.
Hidekazu Furusho (Nagoya University)
Betti side of the double shuﬄe theory
This is on my joint work with Benjamin Enriquez. The double shuﬄerelation is one of the most important algebraic relations among multiple
zeta values. In 2002, Racinet gave its nice description by using certain16

Hopf algebras constructed from the de Rham fundamental group of the
projective line minus three points. My talk is on its Betti counterpart of his work. I will introduce a new coproduct on a Hopf algebra
constructed from the Betti fundamental group and by using this I will
explain how the double shuﬄe relation follows from the associator relation. The proof relies on an interpretation of the harmonic coproductin terms of inﬁnitesimal braid Lie algebras, which is implicit in the unpublished work of Deligne and Terasoma from 2005.
Hikaru Hirano (Kyushu University)
On arithmetic Chern-Simons-Kim invariants for any number rings
Recently, Minhyong Kim introduced the notion of arithmetic Chern- Simons invariants for totally imaginary number ﬁelds, which is based on
analogies with Dijkgraaf-Witten theory for 3-manifolds. In this lecture,
I will extend Kim’s invariants for any number ﬁelds, using modiﬁedétale cohomology which takes real places into account, and then calculate abelian Chern-Simons-Kim invariants concretely for certain real
quadratic number ﬁleds.
Tomoyoshi Ibukiyama (Osaka University)
Graded rings of modular forms of rational weight
We construct a system of modular forms of one variable of rational
weight belonging to Γ(N ) and write the automorphy factor explicitly.
Then we consider the graded rings of modular forms of integral multiples of that weight and ask if the forms we constructed give generatorsof the ring. We show that the answer is yes for small N by using Mumford criterion of normal generation, and give fundamental relations of
the generators for such N .
Masanobu Kaneko (Kyushu University)
On a variant of multiple zeta values of level two
We introduce and discuss a variant of multiple zeta values of level 2,
which forms a subspace of the space of alternating multiple zeta values.
This variant exhibits nice properties such as duality, shuﬄe product,
parity results like ordinary multiple zeta values. We give some conjectures on relations between our values and ordinary multiple zeta values.
Toshiki Matsusaka (Kyushu University)
A Kronecker limit type formula for hyperbolic Eisenstein series
In 1979, Kudla and Millson introduced a form-valued hyperbolic Eisen-17

stein series associated to any closed geodesic on the Riemann surfaceΓ\H. More recently, Jorgenson, Kramer, and Pippich in 2010 deﬁned ascalar-valued analog of the hyperbolic Eisenstein series, and establishedthe meromorphic continuation to the whole s-plane. As in the classicalparabolic case, they studied the Laurent expansion at the special points = 0. In the particular case of Γ = SL2 (Z), this Eisenstein series has
double zero at s = 0. In this talk, we give the second order coeﬃcient
explicitly.
Ken Ono (Emory University)
Polya’s Program for the Riemann Hypothesis and Related Problems
In 1927 Polya proved that the Riemann Hypothesis is equivalent to thehyperbolicity of Jensen polynomials for Riemann’s Xi-function. This
hyperbolicity has only been proved for degrees d=1, 2, 3. For each d we
prove the hyperbolicity of all but (perhaps) ﬁnitely many Jensen polynomials. We obtain a general theorem which models such polynomials
by Hermite polynomials. This theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function. This result
can be thought of as a proof of GUE for the Riemann zeta function inderivative aspect. This is joint work with Michael Griﬃn, Larry Rolen,
and Don Zagier.
Akira Sarashina (RIMS, Kyoto University)
Reconstruction of one-punctured elliptic curves in positive characteristic by their geometric fundamental groups
The principal theme for anabelian geometry is the reconstruction of
the geometry of algebraic varieties by their étale fundamental groups.
By G.A.G.A. theorems, the isomorphism class of the étale fundamental group of a curve over an algebraically closed ﬁeld of characteristic
0 is determined by the genus and the cardinality of cusps. But thatof a curve over an algebraically closed ﬁeld of positive characteristic is
not determined by easy invariants. In other words, it has much information. Tamagawa proved that the isomorphism class as a scheme of
curves over Fp can be reconstructed by their étale fundamental groups
when the genus is 0. In this talk, we will discuss the genus 1 case, andprove a similar result when the genus is 1, the cardinality of cusps is 1
and the characteristic is not equal to 2.
Densuke Shiraishi (Osaka University)
Galois actions on fundamental groups of P1 \{0, 1, ∞} and triple ℓ-th18

