## Seminar information archive

Seminar information archive ～02/19｜Today's seminar 02/20 | Future seminars 02/21～

### 2017/01/27

#### Algebraic Geometry Seminar

14:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On the number and boundedness of minimal models of general type (English)

Adjoint dimension of foliations (English)

https://www.dpmms.cam.ac.uk/~rs872/

**Luca Tasin**(Roma Tre University) 14:00-15:30On the number and boundedness of minimal models of general type (English)

[ Abstract ]

In this talk I will explain that the number of minimal models yields a constructible function on the base of any family of varieties of general type. From this it follows that the number of minimal models of a variety of general type can be bounded in terms of its volume. I will also show that in any dimension minimal models of general type and bounded volume form a bounded family. This is based on a joint work with D. Martinelli and S. Schreieder.

In this talk I will explain that the number of minimal models yields a constructible function on the base of any family of varieties of general type. From this it follows that the number of minimal models of a variety of general type can be bounded in terms of its volume. I will also show that in any dimension minimal models of general type and bounded volume form a bounded family. This is based on a joint work with D. Martinelli and S. Schreieder.

**Roberto Svaldi**(University of Cambridge/SISSA) 16:00-17:30Adjoint dimension of foliations (English)

[ Abstract ]

The classification of foliated surfaces by Brunella, McQuillan and Mendes carries many similarities with Enriques-Kodaira classification of surfaces but also many important differences. I will discuss an alternative classification scheme where the role of differential forms along the leaves is replaced by differential forms along the leaves with values in fractional powers of the conormal bundle of the foliation. In this alternative setup one obtains a classification of foliated surfaces closer to the usual Enriques-Kodaira classification. If time permits, I will show how to apply this alternative classification to describe the Zariski closure of the set foliations which admit rational first integral of bounded genus in families of foliated surfaces. Joint work with Jorge Vitorio Pereira.

[ Reference URL ]The classification of foliated surfaces by Brunella, McQuillan and Mendes carries many similarities with Enriques-Kodaira classification of surfaces but also many important differences. I will discuss an alternative classification scheme where the role of differential forms along the leaves is replaced by differential forms along the leaves with values in fractional powers of the conormal bundle of the foliation. In this alternative setup one obtains a classification of foliated surfaces closer to the usual Enriques-Kodaira classification. If time permits, I will show how to apply this alternative classification to describe the Zariski closure of the set foliations which admit rational first integral of bounded genus in families of foliated surfaces. Joint work with Jorge Vitorio Pereira.

https://www.dpmms.cam.ac.uk/~rs872/

### 2017/01/26

#### Seminar on Probability and Statistics

13:00-16:00 Room #052 (Graduate School of Math. Sci. Bldg.)

High-frequency financial data : trades and quotes databases, order flows and time resolution I, II, III

**Ioane Muni Toke**(Centrale Supelec Paris)High-frequency financial data : trades and quotes databases, order flows and time resolution I, II, III

[ Abstract ]

I present some of the challenges associated with preparing high-frequency trades and quotes databases for statistics purposes. In a first part, I investigate TRTH tick-by-tick data on three exchanges (Paris, London and Frankfurt) and on a five-year span. I analyse the performances of a procedure of reconstruction of orders flows. This turns out to be a forensic tool assessing the quality of the database: significant technical changes affecting the exchanges are tracked through the data. Moreover, the choices made when reconstructing order flows may have consequences on the quantitative models that are calibrated afterwards on such data. I also provide a refined look at the Lee–Ready procedure and its optimal lags. Findings are in line with both financial reasoning and the analysis of an illustrative Poisson model. In a second part, I investigate Nikkei-packaged Tokyo-traded ETF data. The application the order flow reconstruction procedure underlines the differences between the TRTH and Nikkei data. In a brief last part, we will discuss the time resolution of these databases and the potential problems arising when modelling a limit order book with simple point processes.

I present some of the challenges associated with preparing high-frequency trades and quotes databases for statistics purposes. In a first part, I investigate TRTH tick-by-tick data on three exchanges (Paris, London and Frankfurt) and on a five-year span. I analyse the performances of a procedure of reconstruction of orders flows. This turns out to be a forensic tool assessing the quality of the database: significant technical changes affecting the exchanges are tracked through the data. Moreover, the choices made when reconstructing order flows may have consequences on the quantitative models that are calibrated afterwards on such data. I also provide a refined look at the Lee–Ready procedure and its optimal lags. Findings are in line with both financial reasoning and the analysis of an illustrative Poisson model. In a second part, I investigate Nikkei-packaged Tokyo-traded ETF data. The application the order flow reconstruction procedure underlines the differences between the TRTH and Nikkei data. In a brief last part, we will discuss the time resolution of these databases and the potential problems arising when modelling a limit order book with simple point processes.

