## Seminar information archive

Seminar information archive ～12/08｜Today's seminar 12/09 | Future seminars 12/10～

#### Lie Groups and Representation Theory

17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Monomial representations of discrete type and differential operators. (English)

**Ali Baklouti**(Faculté des Sciences de Sfax)Monomial representations of discrete type and differential operators. (English)

[ Abstract ]

Let $G$ be an exponential solvable Lie group and $\tau$ a monomial representation of $G$, an induced representation from a connected closed subgroup of $G$ of a unitary character. It is well known that $\tau$ disintegrates into irreducible factors and the multiplicities of each isotypic component are explicitly determined. In the case where $G$ is nilpotent, these multiplicities are either finite or infinite almost everywhere, with respect to the disintegration's measure. We associate to $\tau$ an algebra of differential operators and it is shown that in the nilpotent case, the commutativity of this algebra is equivalent to the finiteness of the multiplicities of $\tau$. In the exponential case, we define the notion of monomial representation of discrete type. In this case, we show that such an equivalence does not hold and this answers a question posed by M. Duflo. This is a joint work with H. Fujiwara and J. Ludwig.

Let $G$ be an exponential solvable Lie group and $\tau$ a monomial representation of $G$, an induced representation from a connected closed subgroup of $G$ of a unitary character. It is well known that $\tau$ disintegrates into irreducible factors and the multiplicities of each isotypic component are explicitly determined. In the case where $G$ is nilpotent, these multiplicities are either finite or infinite almost everywhere, with respect to the disintegration's measure. We associate to $\tau$ an algebra of differential operators and it is shown that in the nilpotent case, the commutativity of this algebra is equivalent to the finiteness of the multiplicities of $\tau$. In the exponential case, we define the notion of monomial representation of discrete type. In this case, we show that such an equivalence does not hold and this answers a question posed by M. Duflo. This is a joint work with H. Fujiwara and J. Ludwig.

### 2018/11/30

#### Colloquium

15:30-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

The theory of viscosity solutions and Aubry-Mather theory

(日本語)

**Hiroyoshi Mitake**(The University of Tokyo)The theory of viscosity solutions and Aubry-Mather theory

(日本語)

[ Abstract ]

In this talk, we give two topics of my recent results.

(i) Asymptotic analysis based on the nonlinear adjoint method: Wepresent two results on the large-time behavior for the Cauchy problem, and the vanishing discount problem for degenerate Hamilton-Jacobiequations.

(ii) Rate of convergence in homogenization of Hamilton-Jacobi equations: The convergence appearing in the homogenization was proved in a famous unpublished paper by Lions, Papanicolaou, Varadhan (1987). In this talk, we present some recent progress in obtaining the optimal rate of convergence $O(¥epsilon)$ in periodic homogenization of Hamilton-Jacobi equations. Our method is completely different from previous pure PDE approaches which only provides $O(¥epsilon^{1/3})$. We have discovered a natural connection between the convergence rate and the underlying Hamiltonian system.

In this talk, we give two topics of my recent results.

(i) Asymptotic analysis based on the nonlinear adjoint method: Wepresent two results on the large-time behavior for the Cauchy problem, and the vanishing discount problem for degenerate Hamilton-Jacobiequations.

(ii) Rate of convergence in homogenization of Hamilton-Jacobi equations: The convergence appearing in the homogenization was proved in a famous unpublished paper by Lions, Papanicolaou, Varadhan (1987). In this talk, we present some recent progress in obtaining the optimal rate of convergence $O(¥epsilon)$ in periodic homogenization of Hamilton-Jacobi equations. Our method is completely different from previous pure PDE approaches which only provides $O(¥epsilon^{1/3})$. We have discovered a natural connection between the convergence rate and the underlying Hamiltonian system.

### 2018/11/28

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Localization of signature for singular fiber bundles

**Mayuko Yamashita**(Univ. Tokyo)Localization of signature for singular fiber bundles

### 2018/11/27

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Frobenius summands and the finite F-representation type (TBA)

**Nobuo Hara**(Tokyo University of Agriculture and Technology)Frobenius summands and the finite F-representation type (TBA)

[ Abstract ]

We are motivated by a question arising from commutative algebra, asking what kind of

graded rings in positive characteristic p have finite F-representation type. In geometric

setting, this is related to the problem to looking out for Frobenius summands. Namely,

given aline bundle L on a projective variety X, we want to know how many and what

kind of indecomposable direct summands appear in the direct sum decomposition of

the iterated Frobenius push-forwards of L. We will consider the problem in the following

two cases, although the present situation in (2) is far from satisfactory.

