## Seminar information archive

Seminar information archive ～09/18｜Today's seminar 09/19 | Future seminars 09/20～

### 2016/10/24

#### Seminar on Geometric Complex Analysis

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

Structure and equivalence of a class of tube domains with solvable groups of automorphisms (JAPANESE)

**Satoru Shimizu**(Tohoku University)Structure and equivalence of a class of tube domains with solvable groups of automorphisms (JAPANESE)

[ Abstract ]

In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of $\frak g(T_{\Omega})$ plays an important role, where $\frak g(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.

In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of $\frak g(T_{\Omega})$ plays an important role, where $\frak g(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.

#### Operator Algebra Seminars

16:45-18:15 Room #117 (Graduate School of Math. Sci. Bldg.)

Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

**Yusuke Isono**(RIMS, Kyoto Univ.)Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

### 2016/10/18

#### Tuesday Seminar on Topology

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

Conformal field theory for extended W-algebras (JAPANESE)

**Yoshitake Hashimoto**(Tokyo City University)Conformal field theory for extended W-algebras (JAPANESE)

[ Abstract ]

This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of

"infinitesimal deformation of parameter" and Jack symmetric polynomials.

In this talk I will discuss the following subjects:

1. "factorization" in conformal field theory,

2. tensor structure of the category of M-modules and "module-bimodule correspondence".

This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of

"infinitesimal deformation of parameter" and Jack symmetric polynomials.

In this talk I will discuss the following subjects:

1. "factorization" in conformal field theory,

2. tensor structure of the category of M-modules and "module-bimodule correspondence".

### 2016/10/17

#### Seminar on Geometric Complex Analysis

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

(JAPANESE)

**Takaaki Nomura**(Kyushu University)(JAPANESE)

#### Operator Algebra Seminars

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

Unitarizability, Maurey-Nikishin factorization and Polish groups of finite type (English)

**Hiroshi Ando**(Chiba Univ.)Unitarizability, Maurey-Nikishin factorization and Polish groups of finite type (English)

### 2016/10/12

#### Number Theory Seminar

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

Filtered de Rham Witt complexes and wildly ramified higher class field theory over finite fields (joint work with Shuji Saito and Yigeng Zhao) (English)

**Uwe Jannsen**(Universität Regensburg, The University of Tokyo)Filtered de Rham Witt complexes and wildly ramified higher class field theory over finite fields (joint work with Shuji Saito and Yigeng Zhao) (English)

[ Abstract ]

We will consider abelian coverings of smooth projective varieties over finite fields which are wildly ramified along a divisor D with normal crossings, and will describe the corresponding abelianized fundamental group via modified logarithmic de Rham-Witt sheaves.

We will consider abelian coverings of smooth projective varieties over finite fields which are wildly ramified along a divisor D with normal crossings, and will describe the corresponding abelianized fundamental group via modified logarithmic de Rham-Witt sheaves.

### 2016/10/11

#### PDE Real Analysis Seminar

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

Global solutions to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations (English)

**Nam Quang Le**(Indiana University)Global solutions to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations (English)

[ Abstract ]

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem under some restrictions on the sign or integrability of the affine mean curvature. In this talk, we explain how to remove these restrictions and obtain global solutions under optimal integrability conditions on the affine mean curvature. Our analysis also covers the case of Abreu's equation arising in complex geometry.

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem under some restrictions on the sign or integrability of the affine mean curvature. In this talk, we explain how to remove these restrictions and obtain global solutions under optimal integrability conditions on the affine mean curvature. Our analysis also covers the case of Abreu's equation arising in complex geometry.

#### Algebraic Geometry Seminar

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

On varieties with splittings of relative Frobenius morphisms of Albanese maps

**Sho Ejiri**(University of Tokyo)On varieties with splittings of relative Frobenius morphisms of Albanese maps

[ Abstract ]

Varieties with splittings of Frobenius morphisms are called F-split varieties, which satisfy strong properties such as Kodaira vanishing. However, some important varieties are not F-split. For example, an abelian variety is F-split if and only if its p-rank is maximum. In this talk, we discuss the class of varieties with splittings of relative Frobenius morphisms of Albanese maps, which includes abelian varieties. As a consequence of Sannai and Tanaka's characterization of ordinary abelian varieties, we see that this class also includes F-split varieties. Furthermore, for varieties in this class, we show that the Kodaira vanishing theorem holds, and that Albanese maps are algebraic fiber spaces.