### 2017/01/24

#### Tuesday Seminar on Topology

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

On the existence of infinitely many non-contractible periodic trajectories in Hamiltonian dynamics on closed symplectic manifolds (JAPANESE)

**Ryuma Orita**(The University of Tokyo)On the existence of infinitely many non-contractible periodic trajectories in Hamiltonian dynamics on closed symplectic manifolds (JAPANESE)

[ Abstract ]

We show that the presence of a non-contractible Hamiltonian one-periodic trajectory in a closed symplectic manifold yields the existence of infinitely many non-contractible periodic trajectories, provided that the symplectic form is aspherical and the fundamental group is virtually abelian. Moreover, we also show that a similar statement holds for closed monotone or negative monotone symplectic manifolds having virtually abelian fundamental groups. These results are certain generalizations of works by Ginzburg and Gurel who proved a similar statement holds for atoroidal or toroidally monotone closed symplectic manifolds. The proof is based on the machinery of filtered Floer--Novikov homology for non-contractible periodic trajectories.

We show that the presence of a non-contractible Hamiltonian one-periodic trajectory in a closed symplectic manifold yields the existence of infinitely many non-contractible periodic trajectories, provided that the symplectic form is aspherical and the fundamental group is virtually abelian. Moreover, we also show that a similar statement holds for closed monotone or negative monotone symplectic manifolds having virtually abelian fundamental groups. These results are certain generalizations of works by Ginzburg and Gurel who proved a similar statement holds for atoroidal or toroidally monotone closed symplectic manifolds. The proof is based on the machinery of filtered Floer--Novikov homology for non-contractible periodic trajectories.

#### Tuesday Seminar on Topology

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

Quantitative shadowing property, shadowable points, and local properties of topological dynamical systems (JAPANESE)

**Noriaki Kawaguchi**(The University of Tokyo)Quantitative shadowing property, shadowable points, and local properties of topological dynamical systems (JAPANESE)

[ Abstract ]

Shadowing property has been one of the key notions in topological hyperbolic dynamics, which is also common since C^0-generic homeomorphisms on a smooth closed manifold satisfy the property for instance. In this talk, the shadowing property in relation to other chaotic or non-chaotic properties of dynamical systems (entropy, sensitivity, equicontinuity, etc.) is discussed. Also, we introduce an idea of localizing and quantifying the shadowing property following the recent work of Morales, and present some of its consequences. The idea is shown to be effective for the description of local properties of dynamical systems.

Shadowing property has been one of the key notions in topological hyperbolic dynamics, which is also common since C^0-generic homeomorphisms on a smooth closed manifold satisfy the property for instance. In this talk, the shadowing property in relation to other chaotic or non-chaotic properties of dynamical systems (entropy, sensitivity, equicontinuity, etc.) is discussed. Also, we introduce an idea of localizing and quantifying the shadowing property following the recent work of Morales, and present some of its consequences. The idea is shown to be effective for the description of local properties of dynamical systems.

### 2017/01/23

#### Seminar on Geometric Complex Analysis

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

A unified proof of Cousin I, II and d-bar equation on domains of holomorphy (JAPANESE)

**Junjiro Noguchi**(The University of Tokyo)A unified proof of Cousin I, II and d-bar equation on domains of holomorphy (JAPANESE)

[ Abstract ]

Oka's J\^oku-Ik\^o says that holomorphic functions on a complex submanifold of a polydisk extend holomorphically to the whole polydisk. By making use of Oka's J\^oku-Ik\^o we give a titled proof with introducing an argument that represents one of the three cases.

The proof is a modification of the cube dimension induction, used in the proof of Oka's Syzygy for coherent sheaves.

Oka's J\^oku-Ik\^o says that holomorphic functions on a complex submanifold of a polydisk extend holomorphically to the whole polydisk. By making use of Oka's J\^oku-Ik\^o we give a titled proof with introducing an argument that represents one of the three cases.

The proof is a modification of the cube dimension induction, used in the proof of Oka's Syzygy for coherent sheaves.

### 2017/01/19

#### Seminar on Probability and Statistics

13:00-15:30 Room #052 (Graduate School of Math. Sci. Bldg.)