(1) two-dimensional normal graded rings (a joint work with Ryo Ohkawa)

(2) the anti-canonical ring of a quintic del Pezzo surface

We are motivated by a question arising from commutative algebra, asking what kind of

graded rings in positive characteristic p have finite F-representation type. In geometric

setting, this is related to the problem to looking out for Frobenius summands. Namely,

given aline bundle L on a projective variety X, we want to know how many and what

kind of indecomposable direct summands appear in the direct sum decomposition of

the iterated Frobenius push-forwards of L. We will consider the problem in the following

two cases, although the present situation in (2) is far from satisfactory.

(1) two-dimensional normal graded rings (a joint work with Ryo Ohkawa)

(2) the anti-canonical ring of a quintic del Pezzo surface

#### Tuesday Seminar on Topology

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fixed points for group actions on non-positively curved spaces (JAPANESE)

**Motoko Kato**(The University of Tokyo)Fixed points for group actions on non-positively curved spaces (JAPANESE)

[ Abstract ]

In this talk, we introduce a fixed point property of groups which is a broad generalization of Serre's property FA, and give a criterion for groups to have such a property. We also apply the criterion to show that various generalizations of Thompson's group T have fixed points whenever they act on finite dimensional non-positively curved metric spaces, including CAT(0) spaces. Since Thompson's group T is known to have fixed point free actions on infinite dimensional CAT(0) spaces, it follows that there is a group which acts on infinite dimensional CAT(0) spaces without global fixed points, but not on finite dimensional ones.

In this talk, we introduce a fixed point property of groups which is a broad generalization of Serre's property FA, and give a criterion for groups to have such a property. We also apply the criterion to show that various generalizations of Thompson's group T have fixed points whenever they act on finite dimensional non-positively curved metric spaces, including CAT(0) spaces. Since Thompson's group T is known to have fixed point free actions on infinite dimensional CAT(0) spaces, it follows that there is a group which acts on infinite dimensional CAT(0) spaces without global fixed points, but not on finite dimensional ones.

### 2018/11/26

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

DGA-Models of variations of mixed Hodge structures (JAPANESE)

**Hisashi Kasuya**(Osaka University)DGA-Models of variations of mixed Hodge structures (JAPANESE)

### 2018/11/21

#### Number Theory Seminar

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Poncelet games, confinement of algebraic integers, and hyperbolic Ax-Schanuel (ENGLISH)

**Yves André**(Université Pierre et Marie Curie)Poncelet games, confinement of algebraic integers, and hyperbolic Ax-Schanuel (ENGLISH)

[ Abstract ]

We shall theorize and exemplify the problem of torsion values of sections of abelian schemes. This « unlikely intersection problem », which arises in various diophantine and algebro-geometric contexts, can be reformulated in a non-trivial way in terms of Kodaira-Spencer maps. A key tool toward its general solution is then provided by recent theorems of Ax-Schanuel type (joint work with P. Corvaja, U. Zannier, and partly Z. Gao).

We shall theorize and exemplify the problem of torsion values of sections of abelian schemes. This « unlikely intersection problem », which arises in various diophantine and algebro-geometric contexts, can be reformulated in a non-trivial way in terms of Kodaira-Spencer maps. A key tool toward its general solution is then provided by recent theorems of Ax-Schanuel type (joint work with P. Corvaja, U. Zannier, and partly Z. Gao).

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Orbit equivalence classes for free actions of free products of infinite abelian groups

**Takaaki Moriyama**(the University of Tokyo)Orbit equivalence classes for free actions of free products of infinite abelian groups

### 2018/11/20

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Artin-Mazur height, Yobuko height and

Hodge-Wittt cohomologies

**Nakkajima Yukiyoshi**(Tokyo Denki University)Artin-Mazur height, Yobuko height and

Hodge-Wittt cohomologies

[ Abstract ]

A few years ago Yobuko has introduced the notion of

a delicate invariant for a proper smooth scheme over a perfect field $k$

of finite characteristic. (We call this invariant Yobuko height.)

