## Seminar information archive

Seminar information archive ～09/14｜Today's seminar 09/15 | Future seminars 09/16～

### 2018/10/31

#### FMSP Lectures

15:00-16:30 Room #122 (Graduate School of Math. Sci. Bldg.)

K-THEORY AND THE DIRAC OPERATOR (4/4)

Lecture 4. BEYOND ELLIPTICITY or K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS (ENGLISH)

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

**Paul Baum**(The Pennsylvania State University)K-THEORY AND THE DIRAC OPERATOR (4/4)

Lecture 4. BEYOND ELLIPTICITY or K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS (ENGLISH)

[ Abstract ]

K-homology is the dual theory to K-theory. The BD (Baum-Douglas) isomorphism of Atiyah-Kasparov K-homology and K-cycle K-homology provides a framework within which the Atiyah-Singer index theorem can be extended to certain differential operators which are hypoelliptic but not elliptic. This talk will consider such a class of differential operators on compact contact manifolds. These operators have been studied by a number of mathematicians (e.g. C.Epstein and R.Melrose).

Operators with similar analytical properties have also been studied (e.g. by Alain Connes and Henri Moscovici --- also Michel Hilsum and Georges Skandalis). Working within the BD framework, the index problem will be solved for these differential operators on compact contact manifolds.

This is joint work with Erik van Erp.

[ Reference URL ]K-homology is the dual theory to K-theory. The BD (Baum-Douglas) isomorphism of Atiyah-Kasparov K-homology and K-cycle K-homology provides a framework within which the Atiyah-Singer index theorem can be extended to certain differential operators which are hypoelliptic but not elliptic. This talk will consider such a class of differential operators on compact contact manifolds. These operators have been studied by a number of mathematicians (e.g. C.Epstein and R.Melrose).

Operators with similar analytical properties have also been studied (e.g. by Alain Connes and Henri Moscovici --- also Michel Hilsum and Georges Skandalis). Working within the BD framework, the index problem will be solved for these differential operators on compact contact manifolds.

This is joint work with Erik van Erp.

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

#### Operator Algebra Seminars

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

Strong Tools in Free Probability Theory

**Tomohiro Hayase**(the University of Tokyo)Strong Tools in Free Probability Theory

### 2018/10/30

#### Tuesday Seminar of Analysis

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

Spectral structure of the Neumann-Poincaré operator in three dimensions: Willmore energy and surface geometry (日本語)

**MIYANISHI Yoshihisa**(Osaka University)Spectral structure of the Neumann-Poincaré operator in three dimensions: Willmore energy and surface geometry (日本語)

[ Abstract ]

The Neumann-Poincaré operator (abbreviated by NP) is a boundary integral operator naturally arising when solving classical boundary value problems using layer potentials. If the boundary of the domain, on which the NP operator is defined, is $C^{1, \alpha}$ smooth, then the NP operator is compact. Thus, the Fredholm integral equation, which appears when solving Dirichlet or Neumann problems, can be solved using the Fredholm index theory.

Regarding spectral properties of the NP operator, the spectrum consists of eigenvalues converging to $0$ for $C^{1, \alpha}$ smooth boundaries. Our main purpose here is to deduce eigenvalue asymptotics of the NP operators in three dimensions. This formula is the so-called Weyl's law for eigenvalue problems of NP operators. Then we discuss relationships among the Weyl's law, the Euler characteristic and the Willmore energy on the boundary surface.

The Neumann-Poincaré operator (abbreviated by NP) is a boundary integral operator naturally arising when solving classical boundary value problems using layer potentials. If the boundary of the domain, on which the NP operator is defined, is $C^{1, \alpha}$ smooth, then the NP operator is compact. Thus, the Fredholm integral equation, which appears when solving Dirichlet or Neumann problems, can be solved using the Fredholm index theory.

Regarding spectral properties of the NP operator, the spectrum consists of eigenvalues converging to $0$ for $C^{1, \alpha}$ smooth boundaries. Our main purpose here is to deduce eigenvalue asymptotics of the NP operators in three dimensions. This formula is the so-called Weyl's law for eigenvalue problems of NP operators. Then we discuss relationships among the Weyl's law, the Euler characteristic and the Willmore energy on the boundary surface.

#### PDE Real Analysis Seminar

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

The least gradient problem in the plain (English)

**Piotr Rybka**(University of Warsaw)The least gradient problem in the plain (English)

[ Abstract ]

The least gradient problem arises in many application, e.g. in the free material design. We show existence of solutions in bounded, strictly convex planar regions, when the data are functions on bounded variation.