Talk 1:Likelihood inference for a continuous time GARCH model

Talk 2:Nonparametric Estimation for Self-Exciting Point Processes: A Parsimonious Approach

**Feng Chen**(University of New South Wales)Talk 1:Likelihood inference for a continuous time GARCH model

Talk 2:Nonparametric Estimation for Self-Exciting Point Processes: A Parsimonious Approach

[ Abstract ]

Talk 1:The continuous time GARCH (COGARCH) model of Kluppelberg, Lindner and Maller (2004) is a natural extension of the discrete time GARCH(1,1) model which preserves important features of the GARCH model in the discrete-time setting. For example, the COGARCH model is driven by a single source of noise as in the discrete time GARCH model, which is a Levy process in the COGARCH case, and both models can produced heavy tailed marginal returns even when the driving noise is light-tailed. However, calibrating the COGARCH model to data is a challenge, especially when observations of the COGARCH process are obtained at irregularly spaced time points. The method of moments has had some success in the case with regularly spaced data, yet it is not clear how to make it work in the more interesting case with irregularly spaced data. As a well-known method of estimation, the maximum likelihood method has not been developed for the COGARCH model, even in the quite simple case with the driving Levy process being compound Poisson, though a quasi-maximum likelihood (QML)method has been proposed. The challenge with the maximum likelihood method in this context is mainly due to the lack of a tractable form for the likelihood. In this talk, we propose a Monte Carlo method to approximate the likelihood of the compound Poisson driven COGARCH model. We evaluate the performance of the resulting maximum likelihood (ML) estimator using simulated data, and illustrate its application with high frequency exchange rate data. (Joint work with Damien Wee and William Dunsmuir).

Talk 2:There is ample evidence that in applications of self-exciting point process (SEPP) models, the intensity of background events is often far from constant. If a constant background is imposed, that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modelling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be

used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis

leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this paper we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam's razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy. Although seldom used for nonparametric function estimation in other settings, this approach is appropriate in the context of SEPP models. (Joint work with the late Peter Hall.)

Talk 1:The continuous time GARCH (COGARCH) model of Kluppelberg, Lindner and Maller (2004) is a natural extension of the discrete time GARCH(1,1) model which preserves important features of the GARCH model in the discrete-time setting. For example, the COGARCH model is driven by a single source of noise as in the discrete time GARCH model, which is a Levy process in the COGARCH case, and both models can produced heavy tailed marginal returns even when the driving noise is light-tailed. However, calibrating the COGARCH model to data is a challenge, especially when observations of the COGARCH process are obtained at irregularly spaced time points. The method of moments has had some success in the case with regularly spaced data, yet it is not clear how to make it work in the more interesting case with irregularly spaced data. As a well-known method of estimation, the maximum likelihood method has not been developed for the COGARCH model, even in the quite simple case with the driving Levy process being compound Poisson, though a quasi-maximum likelihood (QML)method has been proposed. The challenge with the maximum likelihood method in this context is mainly due to the lack of a tractable form for the likelihood. In this talk, we propose a Monte Carlo method to approximate the likelihood of the compound Poisson driven COGARCH model. We evaluate the performance of the resulting maximum likelihood (ML) estimator using simulated data, and illustrate its application with high frequency exchange rate data. (Joint work with Damien Wee and William Dunsmuir).

Talk 2:There is ample evidence that in applications of self-exciting point process (SEPP) models, the intensity of background events is often far from constant. If a constant background is imposed, that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modelling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be

used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis

leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this paper we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam's razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy. Although seldom used for nonparametric function estimation in other settings, this approach is appropriate in the context of SEPP models. (Joint work with the late Peter Hall.)

### 2017/01/17

#### Tuesday Seminar on Topology

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

On an application of the Fukaya categories to the Koszul duality (JAPANESE)

**Satoshi Sugiyama**(The University of Tokyo)On an application of the Fukaya categories to the Koszul duality (JAPANESE)

[ Abstract ]

In this talk, we compute an A

The Koszul duality is originally a duality between certain quadratic algebras called Koszul algebras. In this talk, we are interested in the case when A is not a quadratic algebra, i.e. the case when A is defined as a quotient algebra of tensor algebra devided by higher degree relations.

The definition of Koszul duals for such algebras, A

In this talk, we compute an A

_{∞}-Koszul dual of path algebras with relations over the directed A_{n}-type quivers via the Fukaya categories of exact Riemann surfaces.The Koszul duality is originally a duality between certain quadratic algebras called Koszul algebras. In this talk, we are interested in the case when A is not a quadratic algebra, i.e. the case when A is defined as a quotient algebra of tensor algebra devided by higher degree relations.

The definition of Koszul duals for such algebras, A

_{∞}-Koszul duals, are given by some people, for example, D. M. Lu, J. H. Palmieri, Q. S. Wu, J. J. Zhang. However, the computation for a concrete examples is hard. In this talk, we use the Fukaya categories of exact Riemann surfaces to compute A_{∞}-Koszul duals. Then, we understand the Koszul duality as a duality between higher products and relations.### 2017/01/16

#### Seminar on Probability and Statistics

16:50-18:00 Room #052 (Graduate School of Math. Sci. Bldg.)

