## Seminar information archive

Seminar information archive ～05/21｜Today's seminar 05/22 | Future seminars 05/23～

### 2010/10/18

#### Seminar on Geometric Complex Analysis

10:30-11:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Limiting behavior of minimal trajectories of parabolic vector fields on the complex projective plane. (ENGLISH)

**Sergey Ivashkovitch**(Univ. de Lille)Limiting behavior of minimal trajectories of parabolic vector fields on the complex projective plane. (ENGLISH)

[ Abstract ]

The classical Poincare-Bendixson theory describes the way a trajectory of a vector field on the real plane behaves when accumulating to the singular locus of the vector field. We shall describe, in the first approximation, the way a minimal trajectory of a parabolic complex polynomial vector field (or, a holomorphic foliation) on the complex projective plane approaches the singular locus. In particular we shall prove that if a holomorphic foliation has an exceptional minimal set then its nef model is necessarily hyperbolic.

The classical Poincare-Bendixson theory describes the way a trajectory of a vector field on the real plane behaves when accumulating to the singular locus of the vector field. We shall describe, in the first approximation, the way a minimal trajectory of a parabolic complex polynomial vector field (or, a holomorphic foliation) on the complex projective plane approaches the singular locus. In particular we shall prove that if a holomorphic foliation has an exceptional minimal set then its nef model is necessarily hyperbolic.

#### Seminar on Geometric Complex Analysis

13:00-14:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Degenerate complex Monge-Ampere equations (ENGLISH)

**Philippe Eyssidieux**(Institut Fourier, Grenoble)Degenerate complex Monge-Ampere equations (ENGLISH)

#### Kavli IPMU Komaba Seminar

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

Quasi-modular forms and Gromov--Witten theory of elliptic orbifold $\\mathbb{P}^1$ (ENGLISH)

**Todor Milanov**(IPMU)Quasi-modular forms and Gromov--Witten theory of elliptic orbifold $\\mathbb{P}^1$ (ENGLISH)

[ Abstract ]

This talk is based on my current work with Y. Ruan. Our project is part of the so called Landau--Ginzburg/Calabi-Yau correspondence. The latter is a conjecture, due to Ruan, that describes the relation between the $W$-spin invariants of a Landau-Ginzburg potential $W$ and the Gromov--Witten invariants of a certain Calabi--Yau orbifold. I am planning first to explain the higher-genus reconstruction formalism of Givental. This formalism together with the work of M. Krawitz and Y. Shen allows us to express the Gromov--Witten invariants of the orbifold $\\mathbb{P}^1$'s with weights $(3,3,3)$, $(2,4,4)$, and $(2,3,6)$ in terms of Saito's Frobenius structure associated with the simple elliptic singularities $P_8$, $X_9$, and $J_{10}$ respectively. After explaining Givental's formalism, my goal would be to discuss the Saito's flat structure, and to explain how its modular behavior fits in the Givental's formalism. This allows us to prove that the Gromov--Witten invariants are quasi-modular forms on an appropriate modular group.

This talk is based on my current work with Y. Ruan. Our project is part of the so called Landau--Ginzburg/Calabi-Yau correspondence. The latter is a conjecture, due to Ruan, that describes the relation between the $W$-spin invariants of a Landau-Ginzburg potential $W$ and the Gromov--Witten invariants of a certain Calabi--Yau orbifold. I am planning first to explain the higher-genus reconstruction formalism of Givental. This formalism together with the work of M. Krawitz and Y. Shen allows us to express the Gromov--Witten invariants of the orbifold $\\mathbb{P}^1$'s with weights $(3,3,3)$, $(2,4,4)$, and $(2,3,6)$ in terms of Saito's Frobenius structure associated with the simple elliptic singularities $P_8$, $X_9$, and $J_{10}$ respectively. After explaining Givental's formalism, my goal would be to discuss the Saito's flat structure, and to explain how its modular behavior fits in the Givental's formalism. This allows us to prove that the Gromov--Witten invariants are quasi-modular forms on an appropriate modular group.