Our main goal is to show existence of solution in convex, but not necessarily strictly convex planar regions. In order to avoid technicalities we consider only continuous data, but BV data will do to. We formulate a set of admissibility conditions. We show that they are sufficient for existence.

This is a joint project with Wojciech Górny and Ahmad Sabra.

The least gradient problem arises in many application, e.g. in the free material design. We show existence of solutions in bounded, strictly convex planar regions, when the data are functions on bounded variation.

Our main goal is to show existence of solution in convex, but not necessarily strictly convex planar regions. In order to avoid technicalities we consider only continuous data, but BV data will do to. We formulate a set of admissibility conditions. We show that they are sufficient for existence.

This is a joint project with Wojciech Górny and Ahmad Sabra.

#### Tuesday Seminar on Topology

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

The quasiconformal equivalence of Riemann surfaces and a universality of Schottky spaces (JAPANESE)

**Hiroshige Shiga**(Tokyo Institute of Technology)The quasiconformal equivalence of Riemann surfaces and a universality of Schottky spaces (JAPANESE)

[ Abstract ]

In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated.

In this talk, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to see a universality of Schottky spaces.

In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated.

In this talk, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to see a universality of Schottky spaces.

#### Seminar on Probability and Statistics

15:30-16:40 Room #126 (Graduate School of Math. Sci. Bldg.)

Asymptotic expansion for random vectors

**Ciprian A. Tudor**(Université de Lille 1, Université de Panthéon-Sorbonne Paris 1)Asymptotic expansion for random vectors

[ Abstract ]

We develop the asymptotic expansion theory for vector-valued sequences $F_{N}$ of random variables. We find the second-order term in the expansion of the density of $F_{N}$, based on assumptions in terms of the convergence of the Stein-Malliavin matrix associated to the sequence $F_{N}$ . Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of $F_{N}$ and we discuss the main ideas on higher order asymptotic expansion. We illustrate our results by several examples.

We develop the asymptotic expansion theory for vector-valued sequences $F_{N}$ of random variables. We find the second-order term in the expansion of the density of $F_{N}$, based on assumptions in terms of the convergence of the Stein-Malliavin matrix associated to the sequence $F_{N}$ . Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of $F_{N}$ and we discuss the main ideas on higher order asymptotic expansion. We illustrate our results by several examples.

### 2018/10/29

#### Tokyo Probability Seminar

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

On Hydrodynamic Limits of Young Diagrams (ENGLISH)

http://math.arizona.edu/~sethuram/

**Sunder Sethuraman**(University of Arizona)On Hydrodynamic Limits of Young Diagrams (ENGLISH)

[ Abstract ]

We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. `Static' scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding `dynamical' limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types of parabolic PDEs, depending on the energy structure.

The talk will be based on the article: https://arxiv.org/abs/1809.03592

[ Reference URL ]We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. `Static' scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding `dynamical' limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types of parabolic PDEs, depending on the energy structure.

The talk will be based on the article: https://arxiv.org/abs/1809.03592

http://math.arizona.edu/~sethuram/

#### Seminar on Geometric Complex Analysis

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

On morphisms of compact Kaehler manifolds with semi-positive holomorphic sectional curvature (JAPANESE)

**Shin-ichi Matsumura**(Tohoku University)On morphisms of compact Kaehler manifolds with semi-positive holomorphic sectional curvature (JAPANESE)

[ Abstract ]

In this talk, we consider a smooth projective variety $X$ with semi-positive holomorphic "sectional" curvature, motivated by generalizing Howard-Smyth-Wu's structure theorem and Mok's result for compact Kaehler manifold with semi-positive "bisectional" curvature.

We prove that, if $X$ admits a holomorphic maximally rationally connected fibration $X ¥to Y$, then the morphism is always smooth (that is, a submersion), that the image $Y$ admits a finite ¥'etale cover $T ¥to Y$ by a complex

torus $T$, and further that all the fibers $F$ are isomorphic.

This gives a structure theorem for $X$ when $X$ is a surface.

Moreover we show that $X$ is rationally connected, if the holomorphic sectional curvature is quasi-positive.

This result gives a generalization of Yau's conjecture.

In this talk, we consider a smooth projective variety $X$ with semi-positive holomorphic "sectional" curvature, motivated by generalizing Howard-Smyth-Wu's structure theorem and Mok's result for compact Kaehler manifold with semi-positive "bisectional" curvature.

We prove that, if $X$ admits a holomorphic maximally rationally connected fibration $X ¥to Y$, then the morphism is always smooth (that is, a submersion), that the image $Y$ admits a finite ¥'etale cover $T ¥to Y$ by a complex

torus $T$, and further that all the fibers $F$ are isomorphic.

