## Seminar information archive

Seminar information archive ～06/17｜Today's seminar 06/18 | Future seminars 06/19～

#### Lie Groups and Representation Theory

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

A Pieri-type formula and a factorization formula for K-k-Schur functions

**Motoki Takigiku**(the University of Tokyo)A Pieri-type formula and a factorization formula for K-k-Schur functions

[ Abstract ]

We give a Pieri-type formula for the sum of K-k-Schur functions \sum_{\mu\le\lambda}g^{(k)}_{\mu} over a principal order ideal of the poset of k-bounded partitions under the strong Bruhat order, which sum we denote by \widetilde{g}^{(k)}_{\lambda}. As an application of this, we also give a k-rectangle factorization formula \widetilde{g}^{(k)}_{R_t\cup\lambda}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_{\lambda}

where R_t=(t^{k+1-t}), analogous to that of k-Schur functions s^{(k)}_{R_t\cup \lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}.

We give a Pieri-type formula for the sum of K-k-Schur functions \sum_{\mu\le\lambda}g^{(k)}_{\mu} over a principal order ideal of the poset of k-bounded partitions under the strong Bruhat order, which sum we denote by \widetilde{g}^{(k)}_{\lambda}. As an application of this, we also give a k-rectangle factorization formula \widetilde{g}^{(k)}_{R_t\cup\lambda}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_{\lambda}

where R_t=(t^{k+1-t}), analogous to that of k-Schur functions s^{(k)}_{R_t\cup \lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}.

### 2018/12/10

#### Tokyo Probability Seminar

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

Random polymer models and classical groups (ENGLISH)

https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/zygouras/

**Nikolaos Zygouras**(University of Warwick)Random polymer models and classical groups (ENGLISH)

[ Abstract ]

The relation between polymer models at zero temperature and characters of the general linear group GL_n(R) has been known since the first breakthroughs in the field around the KPZ universality through the works of Johansson, Baik, Rains, Okounkov and others. Later on, geometric liftings of the GL_n(R) characters appeared in the study of positive temperature polymer models in the form of GL_n(R)-Whittaker functions. In this talk I will describe joint works with E. Bisi where we have established that Whittaker functions associated to the orthogonal group SO_{2n+1}(R) can be used to describe laws of positive temperature polymers when their end point is free to lie on a line. Going back to zero temperature, we will also see that characters of other classical groups such as SO_{2n+1}(R); Sp_{2n}(R); SO_{2n}(R) do play a role in describing laws of polymers in various geometries. This occurence might be surprising given the length of time these models have been studied.

[ Reference URL ]The relation between polymer models at zero temperature and characters of the general linear group GL_n(R) has been known since the first breakthroughs in the field around the KPZ universality through the works of Johansson, Baik, Rains, Okounkov and others. Later on, geometric liftings of the GL_n(R) characters appeared in the study of positive temperature polymer models in the form of GL_n(R)-Whittaker functions. In this talk I will describe joint works with E. Bisi where we have established that Whittaker functions associated to the orthogonal group SO_{2n+1}(R) can be used to describe laws of positive temperature polymers when their end point is free to lie on a line. Going back to zero temperature, we will also see that characters of other classical groups such as SO_{2n+1}(R); Sp_{2n}(R); SO_{2n}(R) do play a role in describing laws of polymers in various geometries. This occurence might be surprising given the length of time these models have been studied.

https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/zygouras/

### 2018/12/07

#### Operator Algebra Seminars

15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)### 2018/12/06

#### Infinite Analysis Seminar Tokyo

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Integrability and TBA in non-equilibrium emergent hydrodynamics (ENGLISH)

**Francesco Ravanini**(University of Bologna)Integrability and TBA in non-equilibrium emergent hydrodynamics (ENGLISH)

[ Abstract ]

The paradigm of investigating non-equilibrium phenomena by considering stationary states of emergent hydrodynamics has attracted a lot of attention in the last years. Recent proposals of an exact approach in integrable cases, making use of TBA techniques, are presented and discussed.

The paradigm of investigating non-equilibrium phenomena by considering stationary states of emergent hydrodynamics has attracted a lot of attention in the last years. Recent proposals of an exact approach in integrable cases, making use of TBA techniques, are presented and discussed.

#### Operator Algebra Seminars

15:00-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)### 2018/12/05

#### Operator Algebra Seminars

17:15-18:45 Room #126 (Graduate School of Math. Sci. Bldg.)

The Gromov-Hausdorff Propinquity

**Frederic Latremoliere**(Univ. Denver)The Gromov-Hausdorff Propinquity

#### Operator Algebra Seminars

15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)#### Seminar on Probability and Statistics

13:00-15:00 Room #156 (Graduate School of Math. Sci. Bldg.)