Varieties with splittings of Frobenius morphisms are called F-split varieties, which satisfy strong properties such as Kodaira vanishing. However, some important varieties are not F-split. For example, an abelian variety is F-split if and only if its p-rank is maximum. In this talk, we discuss the class of varieties with splittings of relative Frobenius morphisms of Albanese maps, which includes abelian varieties. As a consequence of Sannai and Tanaka's characterization of ordinary abelian varieties, we see that this class also includes F-split varieties. Furthermore, for varieties in this class, we show that the Kodaira vanishing theorem holds, and that Albanese maps are algebraic fiber spaces.

#### Tuesday Seminar on Topology

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

The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebra in genus zero (JAPANESE)

**Nariya Kawazumi**(The University of Tokyo)The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebra in genus zero (JAPANESE)

[ Abstract ]

In view of results of Goldman and Turaev, the free vector space over the free loops on an oriented surface has a natural Lie bialgebra structure. The Goldman bracket has a formal description by using a special (or symplectic) expansion of the fundamental group of the surface. It is natural to ask for a formal description of the Turaev cobracket. We will show how to obtain a formal description of the Goldman-Turaev Lie bialgebra for genus 0 using a solution of the Kashiwara-Vergne problem. A similar description was recently obtained by Massuyeau using the Kontsevich integral. Moreover we propose a generalization of the Kashiwara-Vergne problem in the context of the Goldman-Turaev Lie bialgebra. This talk is based on a joint work with A. Alekseev, Y. Kuno and F. Naef.

In view of results of Goldman and Turaev, the free vector space over the free loops on an oriented surface has a natural Lie bialgebra structure. The Goldman bracket has a formal description by using a special (or symplectic) expansion of the fundamental group of the surface. It is natural to ask for a formal description of the Turaev cobracket. We will show how to obtain a formal description of the Goldman-Turaev Lie bialgebra for genus 0 using a solution of the Kashiwara-Vergne problem. A similar description was recently obtained by Massuyeau using the Kontsevich integral. Moreover we propose a generalization of the Kashiwara-Vergne problem in the context of the Goldman-Turaev Lie bialgebra. This talk is based on a joint work with A. Alekseev, Y. Kuno and F. Naef.

#### Seminar on Probability and Statistics

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

### 2016/10/04

#### Algebraic Geometry Seminar

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

Higher order minimal families of rational curves and Fano manifolds with nef Chern characters (Japanese. Writing in English. )

**Taku Suzuki**(Waseda University)Higher order minimal families of rational curves and Fano manifolds with nef Chern characters (Japanese. Writing in English. )

[ Abstract ]

In this talk, we introduce higher order minimal families $H_i$ of rational curves

associated to Fano manifolds $X$. We prove that $H_i$ is also a Fano manifold

if the Chern characters of $X$ satisfy some positivity conditions. We also provide

a sufficient condition for Fano manifolds to be covered by higher rational manifolds.

In this talk, we introduce higher order minimal families $H_i$ of rational curves

associated to Fano manifolds $X$. We prove that $H_i$ is also a Fano manifold

if the Chern characters of $X$ satisfy some positivity conditions. We also provide

a sufficient condition for Fano manifolds to be covered by higher rational manifolds.

#### Colloquium

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

Waning and boosting : on the dynamics of immune status (ENGLISH)

http://www.uu.nl/staff/ODiekmann

**Odo Diekmann**(Utrecht University)Waning and boosting : on the dynamics of immune status (ENGLISH)

[ Abstract ]

A first aim is to briefly review various mathematical models of infectious disease dynamics that incorporate waning and boosting of immunity. The focus will be on models that are described by delay equations, in particular renewal equations [1]. Concerning within-host dynamics, we limit ourselves to the rather caricatural models of Aron [2] and de Graaf e.a. [3].From a biomedical point of view the main conclusion is that a higher force of infection may lead to less disease,see [4] and the references given there.