Profile likelihood approach to a large sample distribution of estimators in joint mixture model of survival and longitudinal ordered data

**広瀬勇一**(University of Wellington)Profile likelihood approach to a large sample distribution of estimators in joint mixture model of survival and longitudinal ordered data

[ Abstract ]

We consider a semiparametric joint model that consists of item response and survival components, where these two components are linked through latent variables. We estimate the model parameters through a profile likelihood and the EM algorithm. We propose a method to derive an asymptotic variance of the estimators in this model.

We consider a semiparametric joint model that consists of item response and survival components, where these two components are linked through latent variables. We estimate the model parameters through a profile likelihood and the EM algorithm. We propose a method to derive an asymptotic variance of the estimators in this model.

#### Seminar on Geometric Complex Analysis

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

A geometric second main theorem (ENGLISH)

**Dinh Tuan Huynh**(Osaka University)A geometric second main theorem (ENGLISH)

[ Abstract ]

Using Ahlfors’ theory of covering surfaces, we establish a Cartan’s type Second Main Theorem in the complex projective plane with 1–truncated counting functions for entire holomorphic curves which cluster on an algebraic curve.

Using Ahlfors’ theory of covering surfaces, we establish a Cartan’s type Second Main Theorem in the complex projective plane with 1–truncated counting functions for entire holomorphic curves which cluster on an algebraic curve.

#### Numerical Analysis Seminar

16:50-18:20 Room #117 (Graduate School of Math. Sci. Bldg.)

Mathematical model for the generation of calcium carbonate scale (日本語)

**Hideo Kawarada**(AMSOK and Chiba University)Mathematical model for the generation of calcium carbonate scale (日本語)

#### Tokyo Probability Seminar

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

Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces (日本語)

**Kazumasa Kuwada**(School of science, Tokyo institute of technology)Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces (日本語)

### 2017/01/12

#### Seminar on Probability and Statistics

13:00-15:00 Room #052 (Graduate School of Math. Sci. Bldg.)

yuimaGUI: a Graphical User Interface for the yuima Package

**Emanuele Guidotti**(Milan University)yuimaGUI: a Graphical User Interface for the yuima Package

[ Abstract ]

The yuimaGUI package provides a user-friendly interface for yuima. It greatly simplifies tasks such as estimation and simulation of stochastic processes and it also includes additional tools. Some of them:

data retrieval: stock prices and economic indicators

time series clustering

change point analysis

lead-lag estimation

After a general overview of the whole interface, the yuimaGUI will be shown in real-time. All the settings and the inner workings will be discussed in detail. During this second part, you are kindly invited to ask questions whenever you feel that some problem may arise.

The yuimaGUI package provides a user-friendly interface for yuima. It greatly simplifies tasks such as estimation and simulation of stochastic processes and it also includes additional tools. Some of them:

data retrieval: stock prices and economic indicators

time series clustering

change point analysis

lead-lag estimation

After a general overview of the whole interface, the yuimaGUI will be shown in real-time. All the settings and the inner workings will be discussed in detail. During this second part, you are kindly invited to ask questions whenever you feel that some problem may arise.

### 2017/01/11

#### Number Theory Seminar

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

Deformation and rigidity of $\ell$-adic sheaves (English)

**Lei Fu**(Tsinghua University)Deformation and rigidity of $\ell$-adic sheaves (English)

[ Abstract ]

Let $X$ be a smooth connected algebraic curve over an algebraically closed field, let $S$ be a finite closed subset in $X$, and let $F_0$ be a lisse $\ell$-torsion sheaf on $X-S$. We study the deformation of $F_0$. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $\overline{Q}_\ell$-sheaf $F$ is irreducible and physically rigid, then it is cohomologically rigid in the sense that $\chi(X,j_*End(F))=2$, where $j:X-S\to X$ is the open immersion.

Let $X$ be a smooth connected algebraic curve over an algebraically closed field, let $S$ be a finite closed subset in $X$, and let $F_0$ be a lisse $\ell$-torsion sheaf on $X-S$. We study the deformation of $F_0$. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $\overline{Q}_\ell$-sheaf $F$ is irreducible and physically rigid, then it is cohomologically rigid in the sense that $\chi(X,j_*End(F))=2$, where $j:X-S\to X$ is the open immersion.

### 2017/01/10

#### Tuesday Seminar on Topology

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

Stability of anti-canonically balanced metrics (JAPANESE)

**Shunsuke Saito**(The University of Tokyo)Stability of anti-canonically balanced metrics (JAPANESE)

[ Abstract ]

Donaldson introduced "anti-canonically balanced metrics" on Fano manifolds, which is a finite dimensional analogue of Kähler-Einstein metrics. It is proved that anti-canonically balanced metrics are critical points of the quantized Ding functional.

