## Seminar information archive

Seminar information archive ～11/14｜Today's seminar 11/15 | Future seminars 11/16～

#### FMSP Lectures

16:40-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

(3)Sparsity and low rank matrix learning. (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Dalalyan.pdf

**Arnak Dalalyan**(ENSAE ParisTech)(3)Sparsity and low rank matrix learning. (ENGLISH)

[ Abstract ]

In this third lecture, we will present extensions of the previously introduced sparse recovery techniques to the problems of machine learning and statistics in which a large matrix should be learned from data. The analogue of the sparsity, in this context, is the low-rankness of the matrix. We will show that such matrices can be effectively learned by minimizing the empirical risk penalized by the nuclear norm. The resulting problem is a problem of semi-definite programming and can be solved efficiently even when the dimension is large. Theoretical guarantees for this method will be established in the case of matrix completion with known sampling distribution.

[ Reference URL ]In this third lecture, we will present extensions of the previously introduced sparse recovery techniques to the problems of machine learning and statistics in which a large matrix should be learned from data. The analogue of the sparsity, in this context, is the low-rankness of the matrix. We will show that such matrices can be effectively learned by minimizing the empirical risk penalized by the nuclear norm. The resulting problem is a problem of semi-definite programming and can be solved efficiently even when the dimension is large. Theoretical guarantees for this method will be established in the case of matrix completion with known sampling distribution.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Dalalyan.pdf

### 2015/12/02

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Drinfeld center and representation theory for monoidal categories

**Makoto Yamashita**(Ochanomizu Univ.)Drinfeld center and representation theory for monoidal categories

#### Seminar on Probability and Statistics

14:55-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Learning theory and sparsity ～ Lasso, Dantzig selector and their statistical properties ～

**Arnak Dalalyan**(ENSAE ParisTech)Learning theory and sparsity ～ Lasso, Dantzig selector and their statistical properties ～

[ Abstract ]

In this second lecture, we will focus on the problem of high dimensional linear regression under the sparsity assumption and discuss the three main statistical problems: denoising, prediction and model selection. We will prove that convex programming based predictors such as the lasso and the Dantzig selector are provably consistent as soon as the dictionary elements are normalized and an appropriate upper bound on the noise-level is available. We will also show that under additional assumptions on the dictionary elements, the aforementioned methods are rate-optimal and model-selection consistent.

In this second lecture, we will focus on the problem of high dimensional linear regression under the sparsity assumption and discuss the three main statistical problems: denoising, prediction and model selection. We will prove that convex programming based predictors such as the lasso and the Dantzig selector are provably consistent as soon as the dictionary elements are normalized and an appropriate upper bound on the noise-level is available. We will also show that under additional assumptions on the dictionary elements, the aforementioned methods are rate-optimal and model-selection consistent.

#### FMSP Lectures

14:55-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

(2)Lasso, Dantzig selector and their statistical properties. (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Dalalyan.pdf

**Arnak Dalalyan**(ENSAE ParisTech)(2)Lasso, Dantzig selector and their statistical properties. (ENGLISH)

[ Abstract ]

In this second lecture, we will focus on the problem of high dimensional linear regression under the sparsity assumption and discuss the three main statistical problems: denoising, prediction and model selection. We will prove that convex programming based predictors such as the lasso and the Dantzig selector are provably consistent as soon as the dictionary elements are normalized and an appropriate upper bound on the noise-level is available. We will also show that under additional assumptions on the dictionary elements, the aforementioned methods are rate-optimal and model-selection consistent.

[ Reference URL ]In this second lecture, we will focus on the problem of high dimensional linear regression under the sparsity assumption and discuss the three main statistical problems: denoising, prediction and model selection. We will prove that convex programming based predictors such as the lasso and the Dantzig selector are provably consistent as soon as the dictionary elements are normalized and an appropriate upper bound on the noise-level is available. We will also show that under additional assumptions on the dictionary elements, the aforementioned methods are rate-optimal and model-selection consistent.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Dalalyan.pdf

#### Mathematical Biology Seminar

14:55-16:40 Room #128 (Graduate School of Math. Sci. Bldg.)