This generalize the notion of the F-splitness due to Mehta-Srinivas.

In this talk we give relations between Artin-Mazur heights

and Yobuko heights. We also give a finiteness result on

Hodge-Witt cohomologies of a proper smooth scheme $X$ over $k$

with finite Yobuko height. If time permits, we give a cofinite type result on

the $p$-primary torsion part of Chow group of of $X$

of codimension 2 if $\dim X=3$.

A few years ago Yobuko has introduced the notion of

a delicate invariant for a proper smooth scheme over a perfect field $k$

of finite characteristic. (We call this invariant Yobuko height.)

This generalize the notion of the F-splitness due to Mehta-Srinivas.

In this talk we give relations between Artin-Mazur heights

and Yobuko heights. We also give a finiteness result on

Hodge-Witt cohomologies of a proper smooth scheme $X$ over $k$

with finite Yobuko height. If time permits, we give a cofinite type result on

the $p$-primary torsion part of Chow group of of $X$

of codimension 2 if $\dim X=3$.

#### Discrete mathematical modelling seminar

15:00-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Torelli group, Johnson kernel and invariants of homology 3-spheres (JAPANESE)

**Takuya Sakasai**(The University of Tokyo)Torelli group, Johnson kernel and invariants of homology 3-spheres (JAPANESE)

[ Abstract ]

There are two filtrations of the Torelli group: One is the lower central series and the other is the Johnson filtration. They are closely related to Johnson homomorphisms as well as finite type invariants of homology 3-spheres. We compare the associated graded Lie algebras of the filtrations and report our explicit computational results. Then we discuss some applications of our computations. In particular, we give an explicit description of the rational abelianization of the Johnson kernel. This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

There are two filtrations of the Torelli group: One is the lower central series and the other is the Johnson filtration. They are closely related to Johnson homomorphisms as well as finite type invariants of homology 3-spheres. We compare the associated graded Lie algebras of the filtrations and report our explicit computational results. Then we discuss some applications of our computations. In particular, we give an explicit description of the rational abelianization of the Johnson kernel. This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

### 2018/11/19

#### Tokyo Probability Seminar

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Two-dimensional stochastic interface growth (ENGLISH)

http://math.univ-lyon1.fr/~toninelli/

**Fabio Toninelli**(University Lyon 1)Two-dimensional stochastic interface growth (ENGLISH)

[ Abstract ]

I will discuss stochastic growth of two-dimensional, discrete interfaces, especially models in the so-called Anisotropic KPZ (AKPZ) class, that has the same large-scale behavior as the Stochastic Heat equation with additive noise. I will focus in particular on: 1) the relation between AKPZ exponents, convexity properties of the speed of growth and the preservation of the Gibbs property; and 2) the relation between singularities of the speed of growth and the occurrence of "smooth" (i.e. non-rough) stationary states.

[ Reference URL ]I will discuss stochastic growth of two-dimensional, discrete interfaces, especially models in the so-called Anisotropic KPZ (AKPZ) class, that has the same large-scale behavior as the Stochastic Heat equation with additive noise. I will focus in particular on: 1) the relation between AKPZ exponents, convexity properties of the speed of growth and the preservation of the Gibbs property; and 2) the relation between singularities of the speed of growth and the occurrence of "smooth" (i.e. non-rough) stationary states.

http://math.univ-lyon1.fr/~toninelli/

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

BCOV invariants of Calabi-Yau varieties (ENGLISH)

**Gerard Freixas i Montplet**(Centre National de la Recherche Scientifique)BCOV invariants of Calabi-Yau varieties (ENGLISH)

[ Abstract ]

