## Seminar information archive

Seminar information archive ～08/18｜Today's seminar 08/19 | Future seminars 08/20～

### 2014/07/14

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Higher dimensional analogues of fake projective planes (ENGLISH)

**Gopal Prasad**(University of Michigan)Higher dimensional analogues of fake projective planes (ENGLISH)

[ Abstract ]

A fake projective plane is a smooth projective complex algebraic surface which is not isomorphic to the complex projective plane but whose Betti numbers are that of the complex projective plane. The fake projective planes are algebraic surfaces of general type and have smallest possible Euler-Poincare characteristic among them. The first fake projective plane was constructed by D. Mumford using p-adic uniformization, and it was known that there can only be finitely many of them. A complete classification of the fake projective planes was obtained by Sai-Kee Yeung and myself. We showed that there are 28 classes of them, and constructed at least one explicit example in each class. Later, using long computer assisted computations, D. Cartwright and Tim Steger found that the 28 families altogether contain precisely 100 fake projective planes. Using our work, they also found a very interesting smooth projective complex algebraic surface whose Euler-Poincare characteristic is 3 but whose first Betti

number is 2. We have a natural notion of higher dimensional analogues of fake projective planes and to a large extent determined them. My talk will be devoted to an exposition of this work.

A fake projective plane is a smooth projective complex algebraic surface which is not isomorphic to the complex projective plane but whose Betti numbers are that of the complex projective plane. The fake projective planes are algebraic surfaces of general type and have smallest possible Euler-Poincare characteristic among them. The first fake projective plane was constructed by D. Mumford using p-adic uniformization, and it was known that there can only be finitely many of them. A complete classification of the fake projective planes was obtained by Sai-Kee Yeung and myself. We showed that there are 28 classes of them, and constructed at least one explicit example in each class. Later, using long computer assisted computations, D. Cartwright and Tim Steger found that the 28 families altogether contain precisely 100 fake projective planes. Using our work, they also found a very interesting smooth projective complex algebraic surface whose Euler-Poincare characteristic is 3 but whose first Betti

number is 2. We have a natural notion of higher dimensional analogues of fake projective planes and to a large extent determined them. My talk will be devoted to an exposition of this work.

### 2014/07/13

#### Infinite Analysis Seminar Tokyo

14:00-15:00 Room #002 (Graduate School of Math. Sci. Bldg.)

From vertex operators to the shuffle algebra (ENGLISH)

**Andrei Negut**(Columbia University, Department of Mathematics)From vertex operators to the shuffle algebra (ENGLISH)

[ Abstract ]

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

### 2014/07/12

#### Lie Groups and Representation Theory

13:20-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Perverse sheaves on hyperplane arrangements (ENGLISH)

Upper global nasis, cluster algebra and simplicity of tensor products of simple modules (ENGLISH)

Branching Problems of Representations of Real Reductive Groups (ENGLISH)

**Mikhail Kapranov**(Kavli IPMU) 13:20-14:20Perverse sheaves on hyperplane arrangements (ENGLISH)

[ Abstract ]

Given an arrangement of hyperplanes in $R^n$, one has the complexified arrangement in $C^n$ and the corresponding category of perverse sheaves (smooth along the strata of the natural stratification).

The talk, based in a joint work with V. Schechtman, will present an explicit description of this category in terms of data associated to the face complex of the real arrangement. Such a description suggests a possibility of categorifying the concept of a oerverse sheaf in this and possibly in more general cases.

Given an arrangement of hyperplanes in $R^n$, one has the complexified arrangement in $C^n$ and the corresponding category of perverse sheaves (smooth along the strata of the natural stratification).

The talk, based in a joint work with V. Schechtman, will present an explicit description of this category in terms of data associated to the face complex of the real arrangement. Such a description suggests a possibility of categorifying the concept of a oerverse sheaf in this and possibly in more general cases.