power residue symbols
In this talk, we discuss relationships between ℓ-adic Galois polylogarithms and triple ℓ-th power residue symbols in some special cases studied by a recent work of Hirano-Morishita. We show that a functional
equation of ℓ-adic Galois polylogarithm by Nakamura-Wojtkowiak implies a reciprocity law of triple ℓ-th power residue symbol.
Anastasiia Tsvietkova (Okinawa Institute of Science and Technology)
Representations of knot groups
We describe a new method of producing equations for the representation variety of a knot group into (P)SL(2,C). Unlike known methods,
this does not involve any polyhedral decomposition or triangulation of
the link complement, and uses only a link diagram satisfying a few mild
restrictions. This results in a simple algorithm that can often be performed by hand, and in many cases, for an inﬁnite family of knots at
once. This is a joint work with Kathleen Peterson, based on an earlier
joint work with Morwen Thislethwaite. The discrete faithful representation gives rise to the invariant trace ﬁeld, which is a topological andcommensurability invariant. We will also show how these ideas allow tocompute the ﬁeld exactly and from a link diagram. This is an earlier
joint work with Walter Neumann.
Jun Ueki (Tokyo Denki University)
Chebotarev link is stably generic
When we study the analogy between knots and prime numbers in “arithmetic topology”, it is a fundamental question to ask “what is the analogue of the set of all prime ideals of a number ﬁeld in a 3-manifold?”
In [NiiboUeki2018], the speaker constructed a “very admissible link”
as a candidate in a 3-manifold, and described the idelic class ﬁeld theory by summing up local theories to describe the global theory. Afterwards, Mihara gave a cohomological interpretation on it, and proposed a “stably generic link” reﬁning our link so that we can study
analogues of ray class ﬁelds in [Mihara2018]. On the other hand, McMullen proved in [McMullen2013] that the closed orbits of a pseudoAnosov ﬂow on a 3-manifold obeys the Chebotarev law in the sense of
B. Mazur [Mazur2012], if ordered by length. For instance, the planetarylink of the ﬁgure eight knot (or the Whitehead link or the Borromeanrings) obeys the Chebotarev law, by Thurston’s classiﬁcation theorem.
In addition, this inﬁnite link contains all isotopy classes of links, due tothe theory of universal template [Ghrist1993]. In this talk, we compare

19these inﬁnite links and prove that Chebotarev link is stably generic. In
addition, as an example of Chebotarev phenomena, we study the decomposition types of knots in an analogue of a quintic ﬁeld.
Ian Wagner (Emory University)
Harmonic Hecke eigenlines and Mazur’s problem
We construct two families of harmonic Maass Hecke eigenforms. Using
these families, we construct p-adic harmonic Maass forms in the sense ofSerre. The p-adic properties of these forms answer a question of Mazur
about the existence of an eigencurve-type object in the world ofharmonic Maass forms.
Wataru Yuasa (Kyoto Univeristy)
Andrews-Gordon type identities for A2 with one-row Young diagrams
In this talk, we derive Andrews-Gordon type identities for the sl3 (false)
theta functions via knot theory. The sl3 tail of a link L is a limit of thesl3 colored Jones polynomials {Jλsl3 (L)}λ∈Λ for Λ is a certain subset oftwo-row Young diagrams. For one-row Young diagrams Λ = {(n, 0)}, we
calculate two types of explicit formulae of Jλsl3 (T (2, m)) where T (2, m)
is a (2, m)-torus link and obtain two explicit formulae of its tail. They
give the Andrews-Gordon type identities for Ramanujan (false) theta
functions.

20

Low dimensional topology and number theory XIII
To the memory of Professor Toshie Takata
IMI Auditorium (413 Rm), 4F West 1st Bd, Kyushu University (Ito
Campus) 15th March, 2022 ? 18th March, 2022
ProgramMarch 15th (Tues)
9:40 Organizer
9:45 – 9:55
Osamu Saeki（Kyushu University, IMI)
10:00 – 11:00
Michihisa Wakui（Kansai University）
A new algorithm of the normalized Jones polynomials of rational links
11:20–12:20
Kimihiko Motegi（Nihon University） Online
The Strong Slope Conjecture for Mazur pattern satellite knots
14:00 – 15:00
Toshitake Kohno（Meiji University/The University of Tokyo) Online
Temperley-Lieb-Jones category and the space of conformal blocks
15:20 – 16:40 To the memory of Professor Toshie Takata:
Haruko Takayama (Nishi)（Josai University) Online
Tomoyoshi Ibukiyama (Osaka University) Online
and some others
17:00 – 18:00
Christian Kassel（Université de Strasbourg/CNRS）Online
Braid groups and Steinberg groups
March 16th (Wedes)
10:00 – 11:00
Jun Murakami（Waseda University）
Volume potential function and its applications