The BCOV invariant of Calabi-Yau threefolds was introduced by Fang-Lu-Yoshikawa, themselves inspired by physicists Bershadsky-Cecotti-Ooguri-Vafa. It is a real number, obtained from a combination of holomorphic analytic torsion, and suitably normalized so that it only depends on the complex structure of the threefold. It is conjecturaly expected to encode genus 1 Gromov-Witten invariants of a mirror Calabi-Yau threefold. In order to confirm this prediction for a remarkable example, Fang-Lu-Yoshikawa studied the asymptotic behavior for degenerating families of Calabi-Yau threefolds acquiring at most ordinary double point (ODP) singularities. Their methods rely on former results by Yoshikawa on the singularities of Quillen metrics, together with more classical arguments in the theory of degenerations of Hodge structures and Hodge metrics. In this talk I will present joint work with Dennis Eriksson (Chalmers) and Christophe Mourougane (Rennes), where we extend the construction of the BCOV invariant to any dimension and we give precise asymptotic formulas for degenerating families of Calabi-Yau manifolds. Under several hypothesis on the geometry of the singularities acquired, our general formulas drastically simplify and prove some conjectures or predictions in the literature (Liu-Xia for semi-stable minimal families in dimension 3, Klemm-Pandharipande for ODP singularities in dimension 4, etc.). For this, we slightly improve Yoshikawa's results on the singularities of Quillen metrics, and we also provide a complement to Schmid's asymptotics of Hodge metrics when the monodromy transformations are non-unipotent.

The BCOV invariant of Calabi-Yau threefolds was introduced by Fang-Lu-Yoshikawa, themselves inspired by physicists Bershadsky-Cecotti-Ooguri-Vafa. It is a real number, obtained from a combination of holomorphic analytic torsion, and suitably normalized so that it only depends on the complex structure of the threefold. It is conjecturaly expected to encode genus 1 Gromov-Witten invariants of a mirror Calabi-Yau threefold. In order to confirm this prediction for a remarkable example, Fang-Lu-Yoshikawa studied the asymptotic behavior for degenerating families of Calabi-Yau threefolds acquiring at most ordinary double point (ODP) singularities. Their methods rely on former results by Yoshikawa on the singularities of Quillen metrics, together with more classical arguments in the theory of degenerations of Hodge structures and Hodge metrics. In this talk I will present joint work with Dennis Eriksson (Chalmers) and Christophe Mourougane (Rennes), where we extend the construction of the BCOV invariant to any dimension and we give precise asymptotic formulas for degenerating families of Calabi-Yau manifolds. Under several hypothesis on the geometry of the singularities acquired, our general formulas drastically simplify and prove some conjectures or predictions in the literature (Liu-Xia for semi-stable minimal families in dimension 3, Klemm-Pandharipande for ODP singularities in dimension 4, etc.). For this, we slightly improve Yoshikawa's results on the singularities of Quillen metrics, and we also provide a complement to Schmid's asymptotics of Hodge metrics when the monodromy transformations are non-unipotent.

#### Discrete mathematical modelling seminar

17:15-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Integrability for four-dimensional recurrence relations

**Dinh T. Tran**(School of Mathematics and Statistics, The University of Sydney)Integrability for four-dimensional recurrence relations

[ Abstract ]

In this talk, we study some fourth-order recurrence relations (or mappings). These recurrence relations were obtained by assuming that they possess two polynomial integrals with certain degree patterns.

For mappings with cubic growth, we will reduce them to three-dimensional ones using a procedure called deflation. These three-dimensional maps have two integrals; therefore, they are integrable in the sense of Liouville-Arnold. Furthermore, we can reduce the obtained three-dimensional maps to two-dimensional maps of Quispel-Roberts-Thompsons (QRT) type. On the other hand, for recurrences with quadratic growth and two integrals, we will show that they are integrable in the sense of Liouville-Arnold by providing their Poisson brackets. Non-degenerate Poisson brackets can be found by using the existence of discrete Lagrangians and a discrete analogue of the Ostrogradsky transformation.

This is joint work with G. Gubbiotti, N. Joshi, and C-M. Viallet.

In this talk, we study some fourth-order recurrence relations (or mappings). These recurrence relations were obtained by assuming that they possess two polynomial integrals with certain degree patterns.

For mappings with cubic growth, we will reduce them to three-dimensional ones using a procedure called deflation. These three-dimensional maps have two integrals; therefore, they are integrable in the sense of Liouville-Arnold. Furthermore, we can reduce the obtained three-dimensional maps to two-dimensional maps of Quispel-Roberts-Thompsons (QRT) type. On the other hand, for recurrences with quadratic growth and two integrals, we will show that they are integrable in the sense of Liouville-Arnold by providing their Poisson brackets. Non-degenerate Poisson brackets can be found by using the existence of discrete Lagrangians and a discrete analogue of the Ostrogradsky transformation.