**Masaki Kashiwara**(RIMS) 14:40-15:40Upper global nasis, cluster algebra and simplicity of tensor products of simple modules (ENGLISH)

[ Abstract ]

One of the motivation of cluster algebras introduced by

Fomin and Zelevinsky is

multiplicative properties of upper global basis.

In this talk, I explain their relations, related conjectures by Besrnard Leclerc and the recent progress by the speaker with Seok-Jin Kang, Myungho Kima and Sejin Oh.

One of the motivation of cluster algebras introduced by

Fomin and Zelevinsky is

multiplicative properties of upper global basis.

In this talk, I explain their relations, related conjectures by Besrnard Leclerc and the recent progress by the speaker with Seok-Jin Kang, Myungho Kima and Sejin Oh.

**Toshiyuki Kobayashi**(the University of Tokyo) 16:00-17:00Branching Problems of Representations of Real Reductive Groups (ENGLISH)

[ Abstract ]

Branching problems ask how irreducible representations π of groups G "decompose" when restricted to subgroups G'.

For real reductive groups, branching problems include various important special cases, however, it is notorious that "infinite multiplicities" and "continuous spectra" may well happen in general even if (G,G') are natural pairs such as symmetric pairs.

By using analysis on (real) spherical varieties, we give a necessary and sufficient condition on the pair of reductive groups for the multiplicities to be always finite (and also to be of uniformly bounded). Further, we discuss "discretely decomposable restrictions" which allows us to apply algebraic tools in branching problems. Some classification results will be also presented.

If time permits, I will discuss some applications of branching laws of Zuckerman's derived functor modules to analysis on locally symmetric spaces with indefinite metric.

Branching problems ask how irreducible representations π of groups G "decompose" when restricted to subgroups G'.

For real reductive groups, branching problems include various important special cases, however, it is notorious that "infinite multiplicities" and "continuous spectra" may well happen in general even if (G,G') are natural pairs such as symmetric pairs.

By using analysis on (real) spherical varieties, we give a necessary and sufficient condition on the pair of reductive groups for the multiplicities to be always finite (and also to be of uniformly bounded). Further, we discuss "discretely decomposable restrictions" which allows us to apply algebraic tools in branching problems. Some classification results will be also presented.

If time permits, I will discuss some applications of branching laws of Zuckerman's derived functor modules to analysis on locally symmetric spaces with indefinite metric.

#### Lie Groups and Representation Theory

09:30-11:45 Room #126 (Graduate School of Math. Sci. Bldg.)

Hypergeometric systems and Kac-Moody root systems (ENGLISH)

Representations of covering groups with multiplicity free K-types (ENGLISH)

**Toshio Oshima**(Josai University) 09:30-10:30Hypergeometric systems and Kac-Moody root systems (ENGLISH)

**Gordan Savin**(the University of Utah) 10:45-11:45Representations of covering groups with multiplicity free K-types (ENGLISH)

[ Abstract ]

Let g be a simple Lie algebra over complex numbers. McGovern has

described an ideal J in the enveloping algebra U such that U/J, considered as a g-module under the adjoint action, is a sum of all self-dual representations of g with multiplicity one. In a joint work with Loke, we prove that all (g,K)-modules annihilated by J have multiplicity free K-types, where K is defined by the Chevalley involution.

Let g be a simple Lie algebra over complex numbers. McGovern has

described an ideal J in the enveloping algebra U such that U/J, considered as a g-module under the adjoint action, is a sum of all self-dual representations of g with multiplicity one. In a joint work with Loke, we prove that all (g,K)-modules annihilated by J have multiplicity free K-types, where K is defined by the Chevalley involution.

### 2014/07/11

#### Colloquium

16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Global Geometry and Analysis on Locally Symmetric Spaces with

Indefinite-metric (JAPANESE)

**Toshiyuki Kobayashi**(Graduate School of Mathematical Sciences, University of Tokyo)Global Geometry and Analysis on Locally Symmetric Spaces with

Indefinite-metric (JAPANESE)

[ Abstract ]

The local to global study of geometries was a major trend of 20th

century geometry,

with remarkable developments achieved particularly in Riemannian geometry.