21

11:20 – 12:20
Yoshiyuki Yokota（Tokyo Metropolitan University）
On Neumann-Zagier matrices and generalized angle structures for hyperbolic knots
14:00 – 15:00
Kazuo Habiro（Kyoto University）
Ribbon Yetter-Drinfeld modules and tangle invariants
15:20 – 16:20 To the memory of Professor Toshie Takata:
Sakie Suzuki（Tokyo Institute of Technology) Online
and some others
16:40 – 17:40
Hitoshi Murakami（Tohoku University） Online
The colored Jones polynomial of the ﬁgure-eight knot
March 17th (Thurs)
10:00 – 11:00
Eiko Kin（Osaka University）
Braids and ﬁbered double branched covers of 3-manifolds
11:20 – 12:20
Hidekazu Furusho（Nagoya University）
Kashiwara-Vergne Lie algebra and Goncharov’s dihedral Lie algebra in
mould theory
14:00 – 15:00
Hisatoshi Kodani（Tohoku University） Online
On partial generalization of Hodge correlators associated with diagrams
allowed to have loops
15:20 – 15:50
Densuke Shiraishi（Osaka University）
On functional equations of ℓ-adic Galois polylogarithms
16:10 – 16:40
Hikaru Hirano（Kyushu University）
Brylinski-McLaughlin’s state space and its arithmetic analogue

22

March 18th（Fri)10:00 – 11:00
Hiroyuki Ochiai（Kyushu University, IMI）
On arithmetic-geometric means and hypergeometric functions containedin Gauss’ Werke
11:20 – 12:20
Takeo Uramoto（Kyushu University, IMI）
Classical class ﬁeld theory meets algebraic language theory
14:00 – 14:30
Ryoto Tange（Waseda University） Online
On adjoint homological Selmer modules for SL(2)-representations ofknot groups
14:50 – 15:20
Hyuga Yoshizaki（Tokyo University of Science） Online
Weber’s class number problem for cyclic covers of knots
15:40 – 16:40
Toshiki Matsusaka（Nagoya University）
Modular transformations of homological blocks for Seifert ﬁbered homology 3-spheres

23

Abstract

March 15th (Tues)
Michihisa Wakui（Kansai University）
A new algorithm of the normalized Jones polynomials of rational links
In connection with cluster algebras, snake graphs and q-integers, Kyungyong Lee and Ralf Schiﬄer recently found a formula for computing the(normalized) Jones polynomials of rational links in terms of continued fraction expansion of rational numbers. Sophie Morier-Genoudand Valentin Ovsienko introduced q-deformed continued fractions, andshowed that by using them each coeﬃcient of the normalized Jones
polynomial counted quiver representations of type An. In this talk we
introduce q-deformed integers deﬁned by coprime pairs of natural numbers, which are motivated by the denominators and the numerators of
their q-deformed continued fractions, and give an eﬃcient algorithm forcomputing the normalized Jones polynomials of rational links.
Kimihiko Motegi（Nihon University）
The Strong Slope Conjecture for Mazur pattern satellite knots
The Slope Conjecture proposed by Garoufalidis asserts that the degreeof the colored Jones polynomial determines a boundary slope, and its
reﬁnement, the Strong Slope Conjecture proposed by Kalfagianni andTran asserts that the linear term in the degree determines the topologyof an essential surface that satisﬁes the Slope Conjecture. Under certain hypotheses, we show that a Mazur pattern satellite knots satisfy
the Strong Slope Conjecture if the original knot does. Consequently,
combining with previous results, any knot obtained by a ﬁnite sequenceof cabling, connected sums, Whitehead doubling and taking Mazur pattern satellites of adequate knots (including alternating knots) or torus
knots satisﬁes the Strong Slope Conjecture. This is joint work with
Kenneth L. Baker (University of Miami) and Toshie Takata.
Toshitake Kohno (Meiji University/The University of Tokyo)
Temperley-Lieb-Jones category and the space of conformal blocks
We show that the set of morphisms of the colored Temperley-Lieb-Jones
category at roots of unity is isomorphic to the space of conformal blocks
in the Wess-Zumino-Witten conformal ﬁeld theory. We describe the24braid group action on these spaces and explain that the above isomorphism is equivariant and that the representations are unitary and irreducible. To prove this we use an expression of the space of conformal
blocks by multi-dimensional hypergeometric integrals. This investigation leads us to reveal a family of unitary and irreducible representationsof the braid groups contained in homological representations at roots ofunity.
Christian Kassel（Université de Strasbourg/CNRS）
Braid groups and Steinberg groups
We construct a homomorphism from the braid group on 2n + 2 strands
to the Steinberg group associated with the Lie type Cn and with integer
coeﬃcients. This homomorphism lifts the well-known symplectic representation of the braid groups. We shall describe its image, expressing
it in terms of the level 2 congruence subgroup of the symplectic modular group. We shall also determine its kernel. This is joint work withFrancois Digne (Amiens).
March 16th (Wedes)
Jun Murakami（Waseda University）
Volume potential function and its applications The volume potentialfunction appears as a certain limit of the colored Jones polynomial in
the study of the volume conjecture. This volume potential function isconjectured to relate the A-polynomial and the twisted Reidemeister
torsion. The relation to the A-polynomial is studied as AJ-conjecture,and the relation to the twisted Reidemeister torsion is conjectured by
Gkukov-H.Murakami and is proved for two-bridge knots by OhtsukiTakata. In this talk, such relations are explained for twisted Whiteheadlinks and some other examples. This is a joint with A. Tran.
Yoshiyuki Yokota（Tokyo Metropolitan University）
On Neumann-Zagier matrices and generalized angle structures for hyperbolic knots
We ﬁrst review the simplectic property of the Neumann-Zagier matrix
obtained from an ideal triangulation of a hyperbolic knot complement
M . We then give a nice parametrization of the generalized angle structures, assignments of real numbers to the pairs of opposite edges of the25