This is joint work with G. Gubbiotti, N. Joshi, and C-M. Viallet.

### 2018/11/15

#### Applied Analysis

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)

**Nakao Hayashi**(Osaka University)Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)

[ Abstract ]

We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.

We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.

### 2018/11/14

#### Number Theory Seminar

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

A motivic construction of ramification filtrations (ENGLISH)

**Shuji Saito**(University of Tokyo)A motivic construction of ramification filtrations (ENGLISH)

[ Abstract ]

We give a new interpretation of Artin conductors of characters in the framework of theory of motives with modulus. It gives a unified way to understand Artin conductors of characters and irregularities of line bundle with integrable connections as well as overconvergent F-isocrystals of rank 1. It also gives rise to new conductors, for example, for G-torsors with G a finite flat group scheme, which specializes to the classical Artin conductor in case G = Z/nZ. We also give a motivic proof of a theorem of Kato and Matsuda on the determination of Artin conductors along divisors on smooth schemes by its restrictions to curves. Its proof is based on a motivic version of a theorem of Gabber-Katz. This is a joint work with Kay Rülling.

We give a new interpretation of Artin conductors of characters in the framework of theory of motives with modulus. It gives a unified way to understand Artin conductors of characters and irregularities of line bundle with integrable connections as well as overconvergent F-isocrystals of rank 1. It also gives rise to new conductors, for example, for G-torsors with G a finite flat group scheme, which specializes to the classical Artin conductor in case G = Z/nZ. We also give a motivic proof of a theorem of Kato and Matsuda on the determination of Artin conductors along divisors on smooth schemes by its restrictions to curves. Its proof is based on a motivic version of a theorem of Gabber-Katz. This is a joint work with Kay Rülling.

### 2018/11/13

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below (English)

**Weichung Chen**(Tokyo)Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below (English)

[ Abstract ]

We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from 0 and the coefficients of the boundaries belong to the set of hyperstandard multiplicities Φ(R) associated to a fixed finite set R form a bounded family. We also show α(X, B)d−1vol(−(KX + B)) are bounded from above for all klt weak Fano pairs (X, B) of a fixed dimension d.

We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from 0 and the coefficients of the boundaries belong to the set of hyperstandard multiplicities Φ(R) associated to a fixed finite set R form a bounded family. We also show α(X, B)d−1vol(−(KX + B)) are bounded from above for all klt weak Fano pairs (X, B) of a fixed dimension d.

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On continuity of drifts of the mapping class group (JAPANESE)

**Hidetoshi Masai**(Tokyo Institute of Technology)On continuity of drifts of the mapping class group (JAPANESE)

[ Abstract ]

When a group is acting on a space isometrically, we may consider the "translation distance" of random walks, which is called the drift of the random walk. In this talk we consider mapping class group acting on the Teichmüller space. We first recall several characterizations of the drift. The drift is determined by the transition probability of the random walk. The goal of this talk is to show that the drift varies continuously with the transition probability measure.

When a group is acting on a space isometrically, we may consider the "translation distance" of random walks, which is called the drift of the random walk. In this talk we consider mapping class group acting on the Teichmüller space. We first recall several characterizations of the drift. The drift is determined by the transition probability of the random walk. The goal of this talk is to show that the drift varies continuously with the transition probability measure.

### 2018/11/12

#### Tokyo Probability Seminar

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Random walk at weak and strong disorder (ENGLISH)

http://www.mat.uc.cl/~aramirez/

**Alejandro Ramirez**(Pontificia Universidad Catolica de Chile)Random walk at weak and strong disorder (ENGLISH)

[ Abstract ]

We consider random walks at low disorder on $\mathbb Z^d$. For dimensions $d\ge 4$, we exhibit a phase transition on the strength of the disorder expressed as an equality between the quenched and annealed rate functions. In dimension $d=2$ we exhibit a universal scaling limit to the stochastic heat equation. This talk is based on joint works with Bazaes, Mukherjee and Saglietti, and with Moreno and Quastel.