[1] O.Diekmann, M.Gyllenberg, J.A.J.Metz, H.R.Thieme, On the formulation and analysis

of general deterministic structured population models. I. Linear theory, J. Math. Biol. (1998) 36 : 349 - 388

[2] J.L. Aron, Dynamics of acquired immunity boosted by exposure to infection, Math. Biosc. (1983) 64 : 249-259

[3] W.F. de Graaf, M.E.E. Kretzschmar, P.M.F. Teunis, O. Diekmann, A two-phase within host model for immune response and its application to seriological profiles of pertussis, Epidemics (2014) 9 : 1-7

[4] A.N. Swart, M. Tomasi, M. Kretzschmar, A.H. Havelaar, O. Diekmann, The protective effect of temporary immunity under imposed infection pressure, Epidemics (2012) 4 : 43-47

[ Reference URL ]A first aim is to briefly review various mathematical models of infectious disease dynamics that incorporate waning and boosting of immunity. The focus will be on models that are described by delay equations, in particular renewal equations [1]. Concerning within-host dynamics, we limit ourselves to the rather caricatural models of Aron [2] and de Graaf e.a. [3].From a biomedical point of view the main conclusion is that a higher force of infection may lead to less disease,see [4] and the references given there.

[1] O.Diekmann, M.Gyllenberg, J.A.J.Metz, H.R.Thieme, On the formulation and analysis

of general deterministic structured population models. I. Linear theory, J. Math. Biol. (1998) 36 : 349 - 388

[2] J.L. Aron, Dynamics of acquired immunity boosted by exposure to infection, Math. Biosc. (1983) 64 : 249-259

[3] W.F. de Graaf, M.E.E. Kretzschmar, P.M.F. Teunis, O. Diekmann, A two-phase within host model for immune response and its application to seriological profiles of pertussis, Epidemics (2014) 9 : 1-7

[4] A.N. Swart, M. Tomasi, M. Kretzschmar, A.H. Havelaar, O. Diekmann, The protective effect of temporary immunity under imposed infection pressure, Epidemics (2012) 4 : 43-47

http://www.uu.nl/staff/ODiekmann

### 2016/10/03

#### Seminar on Geometric Complex Analysis

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

Visualizing the radial Loewner flow and the evolution family (JAPANESE)

**Hirokazu Shimauchi**(Yamanashi Eiwa College)Visualizing the radial Loewner flow and the evolution family (JAPANESE)

#### Operator Algebra Seminars

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

$C^*$-tensor categories and subfactors for totally disconnected groups

(English)

**Yuki Arano**(Univ. Tokyo)$C^*$-tensor categories and subfactors for totally disconnected groups

(English)

#### Tokyo Probability Seminar

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

Coupled KPZ equations and complex-valued stochastic Ginzburg-Landau equation (日本語)

**Masato Hoshino**(Graduate School of Mathematical Science, the University of Tokyo)Coupled KPZ equations and complex-valued stochastic Ginzburg-Landau equation (日本語)

### 2016/09/27

#### Tuesday Seminar on Topology

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

CAT(0) properties for orthoscheme complexes (JAPANESE)

**Shouta Tounai**(The University of Tokyo)CAT(0) properties for orthoscheme complexes (JAPANESE)

[ Abstract ]

Gromov showed that a cubical complex is locally CAT(0) if and only if the link of every vertex is a flag complex. Brady and MacCammond introduced an orthoscheme complex as a generalization of cubical complexes. It is, however, difficult to tell whether an orthoscheme complex is (locally) CAT(0) or not. In this talk, I will discuss a translation of Gromov's characterization for orthoscheme complexes. As a generalization of Gromov's characterization, I will show that the orthoscheme complex of locally distributive semilattice is CAT(0) if and only if it is a flag semilattice.

Gromov showed that a cubical complex is locally CAT(0) if and only if the link of every vertex is a flag complex. Brady and MacCammond introduced an orthoscheme complex as a generalization of cubical complexes. It is, however, difficult to tell whether an orthoscheme complex is (locally) CAT(0) or not. In this talk, I will discuss a translation of Gromov's characterization for orthoscheme complexes. As a generalization of Gromov's characterization, I will show that the orthoscheme complex of locally distributive semilattice is CAT(0) if and only if it is a flag semilattice.

### 2016/09/26

#### Operator Algebra Seminars

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

TBA

(English)

**Sorin Popa**(UCLA)TBA

(English)

#### FMSP Lectures

16:00-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Mathematical Aesthetic Principles and Nonintegrable Systems (ENGLISH)

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

**Murray Muraskin**(University of North Dakota, Grand Forks)Mathematical Aesthetic Principles and Nonintegrable Systems (ENGLISH)

[ Abstract ]

The discussion presents a study of a set of mathematical principles that can be classified as "aesthetic”and shows that these principles can be cast into a set of nonlinear equations. The system of equations is nonintegrable in general. New techniques to handle the nonintegrability feature are discussed. We then illustrate how this system of equations leads to sinusoidal solutions, sine within sine solutions, the phenomenon known as beats, random type oscillations, two and three dimensional lattices, as well as multi wave packet systems. The sinusoidal solutions occur when the arbitrary data associated with the equations causes the equations to be linearized. The sinusoidal behavior totally disappears once the integrability equations are satisfied, illustrating how important the nonintegrability concept is to the development.