Network centrality measure based on sensitivity analysis of the basic reproductive ratio

[ Reference URL ]

http://www.soken.ac.jp/

**Kenta Yajima**( The Graduate University for Advanced Studies (Sokendai), School of Advanced Sciences)Network centrality measure based on sensitivity analysis of the basic reproductive ratio

[ Reference URL ]

http://www.soken.ac.jp/

### 2015/12/01

#### Tuesday Seminar of Analysis

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

On complex singularity analysis for some linear partial differential equations

**Stéphane Malek**(Université de Lille, France)On complex singularity analysis for some linear partial differential equations

[ Abstract ]

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc outside some singular set S. The coefficients are written as linear combinations of powers of a solution X of some first order nonlinear partial differential equation following an idea :we have initiated in a previous joint work with C. Stenger. The solutions Y are shown to develop singularities along the singular set S with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series. (Joint work with A. Lastra and C. Stenger).

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc outside some singular set S. The coefficients are written as linear combinations of powers of a solution X of some first order nonlinear partial differential equation following an idea :we have initiated in a previous joint work with C. Stenger. The solutions Y are shown to develop singularities along the singular set S with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series. (Joint work with A. Lastra and C. Stenger).

#### Tuesday Seminar on Topology

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

Monodromies of splitting families for singular fibers (JAPANESE)

**Takayuki Okuda**(The University of Tokyo)Monodromies of splitting families for singular fibers (JAPANESE)

[ Abstract ]

A degeneration of Riemann surfaces is a family of complex curves

over a disk allowed to have a singular fiber.

A singular fiber may split into several simpler singular fibers

under a deformation family of such families,

which is called a splitting family for the singular fiber.

We are interested in the topology of splitting families.

For the topological types of degenerations of Riemann surfaces,

it is known that there is a good relationship with

the surface mapping classes, via topological monodromy.

In this talk,

we introduce the "topological monodromies of splitting families",

and give a description of those of certain splitting families.

A degeneration of Riemann surfaces is a family of complex curves

over a disk allowed to have a singular fiber.

A singular fiber may split into several simpler singular fibers

under a deformation family of such families,

which is called a splitting family for the singular fiber.

We are interested in the topology of splitting families.

For the topological types of degenerations of Riemann surfaces,

it is known that there is a good relationship with

the surface mapping classes, via topological monodromy.

In this talk,

we introduce the "topological monodromies of splitting families",

and give a description of those of certain splitting families.

### 2015/11/30

#### Seminar on Geometric Complex Analysis

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

Extension of holomorphic functions defined on non reduced analytic subvarieties (English)

**Jean-Pierre Demailly**(Univ. de Grenoble I)Extension of holomorphic functions defined on non reduced analytic subvarieties (English)

[ Abstract ]

The goal of this talk will be to discuss $L^2$ extension properties of holomorphic sections of vector bundles satisfying weak semi-positivity properties. Using techniques borrowed from recent proofs of the Ohsawa-Takegoshi extension theorem, we obtain several new surjectivity results for the restriction morphism to a non necessarily reduced subvariety, provided the latter is defined as the zero variety of a multiplier ideal sheaf. These extension results are derived from $L^2$ approximation techniques, and they hold under (probably) optimal curvature conditions.

The goal of this talk will be to discuss $L^2$ extension properties of holomorphic sections of vector bundles satisfying weak semi-positivity properties. Using techniques borrowed from recent proofs of the Ohsawa-Takegoshi extension theorem, we obtain several new surjectivity results for the restriction morphism to a non necessarily reduced subvariety, provided the latter is defined as the zero variety of a multiplier ideal sheaf. These extension results are derived from $L^2$ approximation techniques, and they hold under (probably) optimal curvature conditions.

#### Algebraic Geometry Seminar

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

Interesting surfaces which are coverings of a rational surface branched over few lines (English)

**Fabrizio Catanese**(Universität Bayreuth)Interesting surfaces which are coverings of a rational surface branched over few lines (English)

[ Abstract ]

Surfaces which are covers of the plane branched over 5 or 6 lines have provided answers to long standing questions, for instance the BCD surfaces for Fujita's question on semiampleness of VHS (Dettweiler-Cat); and examples of ball quotients (Hirzebruch), automorphisms acting trivially on integral cohomology (Cat-Gromadtzki), canonical maps with high degree or image-degree (Pardini, Bauer-Cat). I shall speak especially about the above Abelian coverings of the plane, the geometry of the del Pezzo surface of degree 5, the rigidity of BCD surfaces, and a criterion for a fibred surface to be a projective classifying space.