tetrahedra, of M , and show that the volume of M is given by a criticalvalue of the volume functional deﬁned on such parametrized space.
Kazuo Habiro (Kyoto University)
Ribbon Yetter-Drinfeld modules and tangle invariants
It is well known that ﬁnite-dimensional Yetter-Drinfeld modules over a
Hopf algebra form a braided, pivotal category, but not necessarily a ribbon category. We introduce a notion of ribbon Yetter-Drinfeld modules
and show that ﬁnite-dimensional ribbon Yetter-Drinfeld modules form
a ribbon category, and therefore give rise to tangle invariants. This is
joint work with Yuka Kotorii.
Hitoshi Murakami (Tohoku University)
The colored Jones polynomial of the ﬁgure-eight knot
For any positive integer N , the colored Jones polynomial of a knot givesa Laurent polynomial in q. When we replace q with exp(u/N ) for a ﬁxed
complex number u, we have a function fu (N ) from positive integers tocomplex numbers. I will talk about relations between the asymptotic
behaviors of fu (N ) of the ﬁgure-eight knot for various u, and representations of the fundamental group of the knot complement to SL(2; C).
Part of this work is a joint work with Anh Tran.
March 17th (Thurs)
Eiko Kin (Osaka University)
Braids and ﬁbered double branched covers of 3-manifolds
The branched virtual ﬁbering theorem by Sakuma states that every
closed orientable 3-manifold M with a Heegaard surface of genus g has
a branched double cover which is a genus g surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen tobe hyperbolic. In a previous result with Susumu Hirose, we proved thatwhen M is the 3-sphere S 3 , the minimal entropy over all hyperbolic,genus g surface bundles as branched double covers of S 3 behaves like
1/g. In this talk, I will explain that inﬁnitely many closed 3-manifolds
have the same property. To prove this claim, we use an alternative construction of surface bundles in Sakuma’s theorem when M is a brancheddouble cover of S 3 branched over a link. A feature of surface bundles

26

coming from our construction is that the monodromies can be read oﬀ
the braids obtained from the links as the branched set. This is a joint
work with Susumu Hirose.
Hidekazu Furusho (Nagoya University)
Kashiwara-Vergne Lie algebra and Goncharov s dihedral Lie algebrain mould theory
We introduce the Kashiwara-Vergne bigraded Lie algebra associatedwith a ﬁnite abelian group. By giving its mould theoretic interpretation, we show that it includes the Goncharov s dihedral Lie algebra,
which generalizes the result of Raphael and Schneps. It is a joint workwith Nao Komiyama.
Hisatoshi Kodani（Tohoku University）
On partial generalization of Hodge correlators associated with diagrams
allowed to have loops
Goncharov’s Hodge correlators are complex numbers given as certain
integrals assigned to Riemann surfaces. They contain a wide varietyof arithmetic invariants such as classical (elliptic) polylogarithms andspecial values of some L-functions, and provide an alternative way todescribe the standard real mixed Hodge structure on the nilpotent completion of the fundamental group of a punctured Riemann surface given
by Chen’s iterated integral. As he showed, the Hodge correlators can bethought of as tree diagrams part of large N asymptotic expansion of amatrix model. However, the structure and properties of loop diagrams
part was left as problem and has been still unknown, since the loop
part could be divergent. In this talk, we explain a partial generalizationof Hodge correlators associated with diagrams allowed to have loops
and their several propeties. This talk is based on a joint work with Y.
Terashima.
Densuke Shiraishi (Osaka University)
On functional equations of ?-adic Galois polylogarithms
The ℓ-adic Galois polylogarithm is an arithmetic function on the absolute Galois group with values in ℓ-adic numbers, which is deﬁned as27