In contrast, in areas such as pseudo-Riemannian geometry, familiar to us

as the space-time of relativity theory, and more generally in

pseudo-Riemannian geometry of general signature, surprising little is

known about global properties of the geometry even if we impose a

locally homogeneous structure.

I plan to explain two programs:

1. (global shape) Existence problem of compact locally homogeneous spaces,

and deformation theory.

2. (spectral analysis) Construction of the spectrum of the Laplacian,

and its stability under the deformation of the geometric structure.

by taking anti-de Sitter manifolds as a typical example.

The local to global study of geometries was a major trend of 20th

century geometry,

with remarkable developments achieved particularly in Riemannian geometry.

In contrast, in areas such as pseudo-Riemannian geometry, familiar to us

as the space-time of relativity theory, and more generally in

pseudo-Riemannian geometry of general signature, surprising little is

known about global properties of the geometry even if we impose a

locally homogeneous structure.

I plan to explain two programs:

1. (global shape) Existence problem of compact locally homogeneous spaces,

and deformation theory.

2. (spectral analysis) Construction of the spectrum of the Laplacian,

and its stability under the deformation of the geometric structure.

by taking anti-de Sitter manifolds as a typical example.

#### FMSP Lectures

15:00-16:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Conditional stability estimate for the Calderon's problem in two dimensional case (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Emanouilov140711.pdf

**Oleg Emanouilov**(Colorado State Univ.)Conditional stability estimate for the Calderon's problem in two dimensional case (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Emanouilov140711.pdf

### 2014/07/08

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The Johnson homomorphism and a family of curve graphs (ENGLISH)

**Ingrid Irmer**(JSPS, the University of Tokyo)The Johnson homomorphism and a family of curve graphs (ENGLISH)

[ Abstract ]

Abstract: A family of curve graphs of an oriented surface $S_{g,1}$ will be defined on which there exists a natural orientation, coming from the orientation of subsurfaces. Distances in these graphs represent commutator lengths in $\\pi_{1}(S_{g,1})$. The displacement of vertices in the graphs under the action of the Torelli group is used to give a combinatorial description of the Johnson homomorphism."

Abstract: A family of curve graphs of an oriented surface $S_{g,1}$ will be defined on which there exists a natural orientation, coming from the orientation of subsurfaces. Distances in these graphs represent commutator lengths in $\\pi_{1}(S_{g,1})$. The displacement of vertices in the graphs under the action of the Torelli group is used to give a combinatorial description of the Johnson homomorphism."

#### Classical Analysis

16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

ABS equations arising from q-P((A2+A1)^{(1)}) (JAPANESE)

**Nakazono Nobutaka**(University of Sydney)ABS equations arising from q-P((A2+A1)^{(1)}) (JAPANESE)

[ Abstract ]

The study of periodic reductions from ABS equations to discrete Painlevé equations have been investigated by many groups. However, there still remain open questions:

(i) How do we identify the discrete Painlevé equation that would result from applying a periodic reduction to an ABS equation?

(ii) Discrete Painlevé equations obtained by periodic reductions often have insufficient number of parameters. How do we obtain the general case with all essential parameters?

To solve these problems, we investigated the periodic reductions from the viewpoint of Painlevé systems.

In this talk, we show how to construct a lattice where ABS equations arise from relationships between $\\tau$ functions of Painlevé systems and explain how this lattice relates to a hyper cube associated with an ABS equation on each face.

In particular, we consider the $q$-Painlevé equations, which have the affine Weyl group symmetry of type $(A_2+A_1)^{(1)}$.

The study of periodic reductions from ABS equations to discrete Painlevé equations have been investigated by many groups. However, there still remain open questions:

(i) How do we identify the discrete Painlevé equation that would result from applying a periodic reduction to an ABS equation?

(ii) Discrete Painlevé equations obtained by periodic reductions often have insufficient number of parameters. How do we obtain the general case with all essential parameters?