Surfaces which are covers of the plane branched over 5 or 6 lines have provided answers to long standing questions, for instance the BCD surfaces for Fujita's question on semiampleness of VHS (Dettweiler-Cat); and examples of ball quotients (Hirzebruch), automorphisms acting trivially on integral cohomology (Cat-Gromadtzki), canonical maps with high degree or image-degree (Pardini, Bauer-Cat). I shall speak especially about the above Abelian coverings of the plane, the geometry of the del Pezzo surface of degree 5, the rigidity of BCD surfaces, and a criterion for a fibred surface to be a projective classifying space.

#### Tokyo Probability Seminar

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

Self-organized criticality in a discrete model of limited aggregation

**Raoul Normand**(Institute of Mathematics, Academia Sinica)Self-organized criticality in a discrete model of limited aggregation

[ Abstract ]

We consider a discrete model of coagulation, where a large number of particles are initially given a prescribed number of arms. We successively choose arms uniformly at random and bind them two by two, unless they belong to "large" clusters. In that sense, the large clusters are frozen and become inactive. We study the graph structure obtained, and describe what a typical cluster looks like. We show that there is a fixed time T such that, before time T, a typical cluster is a subcritical Galton-Watson tree, whereas after time T, a typical cluster is a critical Galton-Watson tree. In that sense, we observe a phenomenon called self-organized criticality.

We consider a discrete model of coagulation, where a large number of particles are initially given a prescribed number of arms. We successively choose arms uniformly at random and bind them two by two, unless they belong to "large" clusters. In that sense, the large clusters are frozen and become inactive. We study the graph structure obtained, and describe what a typical cluster looks like. We show that there is a fixed time T such that, before time T, a typical cluster is a subcritical Galton-Watson tree, whereas after time T, a typical cluster is a critical Galton-Watson tree. In that sense, we observe a phenomenon called self-organized criticality.

### 2015/11/27

#### Colloquium

16:50-17:50 Room #056 (Graduate School of Math. Sci. Bldg.)

Recent development in amenable groups (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kida/

**Yoshikata Kida**(Graduate School of Mathematical Sciences, University of Tokyo)Recent development in amenable groups (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kida/

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The strong version of K-stability derived from the coercivity property of the K-energy (Japanese)

**Hisamoto Tomoyuki**(Nagoya University)The strong version of K-stability derived from the coercivity property of the K-energy (Japanese)

[ Abstract ]

This is a joint work with S. Boucksom and M. Jonsson. We introduced the notion of J-uniform K-stability in relation to the coercivity property of the K-energy. As a result, one can see that any Kähler-Einstein

manifolds with no non-zero holomorphic vector field is J-uniformly K-stable.

This is a joint work with S. Boucksom and M. Jonsson. We introduced the notion of J-uniform K-stability in relation to the coercivity property of the K-energy. As a result, one can see that any Kähler-Einstein

manifolds with no non-zero holomorphic vector field is J-uniformly K-stable.

### 2015/11/26

#### Lie Groups and Representation Theory

17:00-18:45 Room # (Graduate School of Math. Sci. Bldg.)

Introduction to the cohomology of discrete groups and modular symbols 2 (English)

**Birgit Speh**(Cornell University)Introduction to the cohomology of discrete groups and modular symbols 2 (English)

[ Abstract ]

The course is an introduction to the cohomology of torsion free discrete subgroups $\Gamma \subset G $ of a semi simple group $G$. The discrete group $\Gamma$ acts freely on the symmetric space $X= G/K$ and we will always assume that $\Gamma \backslash G/K$ is compact or has finite volume. An example is a torsion free subgroup $\Gamma_n $ of finite index n in Sl(2,Z) acting on $Sl(2.R)/SO(2) \simeq {\mathcal H}=\{z=x+iy \in C| y >0 \}$ by fractional linear transformations. $\Gamma_n \backslash {\mathcal H}$ can be determined explicitly and it can be visualized as an area in the upper half plane glued at the boundary. It is easy to see that it has some nice compactifications.