a certain coeﬃcient of the ℓ-adic associator arising from the Galois action on the etale fundamental groupoid of the projective line minus
three points. This function was introduced by Wojtkowiak as an ℓ-adic
Galois analogue of the classical polylogarithm, and typical functional
equations of ℓ-adic Galois dilogarithms are proved by Nakamura andWojtkowiak. In this talk, we discuss some new functional equations forℓ-adic Galois polylogarithms that are not found in previous works.
Hikaru Hirano (Kyushu University)
Brylinski-McLaughlin s state space and its arithmetic analogue
Brylinski and McLaughlin introduced a certain state space as a variantof the space of conformal blocks in (2+1)-dim Dijkgraaf-Witten TQFT,
and posed a question on their space. We give an answer to it and study
an arithmetic analogue of the state space.
March 18th（Fri）
Hiroyuki Ochiai（Kyushu University, IMI）
On arithmetic-geometric means and hypergeometric functions containedin Gauss’ Werke
There are contained some numerical examples of arithmetic-geometricmeans in Gauss’ Werke. I will talk about my discovery on those numerical values. It is known that arithmetic-geometric means can be
expressed by the special values of hypergeometric functions. I will also
talk about my observation on some family of special functions related
to those hypergeometric functions.
Takeo Uramoto（Kyushu University, IMI）
Classical class ﬁeld theory meets algebraic language theory
Algebraic language theory is a subﬁeld of the theory of computation,and since the 1960s, has developed semigroup-theoretic methodology
for solving several logical/combinatorial decision problems concerning
regular languages (= sets of ﬁnite words accepted by ﬁnite automata).
In our recent works, we observed that algebraic language theory can be
uniﬁed with galois theory in a certain precise categorical sense, and more
importantly, that this uniﬁcation sheds a new light on classical class ﬁeld
theory and complex multiplication. In this talk, we will overview these28

recent results and discuss some future perspective.
Ryoto Tange（Waseda University）
On adjoint homological Selmer modules for SL(2)-representations ofknot groups
We introduce the adjoint homological Selmer module for an SL(2)representation of a knot group, which may be seen as an analogue ofthe adjoint Selmer module for a Galois representation in number theory. This is joint work with Takahiro Kitayama, Masanori Morishita,
and Yuji Terashima.
Hyuga Yoshizaki (Tokyo University of Science)
Weber’s class number problem for cyclic covers of knots
We study an analogue of Weber s class number problem for cyclic coversof knots in the spirit of arithmetic topology. Let p be a prime number.
Webers problem is an unsolved problem with a long history in number
theory asking the class numbers of certain cyclic pn -th extensions of therationals Q. Since the analogue of the class numbers of number ﬁelds
is the sizes of the ﬁrst homology groups of 3-manifolds, we study thesizes of the ﬁrst homology groups of cyclic pn -th covers of knots in S 3 .
In number theory side, the speaker previously proved the p-adic convergence of the class numbers, but their limit value is not yet known forany example. In this talk, on the other hand, we prove an analogous
result in knot theory side and concretely calculate their limit for thetorus knots and the twist knots. This talk is based on a joint work with
Jun Ueki (Tokyo Denki University).
Toshiki Matsusaka（Nagoya University）
Modular transformations of homological blocks for Seifert ﬁbered homology 3-spheres
Recently, Gukov-Pei-Putrov-Vafa introduced q-series-valued invariants
called homological blocks for any plumbed 3-manifolds. In this talk, for
any Seifert ﬁbered homology 3-sphere, we give modular transformation
formulas of homological blocks. Moreover, based on the modular transformation formulas, we have explicit asymptotic expansion formulas for
the Witten-ReshetikhinTuraev invariants, which give a new proof of aversion by Andersen of the Witten asymptotic conjecture. This is joint

29

work with Yuji Terashima (Tohoku University).