We first study the slope at infinity of the quantized Ding functional along Bergman geodesic rays. Then, we introduce a new algebro-geometric stability of Fano manifolds based on the slope formula, and show that the existence of anti-canonically balanced metrics implies our stability. The relationship between the stability and others is also discussed.

This talk is based on a joint work with R. Takahashi (Tohoku Univ).

Donaldson introduced "anti-canonically balanced metrics" on Fano manifolds, which is a finite dimensional analogue of Kähler-Einstein metrics. It is proved that anti-canonically balanced metrics are critical points of the quantized Ding functional.

We first study the slope at infinity of the quantized Ding functional along Bergman geodesic rays. Then, we introduce a new algebro-geometric stability of Fano manifolds based on the slope formula, and show that the existence of anti-canonically balanced metrics implies our stability. The relationship between the stability and others is also discussed.

This talk is based on a joint work with R. Takahashi (Tohoku Univ).

#### Tuesday Seminar on Topology

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

Topological Invariants and Corner States for Hamiltonians on a Three Dimensional Lattice (JAPANESE)

**Shin Hayashi**(The University of Tokyo)Topological Invariants and Corner States for Hamiltonians on a Three Dimensional Lattice (JAPANESE)

[ Abstract ]

In condensed matter physics, a correspondence between two topological invariants defined for a gapped Hamiltonian is well-known. One is defined for such a Hamiltonian on a lattice (bulk invariant), and the other is defined for its restriction onto a subsemigroup (edge invariant). The edge invariant is related to the wave functions localized near the edge. This correspondence is known as the bulk-edge correspondence. In this talk, we consider a variant of this correspondence. We consider a periodic Hamiltonian on a three dimensional lattice (bulk) and its restrictions onto two subsemigroups (edges) and their intersection (corner). We will show that, if our Hamiltonian is "gapped" in some sense, we can define a topological invariant for the bulk and edges. We will also define another topological invariant related to the wave functions localized near the corner. We will explain that there is a correspondence between these two topological invariants by using the six-term exact sequence associated to the quarter-plane Toeplitz extension obtained by E. Park.

In condensed matter physics, a correspondence between two topological invariants defined for a gapped Hamiltonian is well-known. One is defined for such a Hamiltonian on a lattice (bulk invariant), and the other is defined for its restriction onto a subsemigroup (edge invariant). The edge invariant is related to the wave functions localized near the edge. This correspondence is known as the bulk-edge correspondence. In this talk, we consider a variant of this correspondence. We consider a periodic Hamiltonian on a three dimensional lattice (bulk) and its restrictions onto two subsemigroups (edges) and their intersection (corner). We will show that, if our Hamiltonian is "gapped" in some sense, we can define a topological invariant for the bulk and edges. We will also define another topological invariant related to the wave functions localized near the corner. We will explain that there is a correspondence between these two topological invariants by using the six-term exact sequence associated to the quarter-plane Toeplitz extension obtained by E. Park.

### 2016/12/22

#### Infinite Analysis Seminar Tokyo

14:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Homology cobordisms over a surface of genus one (JAPANESE)

Generalization of Schur partition theorem (JAPANESE)

**Yuta Nozaki**(Graduate School of Mathematical Sciences, the University of Tokyo) 14:00-15:30Homology cobordisms over a surface of genus one (JAPANESE)

[ Abstract ]

Morimoto showed that some lens spaces have no genus one fibered knot,

and Baker completely determined such lens spaces.

In this talk, we introduce our results for the corresponding problem

formulated in terms of homology cobordisms.

The Chebotarev density theorem and binary quadratic forms play a key

role in the proof.

Morimoto showed that some lens spaces have no genus one fibered knot,

and Baker completely determined such lens spaces.

In this talk, we introduce our results for the corresponding problem

formulated in terms of homology cobordisms.

The Chebotarev density theorem and binary quadratic forms play a key

role in the proof.

**Shunsuke Tsuchioka**(Graduate School of Mathematical Sciences, the University of Tokyo) 16:00-17:30Generalization of Schur partition theorem (JAPANESE)

[ Abstract ]

The celebrated Rogers-Ramanujan partition theorem (RRPT) claims that

the number of partitions of n whose parts are ¥pm1 modulo 5

is equinumerous to the number of partitions of n whose successive

differences are

at least 2. Schur found a mod 6 analog of RRPT in 1926.

We will report a generalization for odd $p¥geq 3$ via representation

theory of quantum groups.