The cohomology $H^*(\Gamma, C)$ of the group $\Gamma$ is equal to the deRham cohomology $H^*_{deRham}(\Gamma \backslash X, C)$ of the manifold $\Gamma\backslash X$. This cohomology is studied by proving that it is isomorphic to the $H^*(g,K,{\mathcal A}(\Gamma \backslash G))$. Here ${\mathcal A}(\Gamma \backslash G)$ of automorphic functions on $\Gamma \backslash G$. In the case $\Gamma_n \subset Sl(2,Z)$ the space ${\mathcal A}(\Gamma \backslash G)$ is the space of classical automorphic functions on the upper half plane containing holomorphic cusp form, Eisenstein series, Maass forms and it is often introduced in an introductory course in analytic number theory.

On the geometric side we will construct some of the cycles (modular symbols) in the homology $H_*(\Gamma\backslash X)$ which are dual to the cohomology classes we constructed. In our example $\Gamma_n\backslash Sl(2,R)/SO(2)$ these cycles correspond to geodesics and can easily be visualized.

In this course I will explain these results and show how to use them to prove vanishing and non vanishing theorem for $H^*_{deRham}(\Gamma \backslash X)$. I will state the results in full generality, but I will prove them only in the classical case: G=SL$(2,R)$ and the subgroup $\Gamma= \Gamma_n$ a congruence subgroup. Some familiarity with Lie groups and Lie algebras is only prerequisite for the course.

The course is an introduction to the cohomology of torsion free discrete subgroups $\Gamma \subset G $ of a semi simple group $G$. The discrete group $\Gamma$ acts freely on the symmetric space $X= G/K$ and we will always assume that $\Gamma \backslash G/K$ is compact or has finite volume. An example is a torsion free subgroup $\Gamma_n $ of finite index n in Sl(2,Z) acting on $Sl(2.R)/SO(2) \simeq {\mathcal H}=\{z=x+iy \in C| y >0 \}$ by fractional linear transformations. $\Gamma_n \backslash {\mathcal H}$ can be determined explicitly and it can be visualized as an area in the upper half plane glued at the boundary. It is easy to see that it has some nice compactifications.

The cohomology $H^*(\Gamma, C)$ of the group $\Gamma$ is equal to the deRham cohomology $H^*_{deRham}(\Gamma \backslash X, C)$ of the manifold $\Gamma\backslash X$. This cohomology is studied by proving that it is isomorphic to the $H^*(g,K,{\mathcal A}(\Gamma \backslash G))$. Here ${\mathcal A}(\Gamma \backslash G)$ of automorphic functions on $\Gamma \backslash G$. In the case $\Gamma_n \subset Sl(2,Z)$ the space ${\mathcal A}(\Gamma \backslash G)$ is the space of classical automorphic functions on the upper half plane containing holomorphic cusp form, Eisenstein series, Maass forms and it is often introduced in an introductory course in analytic number theory.

On the geometric side we will construct some of the cycles (modular symbols) in the homology $H_*(\Gamma\backslash X)$ which are dual to the cohomology classes we constructed. In our example $\Gamma_n\backslash Sl(2,R)/SO(2)$ these cycles correspond to geodesics and can easily be visualized.

In this course I will explain these results and show how to use them to prove vanishing and non vanishing theorem for $H^*_{deRham}(\Gamma \backslash X)$. I will state the results in full generality, but I will prove them only in the classical case: G=SL$(2,R)$ and the subgroup $\Gamma= \Gamma_n$ a congruence subgroup. Some familiarity with Lie groups and Lie algebras is only prerequisite for the course.

### 2015/11/25

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Combining Pseudodifferential and Vector Bundle Techniques, and Their Applications to Topological Insulators

**Max Lein**(AIMR, Tohoku Univ.)Combining Pseudodifferential and Vector Bundle Techniques, and Their Applications to Topological Insulators

#### Seminar on Probability and Statistics

14:55-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Learning theory and sparsity ～ Introduction into sparse recovery and compressed sensing ～

**Arnak Dalalyan**(ENSAE ParisTech)Learning theory and sparsity ～ Introduction into sparse recovery and compressed sensing ～

[ Abstract ]

In this introductory lecture, we will present the general framework of high-dimensional statistical modeling and its applications in machine learning and signal processing. Basic methods of sparse recovery, such as the hard and the soft thresholding, will be introduced in the context of orthonormal dictionaries and their statistical accuracy will be discussed in detail. We will also show the relation of these methods with compressed sensing and convex programming based procedures.

In this introductory lecture, we will present the general framework of high-dimensional statistical modeling and its applications in machine learning and signal processing. Basic methods of sparse recovery, such as the hard and the soft thresholding, will be introduced in the context of orthonormal dictionaries and their statistical accuracy will be discussed in detail. We will also show the relation of these methods with compressed sensing and convex programming based procedures.