30

Low dimensional topology and number theory XIV
IMI Auditorium (413 Rm), 4F West 1st Bd, Kyushu University (Ito
Campus)27th March, 2023 ∼ 30th March, 2023
ProgramMarch 27th (Mon)9:30 – 10:30
Jun Ueki（Ochanomizu University）
Non-acyclic SL2 -representations, surgeries, and L-functions of the twisted
Whitehead links
10:50 – 11:50
Sohei Tateno（Nagoya University)
The Iwasawa invariants of Zdp -covers of links
13:40 – 14:40
Takefumi Nosaka（Tokyo Institute of Technology）
Reciprocity of the Chern-Simons invariants of 3-manifolds
March 28th (Tues)
9:30 – 10:30
Naganori Yamaguchi（RIMS, Kyoto University）
On the development of anabelian geometry using the maximal geometrically m-step solvable quotient of arithmetic fundamental groups
10:50 – 11:50
Shun Ishii（RIMS, Kyoto University）
On pro-p outer Galois representations associated to once-punctured CM
elliptic curves
13:40 – 14:40
Takeo Uramoto（IMI, Kyushu University)
On the modularity theorem for algebraic Witt vectors
15:00 – 16:00
James Borger (Australian National University)
Report on scheme theory over semirings

31March 29th (Wednes)
9:30 – 10:30
Yosuke Morita (Kyoto University）
A new framework for Conley index theory
10:50 – 11:50
Jonathan Beardsley（University of Nevada）
Toward Higher Algebra Over F1
13:40 – 14:40
Takuya Takeishi (Kyoto Institute of Technology）
Rigidity theorems of C*-algebras arising from number theory
15:00 – 16:00
Takeshi Shinohara and Nao Komiyama (Nagoya University)
Shuﬄe product of desingularized multiple zeta functions at integer points
March 30th（Thurs)
10:00 – 11:00
Atsushi Katsuda（Kyushu University）
Closed geodesics, Heat kernels and Hofstadter butterﬂy
13:00 – 14:00
Jesus Ántonio Alvarz López (Universidade de Santiago de Compostela)
A trace formula for foliated ﬂows
14:20 – 15:20
Dohyeong Kim（Seoul National University）
Iwasawa’s invariant and its Diophantine application

32

Abstract• Jesús A. Álvarez López (University of Santiago de Compostela)

A trace formula for foliated ﬂows
Let (M, F, ϕt ) be a smooth compact manifold equipped with codimension one foliation and a foliated ﬂow (the ﬂow maps leaves to leaves).
Assume ϕt has simple closed orbits and transversely simple preserved
leaves. Then there are ﬁnitely many leaves preserved by the ﬂow, whose
union is a compact submanifold M 0 , and a precise description of thetransverse structure of F can be given. The distributional leafwise
diﬀerential forms conormal to M 0 form a complex with the de Rhamderivative of the leaves, giving rise to the conormal leafwise reduced
cohomology H̄I(F). We deﬁne a Leftchetz distribution Ldis (ϕt ) on R of
the induced action ϕt ∗ on H̄I(F). Then we prove a distributional Lefschetz trace formula describing Ldis (ϕt ) in terms of inﬁnitesimal data of
the closed orbits and preserved leaves. This kind of distributional trace
formula was conjectured by Christopher Deninger, motivated by possible arithmetic interpretations. In the case where M 0 = ∅ (when F is
Riemannian), this formula was proved using smooth leafwise diﬀerential
forms. This is joint work with Yuri Kordyukov and Eric Leichtnam.
• Jonathan Beardsley (University of Nevada)

Toward Higher Algebra Over F1
Abstract: In a series of papers, Alain Connes and Caterina Consani
have introduced a framework for doing ”algebra in characteristic one.”
Their approach has strong connections to Segal’s approach to stablehomotopy theory via so-called Γ-spaces. I will describe some ﬁrst steps
in further developing this analogy. In particular I will show that within
their framework there is a natural notion of the ”classifying space” ofan F1 -module. If A is the F1 -module associated to an abelian group,
commutative monoid, or abelian hypergroup, with classifying space BA,then taking the loop space of BA (for a slightly non-standard notionof loop space described by Connes and Consani) recovers A. This suggests that higher deloopings and possibly Eilenberg-MacLane spectra of
F1 -modules may be possible, though they would require working witha suitable model for (∞,∞)-categories. This is joint work with Joe
Moeller.
• James Borger (Australian National University)33