[ Reference URL ]We consider random walks at low disorder on $\mathbb Z^d$. For dimensions $d\ge 4$, we exhibit a phase transition on the strength of the disorder expressed as an equality between the quenched and annealed rate functions. In dimension $d=2$ we exhibit a universal scaling limit to the stochastic heat equation. This talk is based on joint works with Bazaes, Mukherjee and Saglietti, and with Moreno and Quastel.

http://www.mat.uc.cl/~aramirez/

### 2018/11/09

#### Seminar on Probability and Statistics

11:00-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Market impact and option hedging in the presence of liquidity costs

**Frédéric Abergel**(CentraleSupélec)Market impact and option hedging in the presence of liquidity costs

[ Abstract ]

The phenomenon of market (or: price) impact is well-known among practicioners, and it has long been recognized as a key feature of trading in electronic markets. In the first part of this talk, I will present some new, recent results on market impact, especially for limit orders. I will then propose a theory for option hedging in the presence of liquidity costs.(Based on joint works with E. Saïd, G. Loeper).

The phenomenon of market (or: price) impact is well-known among practicioners, and it has long been recognized as a key feature of trading in electronic markets. In the first part of this talk, I will present some new, recent results on market impact, especially for limit orders. I will then propose a theory for option hedging in the presence of liquidity costs.(Based on joint works with E. Saïd, G. Loeper).

### 2018/11/08

#### Tuesday Seminar on Topology

10:30-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Deformations of diagonal representations of knot groups into $\mathrm{SL}(n,\mathbb{C})$ (ENGLISH)

**Michael Heusener**(Université Clermont Auvergne)Deformations of diagonal representations of knot groups into $\mathrm{SL}(n,\mathbb{C})$ (ENGLISH)

[ Abstract ]

This is joint work with Leila Ben Abdelghani, Monastir (Tunisia).

Given a manifold $M$, the variety of representations of $\pi_1(M)$ into $\mathrm{SL}(2,\mathbb{C})$ and the variety of characters of such representations both contain information of the topology of $M$. Since the foundational work of W.P. Thurston and Culler & Shalen, the varieties of $\mathrm{SL}(2,\mathbb{C})$-characters have been extensively studied. This is specially interesting for $3$-dimensional manifolds, where the fundamental group and the geometrical properties of the manifold are strongly related.

However, much less is known of the character varieties for other groups, notably for $\mathrm{SL}(n,\mathbb{C})$ with $n\geq 3$. The $\mathrm{SL}(n,\mathbb{C})$-character varieties for free groups have been studied by S. Lawton and P. Will, and the $\mathrm{SL}(3,\mathbb{C})$-character variety of torus knot groups has been determined by V. Munoz and J. Porti.

In this talk I will present some results concerning the deformations of diagonal representations of knot groups in basic notations and some recent results concerning the representation and character varieties of $3$-manifold groups and in particular knot groups. In particular, we are interested in the local structure of the $\mathrm{SL}(n,\mathbb{C})$-representation variety at the diagonal representation.

This is joint work with Leila Ben Abdelghani, Monastir (Tunisia).

Given a manifold $M$, the variety of representations of $\pi_1(M)$ into $\mathrm{SL}(2,\mathbb{C})$ and the variety of characters of such representations both contain information of the topology of $M$. Since the foundational work of W.P. Thurston and Culler & Shalen, the varieties of $\mathrm{SL}(2,\mathbb{C})$-characters have been extensively studied. This is specially interesting for $3$-dimensional manifolds, where the fundamental group and the geometrical properties of the manifold are strongly related.

However, much less is known of the character varieties for other groups, notably for $\mathrm{SL}(n,\mathbb{C})$ with $n\geq 3$. The $\mathrm{SL}(n,\mathbb{C})$-character varieties for free groups have been studied by S. Lawton and P. Will, and the $\mathrm{SL}(3,\mathbb{C})$-character variety of torus knot groups has been determined by V. Munoz and J. Porti.

In this talk I will present some results concerning the deformations of diagonal representations of knot groups in basic notations and some recent results concerning the representation and character varieties of $3$-manifold groups and in particular knot groups. In particular, we are interested in the local structure of the $\mathrm{SL}(n,\mathbb{C})$-representation variety at the diagonal representation.

### 2018/11/06

#### Tuesday Seminar of Analysis

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Global behavior of bifurcation curves and related topics (日本語)

**SHIBATA Tetsutaro**(Hiroshima University)Global behavior of bifurcation curves and related topics (日本語)

[ Abstract ]

In this talk, we consider the asymptotic behavior of bifurcation curves for ODE with oscillatory nonlinear term. First, we study the global and local behavior of oscillatory bifurcation curves. We also consider the bifurcation problems with nonlinear diffusion.