[ Reference URL ]The discussion presents a study of a set of mathematical principles that can be classified as "aesthetic”and shows that these principles can be cast into a set of nonlinear equations. The system of equations is nonintegrable in general. New techniques to handle the nonintegrability feature are discussed. We then illustrate how this system of equations leads to sinusoidal solutions, sine within sine solutions, the phenomenon known as beats, random type oscillations, two and three dimensional lattices, as well as multi wave packet systems. The sinusoidal solutions occur when the arbitrary data associated with the equations causes the equations to be linearized. The sinusoidal behavior totally disappears once the integrability equations are satisfied, illustrating how important the nonintegrability concept is to the development.

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

### 2016/08/29

#### PDE Real Analysis Seminar

10:30-11:30 Room #268 (Graduate School of Math. Sci. Bldg.)

The Navier-Stokes equations: stationary existence, conditional regularity, and self-similar singularities (English)

https://www.math.lsu.edu/~pcnguyen/

**Nguyen Cong Phuc**(Louisiana State University)The Navier-Stokes equations: stationary existence, conditional regularity, and self-similar singularities (English)

[ Abstract ]

In this talk, both stationary and time-dependent Navier-Stokes equations are discussed. The common theme is that the quadratic nonlinearity and the pressure are both treated as weights generally belonging to a Sobolev space of negative order. We obtain the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces that belong to a critical space. This result can be viewed as the stationary counterpart of an existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with small initial data in $BMO^{-1}$. In another direction, some new local energy bounds are obtained for the time-dependent Navier-Stokes equations which imply the regularity condition $L_{t}^{\infty}(X)$, where $X$ is a non-endpoint borderline Lorentz space $X=L_{x}^{3, q}, q\not=\infty$. The analysis also allows us to rule out the existence of Leray's backward self-similar solutions to the Navier–Stokes equations with profiles in $L^{12/5}(\mathbb{R}^3)$ or in the Marcinkiewicz space $L^{q, \infty}(\mathbb{R}^{3})$ for any $q \in (12/5, 6)$.

This talk is based on joint work with Tuoc Van Phan and Cristi Guevara.

[ Reference URL ]In this talk, both stationary and time-dependent Navier-Stokes equations are discussed. The common theme is that the quadratic nonlinearity and the pressure are both treated as weights generally belonging to a Sobolev space of negative order. We obtain the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces that belong to a critical space. This result can be viewed as the stationary counterpart of an existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with small initial data in $BMO^{-1}$. In another direction, some new local energy bounds are obtained for the time-dependent Navier-Stokes equations which imply the regularity condition $L_{t}^{\infty}(X)$, where $X$ is a non-endpoint borderline Lorentz space $X=L_{x}^{3, q}, q\not=\infty$. The analysis also allows us to rule out the existence of Leray's backward self-similar solutions to the Navier–Stokes equations with profiles in $L^{12/5}(\mathbb{R}^3)$ or in the Marcinkiewicz space $L^{q, \infty}(\mathbb{R}^{3})$ for any $q \in (12/5, 6)$.

This talk is based on joint work with Tuoc Van Phan and Cristi Guevara.

https://www.math.lsu.edu/~pcnguyen/

### 2016/08/12

#### FMSP Lectures

15:00-16:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Multiscale simulations of waves and applications (ENGLISH)

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

**Eric Chung**(Chinese Univ. of Hong Kong)Multiscale simulations of waves and applications (ENGLISH)

[ Abstract ]

Numerical simulations of wave propagation in heterogeneous media are important in many applications such as seismic propagation and seismic inversion.

In this talk, we will present a new multiscale approach for seismic wave propagation.

The method is able to compute the solution with much fewer degrees of freedom compared with fine mesh simulation.

The idea is to capture the multiscale features of the solutions by carefully designed multiscale basis functions.

We will also present applications to inverse problems.