Report on scheme theory over semirings
Usually in algebraic geometry, one works with schemes deﬁned over baseﬁelds. But for arithmetic applications, where transferring information
between positive characteristic and characteristic zero is necessary, it’sconvenient to work with schemes deﬁned over base rings, such as theintegers or the p-adic integers. Indeed, since this formalism became
available in the mid-20th century, it has become standard.
It is however possible to go even deeper and set up scheme theory
over any base semiring. This includes objects like projective space overthe nonnegative reals, the Boolean numbers, or the natural numbers,
which is the deepest subring of all. Doing so allows us to preserve
positivity information in the underlying theory, much as the passage
from base ﬁelds to base rings allowed for the preservation of integrality
information.
Scheme theory over semirings is however in its infancy. In this talk,I’ll report on some areas of scheme theory over rings which have been recently extended to semirings, such as the etale fundamental group (joint
with Robert Culling) and the Picard stack (joint with Jaiung Jun). One
notable point that came out of the latter work is that the various familiar deﬁnitions of vector bundle in scheme theory over rings do not
remain equivalent over semirings but that all the familiar deﬁnitions ofline bundle do remain equivalent.
• Shun Ishii (RIMS, Kyoto University)

On pro-p outer Galois representations associated to once-punctured CM
elliptic curves
Let p be an odd prime. Shariﬁ proved that, under the Deligne-Ihara
conjecture (now a theorem of Brown), the kernel of the pro-p outer
Galois representation associated to the thrice-punctured projective line
corresponds to the maximal pro-p extension of Q(µp ) unramiﬁed outside
p if p is regular. In this talk, we discuss an analogue of his result for
imaginary quadratic ﬁelds by considering once-punctured CM elliptic
curves instead of the thrice-punctured projective line.
• Atsushi Katsuda (Kyushu University)

Closed geodesics, Heat kernels and Hofstadter butterﬂy
The Bloch-Floquet theory are popular tools for the investigation of materials with periodic structures. For example, we can show that the
spectrum of periodic Schrödinger operators have band structures. In

34the context of this talk, this theory was applied to the following problems in the case of abelian extensions:(1) A geometric analogue of the Chebotarev density theorem for prime
closed geodesics in a compact Riemannian manifold with negative
curvature(2) A long time asymptotic expansion of the heat kernels of covering
manifolds of compact Riemannian manifolds.
Here, we shall extend the above results by applying our version of the
Bloch-Floquet theory for the Heisenberg group, which is the simplest,
but non-trivial non-abelian inﬁnite group. Our method is based on a
combination of the representation theory for discrete Heisenberg groups
especially due to Pytlik and that of the Heisenberg Lie group.
Moreover, there is a relation to the our methods for the Heisenberg
group and the analysis of the discrete magnetic Laplacian or the Harper
operator on the square lattice Z2 . Spectrum of the latter operators are
expressed by the celebrated Hofstadter butterﬂy as the following ﬁgure:

Figure 1: the Hofstadter’s butterﬂy (created by Hisashi Naito).
Especially, our arguments give an another mathematical justiﬁcation of semi-classical asymptotic expansion formula for spectrum of theHarper operator due to Wilkinson, which is originally done by HelﬀerSjöstrand.
En = −4 + (2n + 1)θ + O(θ2 )n = 0, 1, 2, . . . l

If time permits, I shall explain generalizations of our method to moregeneral nilpotent groups and further applications.

35• Dohyeong Kim (Seoul National University)

Iwasawa’s invariant and its Diophantine application
We review the cyclotomic Iwasawa theory and apply the vanishing of
the mu-invariant to some Diophantine problems. I will compare thevanishing of mu-invariants to the monicity of twisted Alexander polynomials.
• Nao Komiyama, Takeshi Shinohara (Nagoya University)

Shuﬄe product of desingularized multiple zeta functions at integer points
We show that the “shuﬄe-type” formula holds for special values of desingularized multiple zeta functions at any integer points. This is provenby giving an iterated/diﬀerential expression for the special values at integer points of the desingularized multiple zeta functions. We explain
how the so-called renormalization method is applied in our proof.
• Yosuke Morita (kyoto University)

A new framework for Conley index theory
The Conley index is, informally speaking, a ‘space’ that describes the
local dynamics around an isolated invariant subset of a topological dynamical system. It can be seen as a spatial reﬁnement of the Morse
index and is used, for instance, in Manolescu s construction of the
Seiberg-Witten-Floer homotopy type. In this talk, I will explain a newframework for Conley index theory, which I think is simpler and more
ﬂexible than the traditional formulation. One important point is that
the Conley index should be deﬁned as a based equivariant ind-(compact
Hausdorﬀ space) (or slightly more generally, a based equivariant condensed set/anima), not as a mere homotopy type of topological spaces.
• Takefumi Nosaka (Tokyo Institute of Technology)