#### FMSP Lectures

14:55-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

(1)Introduction into sparse recovery and compressed sensing. (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Dalalyan.pdf

**Arnak Dalalyan**(ENSAE ParisTech)(1)Introduction into sparse recovery and compressed sensing. (ENGLISH)

[ Abstract ]

In this introductory lecture, we will present the general framework of high-dimensional statistical modeling and its applications in machine learning and signal processing. Basic methods of sparse recovery, such as the hard and the soft thresholding, will be introduced in the context of orthonormal dictionaries and their statistical accuracy will be discussed in detail. We will also show the relation of these methods with compressed sensing and convex programming based procedures.

[ Reference URL ]In this introductory lecture, we will present the general framework of high-dimensional statistical modeling and its applications in machine learning and signal processing. Basic methods of sparse recovery, such as the hard and the soft thresholding, will be introduced in the context of orthonormal dictionaries and their statistical accuracy will be discussed in detail. We will also show the relation of these methods with compressed sensing and convex programming based procedures.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Dalalyan.pdf

### 2015/11/24

#### Tuesday Seminar of Analysis

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

A local analysis of the swirling flow to the axi-symmetric Navier-Stokes equations near a saddle point and no-slip flat boundary (English)

**Pen-Yuan Hsu**(Graduate School of Mathematical Sciences, the University of Tokyo)A local analysis of the swirling flow to the axi-symmetric Navier-Stokes equations near a saddle point and no-slip flat boundary (English)

[ Abstract ]

As one of the violent flow, tornadoes occur in many place of the world. In order to reduce human losses and material damage caused by tornadoes, there are many research methods. One of the effective methods is numerical simulations. The swirling structure is significant both in mathematical analysis and the numerical simulations of tornado. In this joint work with H. Notsu and T. Yoneda we try to clarify the swirling structure. More precisely, we do numerical computations on axi-symmetric Navier-Stokes flows with no-slip flat boundary. We compare a hyperbolic flow with swirl and one without swirl and observe that the following phenomenons occur only in the swirl case: The distance between the point providing the maximum velocity magnitude $|v|$ and the $z$-axis is drastically changing around some time (which we call it turning point). An ``increasing velocity phenomenon'' occurs near the boundary and the maximum value of $|v|$ is obtained near the axis of symmetry and the boundary when time is close to the turning point.

As one of the violent flow, tornadoes occur in many place of the world. In order to reduce human losses and material damage caused by tornadoes, there are many research methods. One of the effective methods is numerical simulations. The swirling structure is significant both in mathematical analysis and the numerical simulations of tornado. In this joint work with H. Notsu and T. Yoneda we try to clarify the swirling structure. More precisely, we do numerical computations on axi-symmetric Navier-Stokes flows with no-slip flat boundary. We compare a hyperbolic flow with swirl and one without swirl and observe that the following phenomenons occur only in the swirl case: The distance between the point providing the maximum velocity magnitude $|v|$ and the $z$-axis is drastically changing around some time (which we call it turning point). An ``increasing velocity phenomenon'' occurs near the boundary and the maximum value of $|v|$ is obtained near the axis of symmetry and the boundary when time is close to the turning point.

#### Tuesday Seminar on Topology

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

On the cohomology ring of the handlebody mapping class group of genus two (JAPANESE)

**Masatoshi Sato**(Tokyo Denki University)On the cohomology ring of the handlebody mapping class group of genus two (JAPANESE)

[ Abstract ]

The genus two handlebody mapping class group acts on a tree

constructed by Kramer from the disk complex,

and decomposes into an amalgamated product of two subgroups.

We determine the integral cohomology ring of the genus two handlebody

mapping class group by examining these two subgroups

and the Mayer-Vietoris exact sequence.

Using this result, we estimate the ranks of low dimensional homology

groups of the genus three handlebody mapping class group.

The genus two handlebody mapping class group acts on a tree

constructed by Kramer from the disk complex,

and decomposes into an amalgamated product of two subgroups.

We determine the integral cohomology ring of the genus two handlebody

mapping class group by examining these two subgroups

and the Mayer-Vietoris exact sequence.

Using this result, we estimate the ranks of low dimensional homology

groups of the genus three handlebody mapping class group.