This gives a structure theorem for $X$ when $X$ is a surface.

Moreover we show that $X$ is rationally connected, if the holomorphic sectional curvature is quasi-positive.

This result gives a generalization of Yau's conjecture.

#### FMSP Lectures

15:00-16:30 Room #117 (Graduate School of Math. Sci. Bldg.)

K-THEORY AND THE DIRAC OPERATOR (3/4)

Lecture 3. THE RIEMANN-ROCH THEOREM (ENGLISH)

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

**Paul Baum**(The Pennsylvania State University)K-THEORY AND THE DIRAC OPERATOR (3/4)

Lecture 3. THE RIEMANN-ROCH THEOREM (ENGLISH)

[ Abstract ]

Topics in this talk :

1. Classical Riemann-Roch

2. Hirzebruch-Riemann-Roch (HRR)

3. Grothendieck-Riemann-Roch (GRR)

4. RR for possibly singular varieties (Baum-Fulton-MacPherson)

[ Reference URL ]Topics in this talk :

1. Classical Riemann-Roch

2. Hirzebruch-Riemann-Roch (HRR)

3. Grothendieck-Riemann-Roch (GRR)

4. RR for possibly singular varieties (Baum-Fulton-MacPherson)

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

### 2018/10/26

#### Colloquium

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

Asymptotic behavior of generalized eigenfunctions and scattering theory

(JAPANESE)

**Kenichi ITO**(The University of Tokyo)Asymptotic behavior of generalized eigenfunctions and scattering theory

(JAPANESE)

### 2018/10/24

#### Operator Algebra Seminars

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

The Mazur-Ulam property for unital C*-algebras (English)

**Michiya Mori**(the University of Tokyo)The Mazur-Ulam property for unital C*-algebras (English)

#### FMSP Lectures

15:00-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)

K-THEORY AND THE DIRAC OPERATOR (2/4)

Lecture 2. THE DIRAC OPERATOR (ENGLISH)

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

**Paul Baum**(The Pennsylvania State University)K-THEORY AND THE DIRAC OPERATOR (2/4)

Lecture 2. THE DIRAC OPERATOR (ENGLISH)

[ Abstract ]

The Dirac operator of R^n will be defined. This is a first order elliptic differential operator with constant coefficients.

Next, the class of differentiable manifolds which come equipped with an order one differential operator which (at the symbol level)is locally isomorphic to the Dirac operator of R^n will be considered. These are the Spin-c manifolds.

Spin-c is slightly stronger than oriented, so Spin-c can be viewed as "oriented plus epsilon". Most of the oriented manifolds that occur in practice are Spin-c. The Dirac operator of a closed Spin-c manifold is the basic example for the Hirzebruch-Riemann-Roch theorem and the Atiyah-Singer index theorem.

[ Reference URL ]The Dirac operator of R^n will be defined. This is a first order elliptic differential operator with constant coefficients.

Next, the class of differentiable manifolds which come equipped with an order one differential operator which (at the symbol level)is locally isomorphic to the Dirac operator of R^n will be considered. These are the Spin-c manifolds.

Spin-c is slightly stronger than oriented, so Spin-c can be viewed as "oriented plus epsilon". Most of the oriented manifolds that occur in practice are Spin-c. The Dirac operator of a closed Spin-c manifold is the basic example for the Hirzebruch-Riemann-Roch theorem and the Atiyah-Singer index theorem.

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

### 2018/10/23

#### PDE Real Analysis Seminar

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

Least action principle for incompressible flow with free boundary (English)

**Jian-Guo Liu**(Duke University)Least action principle for incompressible flow with free boundary (English)

[ Abstract ]

In this talk I will describe a connection between Arnold's least-action principle for incompressible flows with free boundary and geodesic paths for Wasserstein distance. The least-action problem for geodesic distance on the "manifold" of fluid-blob shapes exhibits instability due to microdroplet formation. Using a conformal map formulation we investigate singularity formation in water-wave dynamics neglecting gravity. A connection with fluid mixture models via a variant of Brenier's relaxed least-action principle for generalized Euler flows will also be discussed.

In this talk I will describe a connection between Arnold's least-action principle for incompressible flows with free boundary and geodesic paths for Wasserstein distance. The least-action problem for geodesic distance on the "manifold" of fluid-blob shapes exhibits instability due to microdroplet formation. Using a conformal map formulation we investigate singularity formation in water-wave dynamics neglecting gravity. A connection with fluid mixture models via a variant of Brenier's relaxed least-action principle for generalized Euler flows will also be discussed.