At p=3, it is Schur's theorem. The statement for p=5 was conjectured by

Andrews in 1970s

in a course of his 3 parameter generalization of RRPT and proved in 1994

by Andrews-Bessenrodt-Olsson with an aid of computer.

This is a joint work with Masaki Watanabe (arXiv:1609.01905).

The celebrated Rogers-Ramanujan partition theorem (RRPT) claims that

the number of partitions of n whose parts are ¥pm1 modulo 5

is equinumerous to the number of partitions of n whose successive

differences are

at least 2. Schur found a mod 6 analog of RRPT in 1926.

We will report a generalization for odd $p¥geq 3$ via representation

theory of quantum groups.

At p=3, it is Schur's theorem. The statement for p=5 was conjectured by

Andrews in 1970s

in a course of his 3 parameter generalization of RRPT and proved in 1994

by Andrews-Bessenrodt-Olsson with an aid of computer.

This is a joint work with Masaki Watanabe (arXiv:1609.01905).

### 2016/12/20

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On the stability of the 3D Couette Flow (English)

**Nader Masmoudi**(Courant Institute, NYU)On the stability of the 3D Couette Flow (English)

[ Abstract ]

We will discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 3D incompressible Navier-Stokes system at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions near this threshold. For rougher data, we obtain an estimate of the stability threshold which agrees closely with numerical experiments. The primary linear stability mechanism is an anisotropic enhanced dissipation resulting from the mixing caused by the large mean shear. There is also a linear inviscid damping similar to the one observed in 2D. The main linear instability is a non-normal instability known as the lift-up effect. There is clearly a competition between these linear effects. Understanding the variety of nonlinear resonances and devising the correct norms to estimate them form the core of the analysis we undertake. This is based on joint works with Jacob Bedrossian and Pierre Germain.

We will discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 3D incompressible Navier-Stokes system at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions near this threshold. For rougher data, we obtain an estimate of the stability threshold which agrees closely with numerical experiments. The primary linear stability mechanism is an anisotropic enhanced dissipation resulting from the mixing caused by the large mean shear. There is also a linear inviscid damping similar to the one observed in 2D. The main linear instability is a non-normal instability known as the lift-up effect. There is clearly a competition between these linear effects. Understanding the variety of nonlinear resonances and devising the correct norms to estimate them form the core of the analysis we undertake. This is based on joint works with Jacob Bedrossian and Pierre Germain.

#### Tuesday Seminar on Topology

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

Deligne-Mostow lattices and cone metrics on the sphere (ENGLISH)

**Irene Pasquinelli**(Durham University)Deligne-Mostow lattices and cone metrics on the sphere (ENGLISH)

[ Abstract ]

Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.

One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.

In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with three folding symmetry.

Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.

One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.

In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with three folding symmetry.

### 2016/12/19

#### Discrete mathematical modelling seminar

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

Discrete Painlevé equations on the affine A3 surface (ENGLISH)

**Anton Dzhamay**(University of Northern Colorado)Discrete Painlevé equations on the affine A3 surface (ENGLISH)

[ Abstract ]

We explain how to construct the birational representation of the extended affine Weyl symmetry group D5 and consider examples of discrete Painlevé equations that correspond to certain translation elements in this group. One of the examples is the famous q-PV equation of Jimbo-Sakai. Some other examples are conjugated to it via explicit change of variables and we explain how representing translation elements as words in the group allows us to see the corresponding change of coordinates explicitly. We also show a new example of a discrete Painlevé equation that is elementary (short translation), but at the same time is different from the q-PVI equation.

We explain how to construct the birational representation of the extended affine Weyl symmetry group D5 and consider examples of discrete Painlevé equations that correspond to certain translation elements in this group. One of the examples is the famous q-PV equation of Jimbo-Sakai. Some other examples are conjugated to it via explicit change of variables and we explain how representing translation elements as words in the group allows us to see the corresponding change of coordinates explicitly. We also show a new example of a discrete Painlevé equation that is elementary (short translation), but at the same time is different from the q-PVI equation.

### 2016/12/17

#### Discrete mathematical modelling seminar

10:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Factorization of Rational Mappings and Geometric Deautonomization (ENGLISH)

From the QRT maps to elliptic difference Painlevé equations (ENGLISH)

The complete degeneration scheme of four-dimensional Painlevé-type equations (ENGLISH)

Degeneration of the Painlevé divisors (ENGLISH)

Rational approximation and Schlesinger transformation (ENGLISH)

Spaces of initial conditions for nonautonomous mappings of the plane (ENGLISH)

**Anton Dzhamay**(University of Northern Colorado) 10:00-10:50Factorization of Rational Mappings and Geometric Deautonomization (ENGLISH)

[ Abstract ]

This talk is the first of two talks describing the joint project with Tomoyuki Takenawa and Stefan Carstea on geometric deautonomization.