#### Lie Groups and Representation Theory

17:00-18:45 Room # (Graduate School of Math. Sci. Bldg.)

Introduction to the cohomology of discrete groups and modular symbols 1 (English)

**Birgit Speh**(Cornell University)Introduction to the cohomology of discrete groups and modular symbols 1 (English)

[ Abstract ]

The course is an introduction to the cohomology of torsion free discrete subgroups $\Gamma \subset G $ of a semi simple group $G$. The discrete group $\Gamma$ acts freely on the symmetric space $X= G/K$ and we will always assume that $\Gamma \backslash G/K$ is compact or has finite volume. An example is a torsion free subgroup $\Gamma_n $ of finite index n in Sl(2,Z) acting on $Sl(2.R)/SO(2) \simeq {\mathcal H}=\{z=x+iy \in C| y >0 \}$ by fractional linear transformations. $\Gamma_n \backslash {\mathcal H}$ can be determined explicitly and it can be visualized as an area in the upper half plane glued at the boundary. It is easy to see that it has some nice compactifications.

The cohomology $H^*(\Gamma, C)$ of the group $\Gamma$ is equal to the deRham cohomology $H^*_{deRham}(\Gamma \backslash X, C)$ of the manifold $\Gamma\backslash X$. This cohomology is studied by proving that it is isomorphic to the $H^*(g,K,{\mathcal A}(\Gamma \backslash G))$. Here ${\mathcal A}(\Gamma \backslash G)$ of automorphic functions on $\Gamma \backslash G$. In the case $\Gamma_n \subset Sl(2,Z)$ the space ${\mathcal A}(\Gamma \backslash G)$ is the space of classical automorphic functions on the upper half plane containing holomorphic cusp form, Eisenstein series, Maass forms and it is often introduced in an introductory course in analytic number theory.

On the geometric side we will construct some of the cycles (modular symbols) in the homology $H_*(\Gamma\backslash X)$ which are dual to the cohomology classes we constructed. In our example $\Gamma_n\backslash Sl(2,R)/SO(2)$ these cycles correspond to geodesics and can easily be visualized.

In this course I will explain these results and show how to use them to prove vanishing and non vanishing theorem for $H^*_{deRham}(\Gamma \backslash X)$. I will state the results in full generality, but I will prove them only in the classical case: G=SL$(2,R)$ and the subgroup $\Gamma= \Gamma_n$ a congruence subgroup. Some familiarity with Lie groups and Lie algebras is only prerequisite for the course.

The course is an introduction to the cohomology of torsion free discrete subgroups $\Gamma \subset G $ of a semi simple group $G$. The discrete group $\Gamma$ acts freely on the symmetric space $X= G/K$ and we will always assume that $\Gamma \backslash G/K$ is compact or has finite volume. An example is a torsion free subgroup $\Gamma_n $ of finite index n in Sl(2,Z) acting on $Sl(2.R)/SO(2) \simeq {\mathcal H}=\{z=x+iy \in C| y >0 \}$ by fractional linear transformations. $\Gamma_n \backslash {\mathcal H}$ can be determined explicitly and it can be visualized as an area in the upper half plane glued at the boundary. It is easy to see that it has some nice compactifications.

The cohomology $H^*(\Gamma, C)$ of the group $\Gamma$ is equal to the deRham cohomology $H^*_{deRham}(\Gamma \backslash X, C)$ of the manifold $\Gamma\backslash X$. This cohomology is studied by proving that it is isomorphic to the $H^*(g,K,{\mathcal A}(\Gamma \backslash G))$. Here ${\mathcal A}(\Gamma \backslash G)$ of automorphic functions on $\Gamma \backslash G$. In the case $\Gamma_n \subset Sl(2,Z)$ the space ${\mathcal A}(\Gamma \backslash G)$ is the space of classical automorphic functions on the upper half plane containing holomorphic cusp form, Eisenstein series, Maass forms and it is often introduced in an introductory course in analytic number theory.

On the geometric side we will construct some of the cycles (modular symbols) in the homology $H_*(\Gamma\backslash X)$ which are dual to the cohomology classes we constructed. In our example $\Gamma_n\backslash Sl(2,R)/SO(2)$ these cycles correspond to geodesics and can easily be visualized.