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Galois extensions and maps on local cohomology (JAPANESE)

**Akiyoshi Sannai**(Univ. of Tokyo)Galois extensions and maps on local cohomology (JAPANESE)

### 2010/10/14

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Nonstandard analysis for operator algebraists (JAPANESE)

**Yasuyuki Kawahigashi**(Univ. Tokyo)Nonstandard analysis for operator algebraists (JAPANESE)

### 2010/10/12

#### Tuesday Seminar on Topology

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

Asymptotics of Morse numbers of finite coverings of manifolds (ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes, The University of Tokyo)Asymptotics of Morse numbers of finite coverings of manifolds (ENGLISH)

[ Abstract ]

Let X be a closed manifold;

denote by m(X) the Morse number of X

(that is, the minimal number of critical

points of a Morse function on X).

Let Y be a finite covering of X of degree d.

In this survey talk we will address the following question

posed by M. Gromov: What are the asymptotic properties

of m(N) as d goes to infinity?

It turns out that for high-dimensional manifolds with

free abelian fundamental group the asymptotics of

the number m(N)/d is directly related to the Novikov homology

of N. We prove this theorem and discuss related results.

Let X be a closed manifold;

denote by m(X) the Morse number of X

(that is, the minimal number of critical

points of a Morse function on X).

Let Y be a finite covering of X of degree d.

In this survey talk we will address the following question

posed by M. Gromov: What are the asymptotic properties

of m(N) as d goes to infinity?

It turns out that for high-dimensional manifolds with

free abelian fundamental group the asymptotics of

the number m(N)/d is directly related to the Novikov homology

of N. We prove this theorem and discuss related results.

### 2010/10/08

#### Colloquium

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

Exceptional Jacobi polynomials as solutions of a Schroedinger

(Sturm-Liouville) equation with $3 +¥ell$ ($¥ell=1,2,¥ldots) regular

singularities (JAPANESE)

**Ryu Sasaki**(Yukawa Institute for Theoretical Physics, Kyoto University)Exceptional Jacobi polynomials as solutions of a Schroedinger

(Sturm-Liouville) equation with $3 +¥ell$ ($¥ell=1,2,¥ldots) regular

singularities (JAPANESE)

[ Abstract ]

Global solutions of Fuchsian differential equations with more than 3 (hypergeometric) or four (Heun) regular singularities had been virtually unkown. Here I present a complete set of eigenfunctions of a Schroedinger (Sturm-Liouville) equation with $3 + ¥ell$ ($¥ell=1,2,¥ldots$) regular singularities. They are deformations of the Darboux-P¥" oschl-Teller potential with the Hamiltonian (Schroedinger operator) ¥[ ¥mathcal{H}=-¥frac{d^2}{dx^2}+¥frac{g(g-1)}{¥sin^2x}+¥frac{h(h-1)} {¥cos^2x}¥] The eigenfunctions consist of the {¥em exceptional Jacobi polynomials} $¥{P_{¥ell,n}(¥eta)¥}$, $n=0,1,2,¥ldots$, with deg($P_{¥ell,n}$)$=n+¥ell$. Thus the restriction due to Bochner's theorem does not apply. The confluent limit produces two sets of the exceptional Laguerre polynomials for $¥ell=1,2,¥ldots$. Similar deformation method provides the exceptional Wilson and Askey-Wilson polynomials for $¥ell=1,2,¥ldots$.

Global solutions of Fuchsian differential equations with more than 3 (hypergeometric) or four (Heun) regular singularities had been virtually unkown. Here I present a complete set of eigenfunctions of a Schroedinger (Sturm-Liouville) equation with $3 + ¥ell$ ($¥ell=1,2,¥ldots$) regular singularities. They are deformations of the Darboux-P¥" oschl-Teller potential with the Hamiltonian (Schroedinger operator) ¥[ ¥mathcal{H}=-¥frac{d^2}{dx^2}+¥frac{g(g-1)}{¥sin^2x}+¥frac{h(h-1)} {¥cos^2x}¥] The eigenfunctions consist of the {¥em exceptional Jacobi polynomials} $¥{P_{¥ell,n}(¥eta)¥}$, $n=0,1,2,¥ldots$, with deg($P_{¥ell,n}$)$=n+¥ell$. Thus the restriction due to Bochner's theorem does not apply. The confluent limit produces two sets of the exceptional Laguerre polynomials for $¥ell=1,2,¥ldots$. Similar deformation method provides the exceptional Wilson and Askey-Wilson polynomials for $¥ell=1,2,¥ldots$.