Lecture 2:Representation results for the Gaussian processes. Financial applications of fractional Brownian motion

**Yuliia Mishura**(The Taras Shevchenko National University of Kiev)Lecture 2:Representation results for the Gaussian processes. Financial applications of fractional Brownian motion

[ Abstract ]

Arbitrage with fBm: why it appears. How to present any contingent claim via self-financing strategy on the financial market involving fBm. Absence of arbitrage for the mixed models. Fractional -Uhlenbeck and fractional Cox-Ingersoll-Ross processes as the models for stochastic volatility.

Arbitrage with fBm: why it appears. How to present any contingent claim via self-financing strategy on the financial market involving fBm. Absence of arbitrage for the mixed models. Fractional -Uhlenbeck and fractional Cox-Ingersoll-Ross processes as the models for stochastic volatility.

#### Seminar on Probability and Statistics

15:00-17:00 Room #156 (Graduate School of Math. Sci. Bldg.)

Lecture 3:Statistical parameter estimation for the diffusion processes and in the models involving fBm

**Yuliia Mishura**(The Taras Shevchenko National University of Kiev)Lecture 3:Statistical parameter estimation for the diffusion processes and in the models involving fBm

[ Abstract ]

Drift parameter estimation in the standard diffusion model and its strong consistency. Hurst and drift parameter estimation in the models involving fBm and in the mixed models. Asymptotic properties. Estimation of the diffusion parameter.

Drift parameter estimation in the standard diffusion model and its strong consistency. Hurst and drift parameter estimation in the models involving fBm and in the mixed models. Asymptotic properties. Estimation of the diffusion parameter.

### 2018/12/04

#### Tuesday Seminar on Topology

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

Slopes and concordance of links (ENGLISH)

**Vincent Florens**(Université de Pau et des Pays de l'Adour)Slopes and concordance of links (ENGLISH)

[ Abstract ]

We define the slope of a link associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima-Yamasaki η-function. It is the ratio of two Conway potentials, provided that the latter makes sense; otherwise, it is a new invariant. We present several examples and discuss the invariance by concordance. Joint with A. Degtyarev and A. Lecuona.

We define the slope of a link associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima-Yamasaki η-function. It is the ratio of two Conway potentials, provided that the latter makes sense; otherwise, it is a new invariant. We present several examples and discuss the invariance by concordance. Joint with A. Degtyarev and A. Lecuona.

#### Operator Algebra Seminars

15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)#### Seminar on Probability and Statistics

15:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Lecture 1: Elements of fractional calculus

How to connect the fractional Brownian motion to the Wiener process. Stochastic integration w.r.t. fBm and stochastic differential equations involving fB

**Yuliia Mishura**(The Taras Shevchenko National University of Kiev)Lecture 1: Elements of fractional calculus

How to connect the fractional Brownian motion to the Wiener process. Stochastic integration w.r.t. fBm and stochastic differential equations involving fB

[ Abstract ]

Fractional integrals and fractional derivatives. Wiener and stochastic integration w.r.t. the fractional Brownian motion. Representations of fBm via a Wiener process and vice versa. Elements of the fractional stochastic calculus. Stochastic differential equations involving fBm: existence, uniqueness, properties of the solutions. Simplest models: fractional Ornstein-Uhlenbeck and fractional Cox-Ingersoll-Ross processes.

Fractional integrals and fractional derivatives. Wiener and stochastic integration w.r.t. the fractional Brownian motion. Representations of fBm via a Wiener process and vice versa. Elements of the fractional stochastic calculus. Stochastic differential equations involving fBm: existence, uniqueness, properties of the solutions. Simplest models: fractional Ornstein-Uhlenbeck and fractional Cox-Ingersoll-Ross processes.

### 2018/12/03

#### Operator Algebra Seminars

15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)#### Seminar on Geometric Complex Analysis

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

Variational theories of complex analysis of several variables (JAPANESE)

**Genki Hosono**(University of Tokyo)Variational theories of complex analysis of several variables (JAPANESE)

[ Abstract ]

In complex analysis, there are some values and functions which are subharmonic under pseudoconvex variations.

For example, the variation of Robin constant (Yamaguchi) and of Bergman kernels (Maitani-Yamaguchi) were studied.

As a generalization, the curvature positivity of spaces of $L^2$ holomorphic functions is proved by Berndtsson.

These theories are known to have some relations with $L^2$ extension theorems.

In this talk, I will explain known results and discuss the variation problem of the Azukawa pseudometric, which is a generalization of the Robin constant.

In complex analysis, there are some values and functions which are subharmonic under pseudoconvex variations.

For example, the variation of Robin constant (Yamaguchi) and of Bergman kernels (Maitani-Yamaguchi) were studied.

As a generalization, the curvature positivity of spaces of $L^2$ holomorphic functions is proved by Berndtsson.

These theories are known to have some relations with $L^2$ extension theorems.

In this talk, I will explain known results and discuss the variation problem of the Azukawa pseudometric, which is a generalization of the Robin constant.

#### 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.

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