In this talk, we consider the asymptotic behavior of bifurcation curves for ODE with oscillatory nonlinear term. First, we study the global and local behavior of oscillatory bifurcation curves. We also consider the bifurcation problems with nonlinear diffusion.

#### Tuesday Seminar on Topology

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Coarsely convex spaces and a coarse Cartan-Hadamard theorem (JAPANESE)

**Shin-ichi Oguni**(Ehime University)Coarsely convex spaces and a coarse Cartan-Hadamard theorem (JAPANESE)

[ Abstract ]

A coarse version of negatively-curved spaces have been very well studied as Gromov hyperbolic spaces. Recently we introduced a coarse version of non-positively curved spaces, named them coarsely convex spaces and showed a coarse version of the Cartan-Hadamard theorem for such spaces in a joint-work with Tomohiro Fukaya (arXiv:1705.05588). Based on the work, I introduce coarsely convex spaces and explain a coarse Cartan-Hadamard theorem, ideas for proof and its applications to differential topology.

A coarse version of negatively-curved spaces have been very well studied as Gromov hyperbolic spaces. Recently we introduced a coarse version of non-positively curved spaces, named them coarsely convex spaces and showed a coarse version of the Cartan-Hadamard theorem for such spaces in a joint-work with Tomohiro Fukaya (arXiv:1705.05588). Based on the work, I introduce coarsely convex spaces and explain a coarse Cartan-Hadamard theorem, ideas for proof and its applications to differential topology.

### 2018/11/05

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On the quasiconformal equivalence of Dynamical Cantor sets (JAPANESE)

**Hiroshige Shiga**(Tokyo Institute of Technology)On the quasiconformal equivalence of Dynamical Cantor sets (JAPANESE)

[ Abstract ]

Let $E$ be a Cantor set in the Riemann sphere $\widehat{\mathbb C}$, that is, a totally disconnected perfect set in $\widehat{\mathbb C}$.

The standard middle one-thirds Cantor set $\mathcal{C}$ is a typical example.

We consider the complement $X_{E}:=\widehat{\mathbb C}\setminus E$ of the Cantor set $E$.

It is an open Riemann surface with uncountable many boundary components.

We are interested in the quasiconformal equivalence of such surfaces.

In this talk, we discuss the quasiconformal equivalence for the complements of Cantor sets given by dynamical systems.

Let $E$ be a Cantor set in the Riemann sphere $\widehat{\mathbb C}$, that is, a totally disconnected perfect set in $\widehat{\mathbb C}$.

The standard middle one-thirds Cantor set $\mathcal{C}$ is a typical example.

We consider the complement $X_{E}:=\widehat{\mathbb C}\setminus E$ of the Cantor set $E$.

It is an open Riemann surface with uncountable many boundary components.

We are interested in the quasiconformal equivalence of such surfaces.

In this talk, we discuss the quasiconformal equivalence for the complements of Cantor sets given by dynamical systems.

#### Numerical Analysis Seminar

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Tosio Kato as an applied mathematician (Japanese)

**Hisashi Okamoto**(Gakushuin University)Tosio Kato as an applied mathematician (Japanese)

[ Abstract ]

Tosio Kato (1917-1999) is nowadays considered to be a rigorous analyst or theorist. Many people consider his contributions in quantum mechanics to be epoch-making, his work on nonlinear partial differential equations elegant and inspiring. However, around the time when he visited USA for the first time in 1954, he was studying problems of applied mathematics, too, notably numerical computation of eigenvalues. I wish to shed light on the historical background of his study of applied mathematics. This is a joint work with Prof. Hiroshi Fujita.

Tosio Kato (1917-1999) is nowadays considered to be a rigorous analyst or theorist. Many people consider his contributions in quantum mechanics to be epoch-making, his work on nonlinear partial differential equations elegant and inspiring. However, around the time when he visited USA for the first time in 1954, he was studying problems of applied mathematics, too, notably numerical computation of eigenvalues. I wish to shed light on the historical background of his study of applied mathematics. This is a joint work with Prof. Hiroshi Fujita.

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