To solve these problems, we investigated the periodic reductions from the viewpoint of Painlevé systems.

In this talk, we show how to construct a lattice where ABS equations arise from relationships between $\\tau$ functions of Painlevé systems and explain how this lattice relates to a hyper cube associated with an ABS equation on each face.

In particular, we consider the $q$-Painlevé equations, which have the affine Weyl group symmetry of type $(A_2+A_1)^{(1)}$.

### 2014/07/07

#### Algebraic Geometry Seminar

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

Balanced line bundles (JAPANESE)

**Sho Tanimoto**(Rice University)Balanced line bundles (JAPANESE)

[ Abstract ]

A conjecture of Batyrev and Manin relates arithmetic properties of

varieties with big anticanonical class to geometric invariants; in

particular, counting functions defined by metrized ample line bundles

and the corresponding asymptotics of rational points of bounded height

are interpreted in terms of cones of effective divisors and certain

thresholds with respect to these cones. This framework leads to the

notion of balanced line bundles, whose counting functions, conjecturally,

capture generic distributions of rational points. We investigate

balanced line bundles in the context of the Minimal Model Program, with

special regard to the classification of Fano threefolds and Mori fiber

spaces.

This is joint work with Brian Lehmann and Yuri Tschinkel.

A conjecture of Batyrev and Manin relates arithmetic properties of

varieties with big anticanonical class to geometric invariants; in

particular, counting functions defined by metrized ample line bundles

and the corresponding asymptotics of rational points of bounded height

are interpreted in terms of cones of effective divisors and certain

thresholds with respect to these cones. This framework leads to the

notion of balanced line bundles, whose counting functions, conjecturally,

capture generic distributions of rational points. We investigate

balanced line bundles in the context of the Minimal Model Program, with

special regard to the classification of Fano threefolds and Mori fiber

spaces.

This is joint work with Brian Lehmann and Yuri Tschinkel.

### 2014/07/03

#### Applied Analysis

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

Parabolic power concavity and parabolic boundary value problems (JAPANESE)

**Kazuhiro Ishige**(Tohoku University)Parabolic power concavity and parabolic boundary value problems (JAPANESE)

#### FMSP Lectures

16:00-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Structure of rational orbits in prehomogeneous spaces. (ENGLISH)

[ Reference URL ]

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

**Gordan Savin**(Univ. of Utah)Structure of rational orbits in prehomogeneous spaces. (ENGLISH)

[ Reference URL ]

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

### 2014/07/01

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Singularities of special Lagrangian submanifolds (JAPANESE)

**Yohsuke Imagi**(Kavli IPMU)Singularities of special Lagrangian submanifolds (JAPANESE)

[ Abstract ]

There are interesting invariants defined by "counting" geometric

objects, such as instantons in dimension 4 and pseudo-holomorphic curves

in symplectic manifolds. To do the counting in a sensible way, however,

we have to care about singularities of the geometric objects. Special

Lagrangian submanifolds seem very difficult to "count" as their

singularities may be very complicated. I'll talk about simple

singularities for which we can make an analogy with instantons in

dimension 4 and pseudo-holomorphic curves in symplectic manifolds. To do

it I'll use some techniques from geometric measure theory and Lagrangian

Floer theory, and the Floer-theoretic part is a joint work with Dominic

Joyce and Oliveira dos Santos.

There are interesting invariants defined by "counting" geometric

objects, such as instantons in dimension 4 and pseudo-holomorphic curves

in symplectic manifolds. To do the counting in a sensible way, however,

we have to care about singularities of the geometric objects. Special

Lagrangian submanifolds seem very difficult to "count" as their

singularities may be very complicated. I'll talk about simple

singularities for which we can make an analogy with instantons in

dimension 4 and pseudo-holomorphic curves in symplectic manifolds. To do

it I'll use some techniques from geometric measure theory and Lagrangian

Floer theory, and the Floer-theoretic part is a joint work with Dominic

Joyce and Oliveira dos Santos.