### 2010/10/06

#### Geometry Seminar

14:45-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Handle attaching in wrapped Floer homology and brake orbits in classical Hamiltonian systems (JAPANESE)

Mirror Symmetry for Weighted Homogeneous Polynomials (JAPANESE)

**Kei Irie**(Kyoto Univ.) 14:45-16:15Handle attaching in wrapped Floer homology and brake orbits in classical Hamiltonian systems (JAPANESE)

[ Abstract ]

In this talk, the term "classical Hamiltonian systems" means special types of Hamiltonian systems, which describe solutions of classical equations of motion. The study of periodic solutions of Hamiltonian systems is an interesting problem, and for classical Hamiltonian systems, the following result is known : for any compact and regular energy surface $S$, there exists a brake orbit (a particular type of periodic solutions) on $S$. This result is first proved by S.V.Bolotin in 1978, and it is a special case of the Arnold chord conjecture. In this talk, I will explain that calculations of wrapped Floer homology (which is a variant of Lagrangian Floer homology) give a new proof of the above result.

In this talk, the term "classical Hamiltonian systems" means special types of Hamiltonian systems, which describe solutions of classical equations of motion. The study of periodic solutions of Hamiltonian systems is an interesting problem, and for classical Hamiltonian systems, the following result is known : for any compact and regular energy surface $S$, there exists a brake orbit (a particular type of periodic solutions) on $S$. This result is first proved by S.V.Bolotin in 1978, and it is a special case of the Arnold chord conjecture. In this talk, I will explain that calculations of wrapped Floer homology (which is a variant of Lagrangian Floer homology) give a new proof of the above result.

**Atsushi Takahashi**(Osaka Univ.) 16:30-18:00Mirror Symmetry for Weighted Homogeneous Polynomials (JAPANESE)

[ Abstract ]

First we give an overview of the algebraic and the geometric aspects of the mirror symmetry conjecture for weighted homogeneous polynomials. Then we concentrate on polynomials in three variables, and show the existence of full (strongly) exceptional collection of categories of maximally graded matrix factorizations for invertible weighted homogeneous polynomials. We will also explain how the mirror symmetry naturally explains and generalizes the Arnold's strange duality between the 14 exceptional unimodal singularities.

First we give an overview of the algebraic and the geometric aspects of the mirror symmetry conjecture for weighted homogeneous polynomials. Then we concentrate on polynomials in three variables, and show the existence of full (strongly) exceptional collection of categories of maximally graded matrix factorizations for invertible weighted homogeneous polynomials. We will also explain how the mirror symmetry naturally explains and generalizes the Arnold's strange duality between the 14 exceptional unimodal singularities.

#### Number Theory Seminar

16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Finite group actions on the affine space (ENGLISH)

**Hélène Esnault**(Universität Duisburg-Essen)Finite group actions on the affine space (ENGLISH)

[ Abstract ]

If $G$ is a finite $\\ell$-group acting on an affine space $\\A^n$ over a

finite field $K$ of cardinality prime to $\\ell$, Serre shows that there

exists a rational fixed point. We generalize this to the case where $K$ is a

henselian discretely valued field of characteristic zero with algebraically

closed residue field and with residue characteristic different from $\\ell$.

We also treat the case where the residue field is finite of cardinality $q$

such that $\\ell$ divides $q-1$. To this aim, we study group actions on weak

N\\'eron models.

(Joint work with Johannes Nicaise)

If $G$ is a finite $\\ell$-group acting on an affine space $\\A^n$ over a

finite field $K$ of cardinality prime to $\\ell$, Serre shows that there

exists a rational fixed point. We generalize this to the case where $K$ is a

henselian discretely valued field of characteristic zero with algebraically

closed residue field and with residue characteristic different from $\\ell$.

We also treat the case where the residue field is finite of cardinality $q$

such that $\\ell$ divides $q-1$. To this aim, we study group actions on weak

N\\'eron models.

(Joint work with Johannes Nicaise)

#### Seminar on Probability and Statistics

15:00-16:10 Room #000 (Graduate School of Math. Sci. Bldg.)

On multiple kernel learning with elasticnet type regularization (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2010/05.html

**SUZUKI, Taiji**(University of Tokyo)On multiple kernel learning with elasticnet type regularization (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2010/05.html

### 2010/10/04

#### Seminar on Geometric Complex Analysis

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

The canonical coordinates associated to homogeneous Kaehler metrics on a homogeneous bounded domain (JAPANESE)

**Hideyuki ISHI**(Nagoya Univ)The canonical coordinates associated to homogeneous Kaehler metrics on a homogeneous bounded domain (JAPANESE)

[ Abstract ]

For a real analytic Kaehler manifold, one can define a canonical coordinate, called the Bochner coordinate, around each point. In this talk, we show that the canonical cooredinate is globally defined for a bounded homogeneous domain with a homogeneous Kaehler manifold, which is not necessarily the Bergman metric.