#### Tuesday Seminar on Topology

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

Co-Minkowski space and hyperbolic surfaces (ENGLISH)

**François Fillastre**(Université de Cergy-Pontoise)Co-Minkowski space and hyperbolic surfaces (ENGLISH)

[ Abstract ]

There are many ways to parametrize two copies of Teichmueller space by constant curvature -1 Riemannian or Lorentzian 3d manifolds (for example the Bers double uniformization theorem). We present the co-Minkowski space (or half-pipe space), which is a constant curvature -1 degenerated 3d space, and which is related to the tangent space of Teichmueller space. As an illustration, we give a new proof of a theorem of Thurston saying that, once the space of measured geodesic laminations on a compact hyperbolic surface is identified with the tangent space of Teichmueller space via infinitesimal earthquake, then the length of laminations is an asymmetric norm. Joint work with Thierry Barbot (Avignon).

There are many ways to parametrize two copies of Teichmueller space by constant curvature -1 Riemannian or Lorentzian 3d manifolds (for example the Bers double uniformization theorem). We present the co-Minkowski space (or half-pipe space), which is a constant curvature -1 degenerated 3d space, and which is related to the tangent space of Teichmueller space. As an illustration, we give a new proof of a theorem of Thurston saying that, once the space of measured geodesic laminations on a compact hyperbolic surface is identified with the tangent space of Teichmueller space via infinitesimal earthquake, then the length of laminations is an asymmetric norm. Joint work with Thierry Barbot (Avignon).

### 2018/10/22

#### Tokyo Probability Seminar

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Limit theorems for random geometric complexes in the critical regime (ENGLISH)

**Trinh Khanh Duy**(Tohoku University)Limit theorems for random geometric complexes in the critical regime (ENGLISH)

[ Abstract ]

Geometric complexes (eg. Cech complexes or Rips complexes) are simplicial complexes defined on a finite set of points in a Euclidean space together with a radius parameter, which can be viewed as a higher dimensional generalization of geometric graphs. This talk concerns with random geometric complexes built over binomial point processes (collections of iid points). Like random geometric graphs, there are three regimes (subcritical(or dust, sparse) regime, critical (or thermodynamic) regime and supercritical regime) which are divided according the growth of the radius parameters in which the limiting behavior of random geometric complexes is totally different. This talk introduces some results on the strong law of large numbers and a central limit theorem in the critical regime.

Geometric complexes (eg. Cech complexes or Rips complexes) are simplicial complexes defined on a finite set of points in a Euclidean space together with a radius parameter, which can be viewed as a higher dimensional generalization of geometric graphs. This talk concerns with random geometric complexes built over binomial point processes (collections of iid points). Like random geometric graphs, there are three regimes (subcritical(or dust, sparse) regime, critical (or thermodynamic) regime and supercritical regime) which are divided according the growth of the radius parameters in which the limiting behavior of random geometric complexes is totally different. This talk introduces some results on the strong law of large numbers and a central limit theorem in the critical regime.

#### Seminar on Geometric Complex Analysis

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

On certain hyperconvex manifolds without non-constant bounded holomorphic functions (JAPANESE)

**Masanori Adachi**(Shizuoka University)On certain hyperconvex manifolds without non-constant bounded holomorphic functions (JAPANESE)

[ Abstract ]

For each compact Riemann surface of genus > 1, we can construct a Riemann sphere bundle over the given Riemann surface using the projective structure induced by its uniformization.

The total space of this bundle is divided into two 1-convex domains by a closed Levi-flat real hypersurface. Although these two domains are not biholomorphic, we see that they have several function theoretic properties in common. In this talk, I would like to explain these common properties: hyperconvexity and expressions for certain Green function, and Liouville property and growth estimates of holomorphic functions.

For each compact Riemann surface of genus > 1, we can construct a Riemann sphere bundle over the given Riemann surface using the projective structure induced by its uniformization.

The total space of this bundle is divided into two 1-convex domains by a closed Levi-flat real hypersurface. Although these two domains are not biholomorphic, we see that they have several function theoretic properties in common. In this talk, I would like to explain these common properties: hyperconvexity and expressions for certain Green function, and Liouville property and growth estimates of holomorphic functions.

#### Numerical Analysis Seminar

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

Residual smoothing technique for short-recurrence Krylov subspace methods (Japanese)

**Kensuke Aihara**(Tokyo City University)Residual smoothing technique for short-recurrence Krylov subspace methods (Japanese)

#### FMSP Lectures

15:00-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)

K-THEORY AND THE DIRAC OPERATOR (1/4)

Lecture 1. WHAT IS K-THEORY AND WHAT IS IT GOOD FOR? (ENGLISH)

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

**Paul Baum**(The Pennsylvania State University)K-THEORY AND THE DIRAC OPERATOR (1/4)

Lecture 1. WHAT IS K-THEORY AND WHAT IS IT GOOD FOR? (ENGLISH)

[ Abstract ]

This talk will consist of four points.