#### Lie Groups and Representation Theory

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

WONDERFUL VARIETIES. REGULARIZED TRACES AND CHARACTERS (ENGLISH)

**Pablo Ramacher**

(Marburg University)WONDERFUL VARIETIES. REGULARIZED TRACES AND CHARACTERS (ENGLISH)

[ Abstract ]

Let G be a connected reductive complex algebraic group with split real form $G^\\sigma$.

In this talk, we introduce a distribution character for the regular representation of $G^\\sigma$ on the real locus of a strict wonderful G-variety X, showing that on a certain open subset of $G^\\sigma$ of transversal elements it is locally integrable, and given by a sum over fixed points.

Let G be a connected reductive complex algebraic group with split real form $G^\\sigma$.

In this talk, we introduce a distribution character for the regular representation of $G^\\sigma$ on the real locus of a strict wonderful G-variety X, showing that on a certain open subset of $G^\\sigma$ of transversal elements it is locally integrable, and given by a sum over fixed points.

### 2014/06/30

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Primitive automorphisms of positive entropy of rational and Calabi-Yau threefolds (JAPANESE)

**Keiji Oguiso**(Osaka University)Primitive automorphisms of positive entropy of rational and Calabi-Yau threefolds (JAPANESE)

#### Algebraic Geometry Seminar

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

Invariant subrings of the Cox rings of K3surfaces by automorphism groups (JAPANESE)

**Akiyoshi Sannai**(University of Tokyo)Invariant subrings of the Cox rings of K3surfaces by automorphism groups (JAPANESE)

[ Abstract ]

Cox rings were introduced by D.Cox and are important rings which appeared in algebraic geometry. One of the main topic related with Cox rings is the finite generation of them. In this talk, we consider the Cox rings of K3 surfaces and answer the following question asked by D. Huybrechts; Are the invariant subrings of the Cox rings of K3 surfaces by automorphism groups finitely generated in general?

Cox rings were introduced by D.Cox and are important rings which appeared in algebraic geometry. One of the main topic related with Cox rings is the finite generation of them. In this talk, we consider the Cox rings of K3 surfaces and answer the following question asked by D. Huybrechts; Are the invariant subrings of the Cox rings of K3 surfaces by automorphism groups finitely generated in general?

#### Kavli IPMU Komaba Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On some quadratic algebras with applications to Topology,

Algebra, Combinatorics, Schubert Calculus and Integrable Systems. (ENGLISH)

**Anatol Kirillov**(RIMS, Kyoto University)On some quadratic algebras with applications to Topology,

Algebra, Combinatorics, Schubert Calculus and Integrable Systems. (ENGLISH)

[ Abstract ]

The main purpose of my talk is to draw attention of the

participants of the seminar to a certain family of quadratic algebras

which has a wide range of applications to the subject mentioned in the

title of my talk.

The main purpose of my talk is to draw attention of the

participants of the seminar to a certain family of quadratic algebras

which has a wide range of applications to the subject mentioned in the

title of my talk.

### 2014/06/28

#### Harmonic Analysis Komaba Seminar

13:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On the multilinear restriction problem (ENGLISH)

John-Nirenberg lemmas for a doubling measure (ENGLISH)

**Neal Bez**(埼玉大学) 13:30-15:00On the multilinear restriction problem (ENGLISH)

[ Abstract ]

I will discuss the multilinear restriction problem for the Fourier transform. This will include an overview of the pioneering work of Bennett, Carbery and Tao on this problem and the very losely connected multilinear Kakeya problem. I will also discuss some of my own work in this area which is connected to nonlinear Brascamp-Lieb inequalities (joint work with Jonathan Bennett).

I will discuss the multilinear restriction problem for the Fourier transform. This will include an overview of the pioneering work of Bennett, Carbery and Tao on this problem and the very losely connected multilinear Kakeya problem. I will also discuss some of my own work in this area which is connected to nonlinear Brascamp-Lieb inequalities (joint work with Jonathan Bennett).