The goal of this project is to develop a systematic approach for deautonomizing discrete integrable mappings, such as the QRT mappings, to non-automonous mappings in the discrete Painlevé family, based on the action of the mapping on the Picard lattice of the surface and a choice of an elliptic fiber. In this talk we will explain the main ideas behind this approach and describe the technique that allows us to recover explicit formulas defining the mapping from the known action on the divisor group (the factorization technique). We illustrate our approach by reconstructing the famous example of the q-PVI equation of Jimbo-Sakai from a simple QRT mapping.

This talk is the first of two talks describing the joint project with Tomoyuki Takenawa and Stefan Carstea on geometric deautonomization.

The goal of this project is to develop a systematic approach for deautonomizing discrete integrable mappings, such as the QRT mappings, to non-automonous mappings in the discrete Painlevé family, based on the action of the mapping on the Picard lattice of the surface and a choice of an elliptic fiber. In this talk we will explain the main ideas behind this approach and describe the technique that allows us to recover explicit formulas defining the mapping from the known action on the divisor group (the factorization technique). We illustrate our approach by reconstructing the famous example of the q-PVI equation of Jimbo-Sakai from a simple QRT mapping.

**Tomoyuki Takenawa**(Tokyo University of Marine Science and Technology) 11:00-11:50From the QRT maps to elliptic difference Painlevé equations (ENGLISH)

[ Abstract ]

This talk is the second part of the joint project with Anton Dzhamay and Stefan Carstea on geometric deautonomization and focuses on the elliptic case and the special symmetry groups. It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated.

In this talk we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Especially, in the case where the fiber is smooth elliptic, imposing certain restrictions on such non autonomous mappings, we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.

This talk is the second part of the joint project with Anton Dzhamay and Stefan Carstea on geometric deautonomization and focuses on the elliptic case and the special symmetry groups. It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated.

In this talk we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Especially, in the case where the fiber is smooth elliptic, imposing certain restrictions on such non autonomous mappings, we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.

**Hiroshi Kawakami**(Aoyama Gakuin University) 13:30-14:20The complete degeneration scheme of four-dimensional Painlevé-type equations (ENGLISH)

[ Abstract ]

In the joint work with H. Sakai and A. Nakamura, we constructed the degeneration scheme of four-dimensional Painlevé-type equations associated with unramified linear equations. In this talk I present the "complete" degeneration scheme of the four-dimensional Painlevé-type equations, which is constructed by means of the degeneration of HTL forms of associated linear equations.

In the joint work with H. Sakai and A. Nakamura, we constructed the degeneration scheme of four-dimensional Painlevé-type equations associated with unramified linear equations. In this talk I present the "complete" degeneration scheme of the four-dimensional Painlevé-type equations, which is constructed by means of the degeneration of HTL forms of associated linear equations.

**Akane Nakamura**(Josai University) 14:30-15:20Degeneration of the Painlevé divisors (ENGLISH)

[ Abstract ]

There are three types of curves associated with 4-dimensional algebraically completely integrable systems, namely the spectral curve, the Painlevé divisors, and the separation curve. I am going to explain these three curves of genus two taking examples derived from the isospectral limit of the 4-dimensional Painlevé-type equations and study the Namikawa-Ueno type degeneration.

There are three types of curves associated with 4-dimensional algebraically completely integrable systems, namely the spectral curve, the Painlevé divisors, and the separation curve. I am going to explain these three curves of genus two taking examples derived from the isospectral limit of the 4-dimensional Painlevé-type equations and study the Namikawa-Ueno type degeneration.

**Teruhisa Tsuda**(Hitotsubashi University) 16:00-16:50Rational approximation and Schlesinger transformation (ENGLISH)

[ Abstract ]

We show how rational approximation problems for functions are related to the construction of Schlesinger transformations. Also we discuss their applications to the theory of isomonodromic deformations or Painlevé equations. This talk is based on a joint work with Toshiyuki Mano.

We show how rational approximation problems for functions are related to the construction of Schlesinger transformations. Also we discuss their applications to the theory of isomonodromic deformations or Painlevé equations. This talk is based on a joint work with Toshiyuki Mano.

**Takafumi Mase**(the University of Tokyo) 17:00-17:50Spaces of initial conditions for nonautonomous mappings of the plane (ENGLISH)

[ Abstract ]

Spaces of initial conditions are one of the most important and powerful tools to analyze mappings of the plane. In this talk, we study the basic properties of general nonautonomous equations that have spaces of initial conditions. We will consider the minimization of spaces of initial conditions for nonautonomous systems and we shall discuss a classification of nonautonomous integrable mappings of the plane with a space of initial conditions.