In this course I will explain these results and show how to use them to prove vanishing and non vanishing theorem for $H^*_{deRham}(\Gamma \backslash X)$. I will state the results in full generality, but I will prove them only in the classical case: G=SL$(2,R)$ and the subgroup $\Gamma= \Gamma_n$ a congruence subgroup. Some familiarity with Lie groups and Lie algebras is only prerequisite for the course.

### 2015/11/18

#### FMSP Lectures

15:00-16:00,16:30-17:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

Crossroads of symplectic rigidity and flexibility (ENGLISH)

http://faculty.ms.u-tokyo.ac.jp/Eliashberg201511.html

**Yakov Eliashberg**(Stanford University)Crossroads of symplectic rigidity and flexibility (ENGLISH)

[ Abstract ]

The development of flexible and rigid sides of symplectic and contact topology towards each other shaped this subject since its inception, and continues shaping its modern development.

In the series of lectures I will discuss the history of this struggle, as well as describe recent breakthroughs on the flexible side.

[ Reference URL ]The development of flexible and rigid sides of symplectic and contact topology towards each other shaped this subject since its inception, and continues shaping its modern development.

In the series of lectures I will discuss the history of this struggle, as well as describe recent breakthroughs on the flexible side.

http://faculty.ms.u-tokyo.ac.jp/Eliashberg201511.html

#### FMSP Lectures

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

Discretising systematically integrable systems (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Ramani1118.pdf

**Alfred Ramani**(Ecole Polytechnique)Discretising systematically integrable systems (ENGLISH)

[ Abstract ]

We present various methods for discretising integrable systerms inspired by the works of Hirota and Mickens. We apply these methods to the systematical discretisation of Painlevé equations.

[ Reference URL ]We present various methods for discretising integrable systerms inspired by the works of Hirota and Mickens. We apply these methods to the systematical discretisation of Painlevé equations.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Ramani1118.pdf

#### Mathematical Biology Seminar

14:55-16:40 Room #128 (Graduate School of Math. Sci. Bldg.)

Spatial population dynamics as a point pattern dynamics (JAPANESE)

http://www.ics.nara-wu.ac.jp/jp/staff/takasu.html

**Fugo TAKASU**(Nara Women's University)Spatial population dynamics as a point pattern dynamics (JAPANESE)

[ Abstract ]

Spatial population dynamics has been conventionally described as

dynamical system where population size (or population density) changes

with time over space as a continuous "real-valued" variable; these are

often given as partial differential equations as reaction-diffusion

models. In this approach, we implicitly assume infinitely large

population thereby population size changes smoothly and

deterministically. In reality, however, a population is a collection of

a certain number of individuals each of which gives birth or dies with

some stochasticity in a space and the population size as the number of

individuals is "integer-valued". In this talk, I introduce an approach

to reconstruct conventional spatial population dynamics in terms of

point pattern dynamics as a stochastic process. I discuss how to

mathematically describe such spatial stochastic processes using the

moments of increasing order of dimension; densities of points, pairs,

and triplets, etc. are described by integro-differential equations.

Quantification of a point pattern is the key issue here. As examples, I

introduce spatial epidemic SIS and SIR models as point pattern dynamics;

each individual has a certain "mark" depending on its health status; a

snapshot of individuals’ distribution over space is represented by a

marked point pattern and this marked point pattern dynamically changes

with time.

[ Reference URL ]Spatial population dynamics has been conventionally described as

dynamical system where population size (or population density) changes

with time over space as a continuous "real-valued" variable; these are

often given as partial differential equations as reaction-diffusion

models. In this approach, we implicitly assume infinitely large

population thereby population size changes smoothly and

deterministically. In reality, however, a population is a collection of

a certain number of individuals each of which gives birth or dies with

some stochasticity in a space and the population size as the number of

individuals is "integer-valued". In this talk, I introduce an approach

to reconstruct conventional spatial population dynamics in terms of

point pattern dynamics as a stochastic process. I discuss how to

mathematically describe such spatial stochastic processes using the

moments of increasing order of dimension; densities of points, pairs,

and triplets, etc. are described by integro-differential equations.