**Hong Yue**(Georgia College and State University) 15:30-17:00John-Nirenberg lemmas for a doubling measure (ENGLISH)

[ Abstract ]

We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderon-Zygmund decomposition in metric spaces and use it to prove the corresponding John-Nirenberg inequality.

We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderon-Zygmund decomposition in metric spaces and use it to prove the corresponding John-Nirenberg inequality.

### 2014/06/26

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Entropic curvature-dimension condition and Bochner’s inequality (JAPANESE)

**Kazumasa Kuwada**(Tokyo Institute of Technology)Entropic curvature-dimension condition and Bochner’s inequality (JAPANESE)

[ Abstract ]

As a characterization of "lower Ricci curvature bound and upper dimension bound”, there appear several conditions which make sense even on singular spaces. In this talk we show the equivalence in complete generality between two major conditions: a reduced version of curvature-dimension bounds of Sturm-Lott-Villani via entropy and optimal transport and Bakry–¥'Emery's one via Markov generator or the associated heat semigroup. More precisely, it holds for metric measure spaces where Cheeger's L^2-energy functional is a quadratic form. In particular, we establish the full Bochner inequality, which originally comes from the Bochner-Weitzenb¥"ock formula, on such spaces. This talk is based on a joint work with M. Erbar and K.-T. Sturm (Bonn).

As a characterization of "lower Ricci curvature bound and upper dimension bound”, there appear several conditions which make sense even on singular spaces. In this talk we show the equivalence in complete generality between two major conditions: a reduced version of curvature-dimension bounds of Sturm-Lott-Villani via entropy and optimal transport and Bakry–¥'Emery's one via Markov generator or the associated heat semigroup. More precisely, it holds for metric measure spaces where Cheeger's L^2-energy functional is a quadratic form. In particular, we establish the full Bochner inequality, which originally comes from the Bochner-Weitzenb¥"ock formula, on such spaces. This talk is based on a joint work with M. Erbar and K.-T. Sturm (Bonn).

### 2014/06/25

#### Operator Algebra Seminars

16:45-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Supersymmetric C*-dynamical systems (JAPANESE)

**Hajime Moriya**(Shibaura Inst. Technology)Supersymmetric C*-dynamical systems (JAPANESE)

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Mathematical modeling and classification of tumor immunity in cell transfer therapy (JAPANESE)

**Shinnji Nakaoka**(理化学研究所統合生命医科学研究センター)Mathematical modeling and classification of tumor immunity in cell transfer therapy (JAPANESE)

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Periods of some two dimensional reducible p-adic representations and non-de Rham B-pairs (JAPANESE)

**Masahiko Takiguchi**(University of Tokyo)Periods of some two dimensional reducible p-adic representations and non-de Rham B-pairs (JAPANESE)

### 2014/06/24

#### PDE Real Analysis Seminar

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

Sudden directional diffusion: counting and watching facets (ENGLISH)

**Piotr Rybka**(University of Warsaw)Sudden directional diffusion: counting and watching facets (ENGLISH)

[ Abstract ]

We study two examples of singular parabolic equations such that the diffusion is so strong that is leads to creation of facets. By facets we mean flat parts of the graphs of solutions with singular slopes. In one of the equations we study there are two singular slopes. The other equation has just one singular slope and the isotropic diffusion term. For both problems we watch and count facet.

For the system with two singular slopes a natural question arises if any solution may have an infinite number of oscillations. We also show that the solutions we constructed are viscosity solutions. This in turn gives estimates on the extinction time based on the comparison principle.

We study two examples of singular parabolic equations such that the diffusion is so strong that is leads to creation of facets. By facets we mean flat parts of the graphs of solutions with singular slopes. In one of the equations we study there are two singular slopes. The other equation has just one singular slope and the isotropic diffusion term. For both problems we watch and count facet.