Reciprocity of the Chern-Simons invariants of 3-manifolds
Given an oriented closed 3-manifold M and a representation π1 (M ) −→SL2 (C), we can deﬁne the Chern-Simons invariant and adjoint Reidemeister torsion. Recently, several physicists and topologists pose andstudy reciprocity conjectures of the torsions. Analogously, I posed reciprocity conjectures of the Chern-Simons invariants of 3-manifolds, anddiscussed some supporting evidence on the conjectures. Especially, I
showed that the conjectures hold if Galois descent of a certain K3 -groupis satisﬁed. In this talk, I will explain the background and the results36

in detail.
• Takuya Takeishi (Kyoto Institute of Technology)

Rigidity theorems of C*-algebras arising from number theory
C*-algebras can be constructed from various mathematical objects, and
it is commonly interested to detect what kind of information is preserved by taking C*-algebras. Usually, a lot of information may vanish,
but it turned out that several classes of C*-algebras constructed fromnumber ﬁelds remember the original number ﬁelds completely. As aconsequence, we can construct several classes of countable (noncommutative) groups which are complete invariants of number ﬁelds. In thistalk, we explain the speaker’s recent contribution to the study of thisphenomenon, together with several related results.
• Sohei Tateno (Nagoya University)

The Iwasawa invariants of Zdp -covers of links
In this talk, we will deﬁne the Iwasawa invariants of links and give two
asymptotic formulae for the ﬁrst homology groups of Zdp -covers of links
in rational homology 3-spheres, which are generalizations of the Iwasawa type formulae proven by Hillman-Matei-Morishita and KadokamiMizusawa. We will also provide examples of these formulae. Moreover,when d = 2, considering the twisted Whitehead links, we will explainthat Iwasawa µ-invariants can be arbitrary non-negative integers. Thisis a joint work with Jun Ueki.
• Jun Ueki (Ochanomizu University)

Non-acyclic SL2 -representations, surgeries, and L-functions of the twisted
Whitehead links
We study the zeros of the SL2 -acyclic torsion functions τ of links. We
ﬁrst study that of the Whitehead link W1 and deduce results for twist
knots simultaneously. Next, we extend the study to every twisted
Whitehead link Wk with k ∈ Z and prove that the zeros of τ havethe multiplicity two on the geometric component of the character variety. Finally, we paraphrase the notion of multiplicity of common zeros
and investigate the L-functions of the universal deformations of residual
representations of Wk . (Based on a joint work with Leo Benard, Ryoto
Tange, and Anh Tran.)• Takeo Uramoto (IMI, Kyushu University)37

On the modularity theorem for algebraic Witt vectors
In my previous work, I proved the (weak/strong) “modularity theorem
for algebraic Witt vectors” over imaginary quadratic ﬁelds K, the strong
one of which claims that they are precisely those generated over K bymodular vectors whose coeﬃcients are special values of deformation
family of Fricke modular functions. Arithmetically this construction
implies, say, congruences between special values of modular functions
living at diﬀerent levels (rather, not necessarily galois-conjugate ones),
and provides natural re-interpretations of classical objects (say, Frickefunctions, Witt vectors) in class ﬁeld theory and the theory of complex
multiplication, albeit it yet seems to be a special facet of some moregeneral theme of deformation theory. In view of this, I would like todevote this talk to some clariﬁcation/discussion on more generic geometric structures behind this speciﬁc phenomenon, extending it to thecase of CM ﬁelds in particular.
• Naganori Yamaguchi (RIMS, Kyoto University)

On the development of anabelian geometry using the maximal geometrically m-step solvable quotient of arithmetic fundamental groups
In anabelian geometry, we have the following conjecture called theGrothendieck conjecture: The geometric information of hyperbolic curves
is reconstructed group-theoretically from their arithmetic fundamentalgroups. This conjecture was proved by Hiroaki Nakamura, Akio Tamagawa, and Shinichi Mochizuki. However, many unresolved problems
still remain around this conjecture. One of these problems is the main
topic of this talk which is called the m-step solvable Grotheindieck conjecture, speciﬁcally: The geometric information of hyperbolic curves isreconstructed group-theoretically from the maximal geometrically mstep solvable quotient of their arithmetic fundamental groups. In thistalk, we will present about this conjecture and a part of its proof by the
speaker.

38