Quantification of a point pattern is the key issue here. As examples, I

introduce spatial epidemic SIS and SIR models as point pattern dynamics;

each individual has a certain "mark" depending on its health status; a

snapshot of individuals’ distribution over space is represented by a

marked point pattern and this marked point pattern dynamically changes

with time.

http://www.ics.nara-wu.ac.jp/jp/staff/takasu.html

#### Seminar on Probability and Statistics

17:00-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Order flow intensities for limit order book modelling

**Ioane Muni Toke**(University of New Caledonia)Order flow intensities for limit order book modelling

[ Abstract ]

Limit order books are at the core of electronic financial markets. Mathematical models of limit order books use point processes to model the arrival of limit, market and cancellation orders in the order book, but it is not clear what a "good" parametric model for the intensities of these point processes should be.

In the first part of the talk, we show that despite their simplicity basic Poisson processes can be used to accurately model a few features of the order book that more advanced models reproduce with volume-dependent intensities.

In the second part of the talk we present ongoing investigations in a more advanced statistical modelling of these order flow intensities using in particular normal mixture distributions and exponential models.

Limit order books are at the core of electronic financial markets. Mathematical models of limit order books use point processes to model the arrival of limit, market and cancellation orders in the order book, but it is not clear what a "good" parametric model for the intensities of these point processes should be.

In the first part of the talk, we show that despite their simplicity basic Poisson processes can be used to accurately model a few features of the order book that more advanced models reproduce with volume-dependent intensities.

In the second part of the talk we present ongoing investigations in a more advanced statistical modelling of these order flow intensities using in particular normal mixture distributions and exponential models.

### 2015/11/17

#### Tuesday Seminar on Topology

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

Topology of some three-dimensional singularities related to algebraic geometry (ENGLISH)

**Atsuko Katanaga**(Shinshu University)Topology of some three-dimensional singularities related to algebraic geometry (ENGLISH)

[ Abstract ]

In this talk, we deal with hypersurface isolated singularities. First, we will recall

some topological results of singularities. Next, we will sketch the classification of

singularities in algebraic geometry. Finally, we will focus on the three-dimensional

case and discuss some results obtained so far.

In this talk, we deal with hypersurface isolated singularities. First, we will recall

some topological results of singularities. Next, we will sketch the classification of

singularities in algebraic geometry. Finally, we will focus on the three-dimensional

case and discuss some results obtained so far.

#### Number Theory Seminar

18:00-19:00 Room #117 (Graduate School of Math. Sci. Bldg.)

The Tamagawa number formula over function fields. (English)

**Dennis Gaitsgory**(Harvard University & IHES)The Tamagawa number formula over function fields. (English)

[ Abstract ]

Let G be a semi-simple and simply connected group and X an algebraic curve. We consider $Bun_G(X)$, the moduli space of G-bundles on X. In their celebrated paper, Atiyah and Bott gave a formula for the cohomology of $Bun_G$, namely $H^*(Bun_G)=Sym(H_*(X)\otimes V)$, where V is the space of generators for $H^*_G(pt)$. When we take our ground field to be a finite field, the Atiyah-Bott formula implies the Tamagawa number conjecture for the function field of X.

The caveat here is that the A-B proof uses the interpretation of $Bun_G$ as the space of connection forms modulo gauge transformations, and thus only works over complex numbers (but can be extend to any field of characteristic zero). In the talk we will outline an algebro-geometric proof that works over any ground field. As its main geometric ingredient, it uses the fact that the space of rational maps from X to G is homologically contractible. Because of the nature of the latter statement, the proof necessarily uses tools from higher category theory. So, it can be regarded as an example how the latter can be used to prove something concrete: a construction at the level of 2-categories leads to an equality of numbers.

Let G be a semi-simple and simply connected group and X an algebraic curve. We consider $Bun_G(X)$, the moduli space of G-bundles on X. In their celebrated paper, Atiyah and Bott gave a formula for the cohomology of $Bun_G$, namely $H^*(Bun_G)=Sym(H_*(X)\otimes V)$, where V is the space of generators for $H^*_G(pt)$. When we take our ground field to be a finite field, the Atiyah-Bott formula implies the Tamagawa number conjecture for the function field of X.

The caveat here is that the A-B proof uses the interpretation of $Bun_G$ as the space of connection forms modulo gauge transformations, and thus only works over complex numbers (but can be extend to any field of characteristic zero). In the talk we will outline an algebro-geometric proof that works over any ground field. As its main geometric ingredient, it uses the fact that the space of rational maps from X to G is homologically contractible. Because of the nature of the latter statement, the proof necessarily uses tools from higher category theory. So, it can be regarded as an example how the latter can be used to prove something concrete: a construction at the level of 2-categories leads to an equality of numbers.

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