[ Reference URL ]Numerical simulations of wave propagation in heterogeneous media are important in many applications such as seismic propagation and seismic inversion.

In this talk, we will present a new multiscale approach for seismic wave propagation.

The method is able to compute the solution with much fewer degrees of freedom compared with fine mesh simulation.

The idea is to capture the multiscale features of the solutions by carefully designed multiscale basis functions.

We will also present applications to inverse problems.

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

#### FMSP Lectures

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

Discrete regularization of parameter identification problems (ENGLISH)

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

**Christian Clason**(University Duisburg-Essen)Discrete regularization of parameter identification problems (ENGLISH)

[ Abstract ]

This talk is concerned with parameter identification problems where a distributed parameter is known a priori to take on values from a given set. This property can be promoted with the aid of a convex regularization term in the Tikhonov functional. We discuss the properties of minimizers of this functional and their numerical computation using a semismooth Newton method.

[ Reference URL ]This talk is concerned with parameter identification problems where a distributed parameter is known a priori to take on values from a given set. This property can be promoted with the aid of a convex regularization term in the Tikhonov functional. We discuss the properties of minimizers of this functional and their numerical computation using a semismooth Newton method.

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

### 2016/08/10

#### FMSP Lectures

10:00-11:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Global-local-integration-based kernel approximation methods: Technical arguments (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hon2.pdf

**Benny Y C Hon**(City Univ. of Hong Kong)Global-local-integration-based kernel approximation methods: Technical arguments (ENGLISH)

[ Abstract ]

We discuss technical details of my talk on 8 Aug. and give also proofs of some main results.

[ Reference URL ]We discuss technical details of my talk on 8 Aug. and give also proofs of some main results.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hon2.pdf

#### Seminar on Probability and Statistics

13:00-14:30 Room #117 (Graduate School of Math. Sci. Bldg.)

**David Nualart**(Kansas University)### 2016/08/09

#### Seminar on Probability and Statistics

13:00-16:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Malliavin calculus and normal approximations

http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=180

**David Nualart**(Kansas University)Malliavin calculus and normal approximations

[ Abstract ]

The purpose of these lectures is to introduce some recent results on the application of Malliavin calculus combined with Stein's method to normal approximation. The Malliavin calculus is a differential calculus on the Wiener space. First, we will present some elements of Malliavin calculus, defining the basic differential operators: the derivative, its adjoint called the divergence operator and the generator of the Ornstein-Uhlenbeck semigroup. The behavior of these operators on the Wiener chaos expansion will be discussed. Then, we will introduce the Stein's method for normal approximation, which leads to general bounds for the Kolmogorov and total variation distances between the law of a Brownian functional and the standard normal distribution. In this context, the integration by parts formula of Malliavin calculus will allow us to express these bounds in terms of the Malliavin operators. We will present the application of this methodology to derive the Fourth Moment Theorem for a sequence of multiple stochastic integrals, and we will discuss some results on the uniform convergence of densities obtained using Malliavin calculus techniques. Finally, examples of functionals of Gaussian processes, such as the fractional Brownian motion, will be discussed.

[ Reference URL ]The purpose of these lectures is to introduce some recent results on the application of Malliavin calculus combined with Stein's method to normal approximation. The Malliavin calculus is a differential calculus on the Wiener space. First, we will present some elements of Malliavin calculus, defining the basic differential operators: the derivative, its adjoint called the divergence operator and the generator of the Ornstein-Uhlenbeck semigroup. The behavior of these operators on the Wiener chaos expansion will be discussed. Then, we will introduce the Stein's method for normal approximation, which leads to general bounds for the Kolmogorov and total variation distances between the law of a Brownian functional and the standard normal distribution. In this context, the integration by parts formula of Malliavin calculus will allow us to express these bounds in terms of the Malliavin operators. We will present the application of this methodology to derive the Fourth Moment Theorem for a sequence of multiple stochastic integrals, and we will discuss some results on the uniform convergence of densities obtained using Malliavin calculus techniques. Finally, examples of functionals of Gaussian processes, such as the fractional Brownian motion, will be discussed.

http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=180

### 2016/08/08

#### FMSP Lectures

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

Global-local-integration-based kernel approximation methods (ENGLISH)

[ Reference URL ]

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

**Benny Y C Hon**(City Univ. of Hong Kong)Global-local-integration-based kernel approximation methods (ENGLISH)

[ Reference URL ]

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

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