Spaces of initial conditions are one of the most important and powerful tools to analyze mappings of the plane. In this talk, we study the basic properties of general nonautonomous equations that have spaces of initial conditions. We will consider the minimization of spaces of initial conditions for nonautonomous systems and we shall discuss a classification of nonautonomous integrable mappings of the plane with a space of initial conditions.

### 2016/12/14

#### Number Theory Seminar

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

On vanishing cycles and duality, after A. Beilinson (English)

**Luc Illusie**(Université Paris-Sud)On vanishing cycles and duality, after A. Beilinson (English)

[ Abstract ]

It was proved by Gabber in the early 1980's that $R\Psi$ commutes with duality, and that R\Phi preserves perversity up to shift. It had been in the folklore since then that this last result was in fact a consequence of a finer one, namely the compatibility of $R\Phi$ with duality. In this talk I'll give a proof of this, using a method explained to me by A. Beilinson.

It was proved by Gabber in the early 1980's that $R\Psi$ commutes with duality, and that R\Phi preserves perversity up to shift. It had been in the folklore since then that this last result was in fact a consequence of a finer one, namely the compatibility of $R\Phi$ with duality. In this talk I'll give a proof of this, using a method explained to me by A. Beilinson.

### 2016/12/13

#### Tuesday Seminar on Topology

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

Plane fields on 3-manifolds and asymptotic linking of the tangential incompressible flows (JAPANESE)

**Yoshihiko Mitsumatsu**(Chuo University)Plane fields on 3-manifolds and asymptotic linking of the tangential incompressible flows (JAPANESE)

[ Abstract ]

This is a report on a project in (a very slow) progress which aims to prove the tightness of contact structures associated with algebraic Anosov flows without using Bennequin's nor Gromov's results.

After introducing an interpretation of asymptotic linking pairing in terms of differential forms, we attach a subspaces of exact 2-forms to each plane field. We analyze this space in the case where the plane field is an algebraic Anosov foliation and explain what can be done using results from foliated cohomology and frameworks for secondary characteristic classes. We also show some explicit computations.

To close the talk, a quantization phenomenon which happens when a foliation is deformed into a contact structure is explained and we state some perspectives on applying the results on foliations to the tightness.

This is a report on a project in (a very slow) progress which aims to prove the tightness of contact structures associated with algebraic Anosov flows without using Bennequin's nor Gromov's results.

After introducing an interpretation of asymptotic linking pairing in terms of differential forms, we attach a subspaces of exact 2-forms to each plane field. We analyze this space in the case where the plane field is an algebraic Anosov foliation and explain what can be done using results from foliated cohomology and frameworks for secondary characteristic classes. We also show some explicit computations.

To close the talk, a quantization phenomenon which happens when a foliation is deformed into a contact structure is explained and we state some perspectives on applying the results on foliations to the tightness.

#### Tuesday Seminar of Analysis

16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)

Distribution of eigenfunction mass on some really simple domains (English)

**Hans Christianson**(North Carolina State University)Distribution of eigenfunction mass on some really simple domains (English)

[ Abstract ]

Eigenfunctions are fundamental objects of study in spectral geometry and quantum chaos. On a domain or manifold, they determine the behaviour of solutions to many evolution type equations using, for example, separation of variables. Eigenfunctions are very sensitive to background geometry, so it is important to understand what the eigenfunctions look like: where are they large and where are they small? There are many different ways to measure what "large" and "small" mean. One can consider local $L^2$ distribution, local and global $L^p$ distribution, as well as restrictions and boundary values. I will give an overview of what is known, and then discuss some very recent works in progress demonstrating that complicated things can happen even in very simple geometric settings.

Eigenfunctions are fundamental objects of study in spectral geometry and quantum chaos. On a domain or manifold, they determine the behaviour of solutions to many evolution type equations using, for example, separation of variables. Eigenfunctions are very sensitive to background geometry, so it is important to understand what the eigenfunctions look like: where are they large and where are they small? There are many different ways to measure what "large" and "small" mean. One can consider local $L^2$ distribution, local and global $L^p$ distribution, as well as restrictions and boundary values. I will give an overview of what is known, and then discuss some very recent works in progress demonstrating that complicated things can happen even in very simple geometric settings.

### 2016/12/12

#### Tokyo Probability Seminar

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

**Takuma Akimoto**(Keio University)#### Seminar on Geometric Complex Analysis

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

(JAPANESE)

**Yu Kawakami**(Kanazawa University)(JAPANESE)

< Previous 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143 Next >