1. The basic definition of K-theory

2. A brief history of K-theory

3. Algebraic versus topological K-theory

4. The unity of K-theory

[ Reference URL ]This talk will consist of four points.

1. The basic definition of K-theory

2. A brief history of K-theory

3. Algebraic versus topological K-theory

4. The unity of K-theory

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

### 2018/10/16

#### Tuesday Seminar of Analysis

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

Positive solutions of Schr\"odinger equations in product form (日本語)

**TSUCHIDA Tetsuo**(Meijo University)Positive solutions of Schr\"odinger equations in product form (日本語)

#### Tuesday Seminar on Topology

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

Resonance varieties and matrix tree theorems (ENGLISH)

**Daniel Matei**(IMAR Bucharest)Resonance varieties and matrix tree theorems (ENGLISH)

[ Abstract ]

We discuss the resonance varieties, encoding vanishing of cohomology cup products, of various classes of finitely presented groups of geometric and combinatorial origin. We describe the ideals defining those varieties in terms spanning trees in a similar vein with the classical matrix tree theorem in graph theory. We present applications of this description to 3-manifold groups and Artin groups.

We discuss the resonance varieties, encoding vanishing of cohomology cup products, of various classes of finitely presented groups of geometric and combinatorial origin. We describe the ideals defining those varieties in terms spanning trees in a similar vein with the classical matrix tree theorem in graph theory. We present applications of this description to 3-manifold groups and Artin groups.

#### Algebraic Geometry Seminar

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

A countable characterisation of smooth algebraic plane curves, and generalisations (English)

**Tuyen Truong**(Oslo)A countable characterisation of smooth algebraic plane curves, and generalisations (English)

[ Abstract ]

Given a smooth algebraic curve X in C^3, I will present a way to construct a sequence of algebraic varieties (whose ideals are explicitly determined from the ideal defining X), whose solution set is non-empty iff the curve X can be algebraically embedded into C^2.

Various other questions, such as whether two given algebraic varieties are birational, can be similarly treated. Some related conjectures are stated.

Given a smooth algebraic curve X in C^3, I will present a way to construct a sequence of algebraic varieties (whose ideals are explicitly determined from the ideal defining X), whose solution set is non-empty iff the curve X can be algebraically embedded into C^2.

Various other questions, such as whether two given algebraic varieties are birational, can be similarly treated. Some related conjectures are stated.

### 2018/10/15

#### Seminar on Geometric Complex Analysis

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

Recent problems on Loewner theory (JAPANESE)

**Ikkei Hotta**(Yamaguchi University)Recent problems on Loewner theory (JAPANESE)

[ Abstract ]

Loewner Theory, which goes back to the parametric representation of univalent functions introduced by Loewner in 1923, has recently undergone significant development in various directions, including Schramm’s stochastic version of the Loewner differential equation and the new intrinsic approach suggested by Bracci, Contreras, Diaz-Madrigal and Gumenyuk.

In this talk, we firstly review the theory of Loewner equations in classical and modern treatments. Then we discuss some recent problems on the theory:

(i) Quasiconformal extensions of Loewner chains;

(ii) Hydrodynamics of multiple SLE.

Loewner Theory, which goes back to the parametric representation of univalent functions introduced by Loewner in 1923, has recently undergone significant development in various directions, including Schramm’s stochastic version of the Loewner differential equation and the new intrinsic approach suggested by Bracci, Contreras, Diaz-Madrigal and Gumenyuk.

In this talk, we firstly review the theory of Loewner equations in classical and modern treatments. Then we discuss some recent problems on the theory:

(i) Quasiconformal extensions of Loewner chains;

(ii) Hydrodynamics of multiple SLE.

#### Numerical Analysis Seminar

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

Möbius invariant discretizations and decomposition of the Möbius energy (Japanese)

**Takeyuki Nagasawa**(Saitama University)Möbius invariant discretizations and decomposition of the Möbius energy (Japanese)

### 2018/10/11

#### Applied Analysis

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

(Japanese)

**Takahito Kashiwabara**(University of Tokyo)(Japanese)

### 2018/10/10

#### Operator Algebra Seminars

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

Extension modules over some conformal algebras related Virasoro algebra (English)

**Kaijing Ling**(Harbin Institute of Technology/Univ. Tokyo)Extension modules over some conformal algebras related Virasoro algebra (English)

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