For the system with two singular slopes a natural question arises if any solution may have an infinite number of oscillations. We also show that the solutions we constructed are viscosity solutions. This in turn gives estimates on the extinction time based on the comparison principle.

#### Tuesday Seminar on Topology

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

On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)

**Takefumi Nosaka**(Faculty of Mathematics, Kyushu University)On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)

[ Abstract ]

本講演では, 群とその群同型の組から定まるカンドルを扱い, 以下の結果を紹介

する. まず, その際Inoue-Kabaya鎖写像が, カンドルホモロジーから群ホモロ

ジーへの写像とし, 定式化される事を見る. 例えば, 有限体上のAlexander

quandleに対し, 望月3-コサイクル全ては, 当写像を通じ, 或る群コホモロジー

から導出され, 殆どがトリプルマッセイ積で解釈できる事をみる. 加えてカンド

ルの普遍中心拡大に対し, Inoue-Kabaya鎖写像が3次において(或る捩れ部分を

除き)同型となる. なお講演内容は当週にある集中講義の聴講を仮定しない.

本講演では, 群とその群同型の組から定まるカンドルを扱い, 以下の結果を紹介

する. まず, その際Inoue-Kabaya鎖写像が, カンドルホモロジーから群ホモロ

ジーへの写像とし, 定式化される事を見る. 例えば, 有限体上のAlexander

quandleに対し, 望月3-コサイクル全ては, 当写像を通じ, 或る群コホモロジー

から導出され, 殆どがトリプルマッセイ積で解釈できる事をみる. 加えてカンド

ルの普遍中心拡大に対し, Inoue-Kabaya鎖写像が3次において(或る捩れ部分を

除き)同型となる. なお講演内容は当週にある集中講義の聴講を仮定しない.

#### Classical Analysis

16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Irreducibility of the discrete Painlev\\'e equation of type $D_7$ (JAPANESE)

**Nishioka Seiji**(Yamagata University)Irreducibility of the discrete Painlev\\'e equation of type $D_7$ (JAPANESE)

### 2014/06/23

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

A remark to the division algorithm in the proof of Oka's First Coherence Theorem (JAPANESE)

**Junjiro Noguchi**(University of Tokyo)A remark to the division algorithm in the proof of Oka's First Coherence Theorem (JAPANESE)

[ Abstract ]

The problem is the local finite generation of a relation sheaf $R(f_1, \ldots, f_q)$ in $\mathcal{O}_n=\mathcal{O}_{C^n}$. After $f_j$ reduced to Weierstrass' polynomials in $z_n$, it is the key to apply the induction in $n$ to show that elements of $R(f_1, \ldots, q)$ are expressed by $z_n$-polynomial-like elements of degree at most $p=\max_j\deg f_j$ over $\mathcal{O}_n$. In that proof one is used to use a divison by $f_j$ of $\deg f_j=p$ (Oka '48, Cartan '50, Hörmander, Demailly, . . .). In this talk we shall confirm that the division abve works by making use of $f_k$ of the minimum degree $\min_j \deg f_j$. This proof is natrually compatible with the simple case when some $f_j$ is a unit, and gives some improvement in the degree estimate of generators.

The problem is the local finite generation of a relation sheaf $R(f_1, \ldots, f_q)$ in $\mathcal{O}_n=\mathcal{O}_{C^n}$. After $f_j$ reduced to Weierstrass' polynomials in $z_n$, it is the key to apply the induction in $n$ to show that elements of $R(f_1, \ldots, q)$ are expressed by $z_n$-polynomial-like elements of degree at most $p=\max_j\deg f_j$ over $\mathcal{O}_n$. In that proof one is used to use a divison by $f_j$ of $\deg f_j=p$ (Oka '48, Cartan '50, Hörmander, Demailly, . . .). In this talk we shall confirm that the division abve works by making use of $f_k$ of the minimum degree $\min_j \deg f_j$. This proof is natrually compatible with the simple case when some $f_j$ is a unit, and gives some improvement in the degree estimate of generators.

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