Then we obtain a standard realization of the homogeneous domain associated to the homogeneous metric.

For a real analytic Kaehler manifold, one can define a canonical coordinate, called the Bochner coordinate, around each point. In this talk, we show that the canonical cooredinate is globally defined for a bounded homogeneous domain with a homogeneous Kaehler manifold, which is not necessarily the Bergman metric.

Then we obtain a standard realization of the homogeneous domain associated to the homogeneous metric.

### 2010/09/28

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Some spectral and resonance properties of quantum graphs (ENGLISH)

**Pavel Exner**(Czech Academy of Sciences)Some spectral and resonance properties of quantum graphs (ENGLISH)

[ Abstract ]

In this talk I will discuss three new results about Schr¨odinger operators

on metric graphs obtained in collaboration with Jiri Lipovskyand Brian Davies.

The first one is related to invalidity of the uniform continuation principle for such

operators. One manifestation of this fact are embedded eigenvalues due to

rational relations of graph edge lengths. This effect is non-generic and we show

how geometric perturbations turn these embedded eigenvalues into resonances.

Then second problem is related to high-energy behavior of resonances: we extend

a recent result of Davies and Pushnitski to graphs with general vertex couplings

and find conditions under which the asymptotics does not have Weyl character.

Finally, the last question addressed here concerns the absolutely continuous spectrum

of radial-tree graphs. In a similar vein, we generalize a recent result by Breuer and

Frank that in case of the free (or Kirhhoff) coupling the ac spectrum is absent

provided the edge length are increasing without a bound along the tree.

We show that the result remains valid for a large class of vertex couplings,

but on the other hand, there are nontrivial couplings leading to an ac spectrum.

In this talk I will discuss three new results about Schr¨odinger operators

on metric graphs obtained in collaboration with Jiri Lipovskyand Brian Davies.

The first one is related to invalidity of the uniform continuation principle for such

operators. One manifestation of this fact are embedded eigenvalues due to

rational relations of graph edge lengths. This effect is non-generic and we show

how geometric perturbations turn these embedded eigenvalues into resonances.

Then second problem is related to high-energy behavior of resonances: we extend

a recent result of Davies and Pushnitski to graphs with general vertex couplings

and find conditions under which the asymptotics does not have Weyl character.

Finally, the last question addressed here concerns the absolutely continuous spectrum

of radial-tree graphs. In a similar vein, we generalize a recent result by Breuer and

Frank that in case of the free (or Kirhhoff) coupling the ac spectrum is absent

provided the edge length are increasing without a bound along the tree.

We show that the result remains valid for a large class of vertex couplings,

but on the other hand, there are nontrivial couplings leading to an ac spectrum.

### 2010/09/14

#### Infinite Analysis Seminar Tokyo

10:30-14:00 Room #117 (Graduate School of Math. Sci. Bldg.)

AGT conjectures and recursion formulas (JAPANESE)

classification of solutions to the reflection equation associated to

trigonometrical $R$-matrix of Belavin (JAPANESE)

**Shintarou Yanagida**(Kobe Univ.) 10:30-11:30AGT conjectures and recursion formulas (JAPANESE)

**Yuji Yamada**(Rikkyo Univ.) 13:00-14:00classification of solutions to the reflection equation associated to

trigonometrical $R$-matrix of Belavin (JAPANESE)

### 2010/09/13

#### Infinite Analysis Seminar Tokyo

10:30-15:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Polynomial representations of DAHA of type $C^¥vee C$ and boundary qKZ equations (JAPANESE)

CFT, Isomonodromy deformations and Nekrasov functions (JAPANESE)

Twisted de Rham theory---resonances and the non-resonance (JAPANESE)

**Masahiro Kasatani**(Tokyo Univ.) 10:30-11:30Polynomial representations of DAHA of type $C^¥vee C$ and boundary qKZ equations (JAPANESE)

[ Abstract ]

First I will review basic facts about

the double affine Hecke algebra of type $C^¥vee C$

and its polynomial representation.

Next I will intrduce a boundary qKZ equation

and construct its solution in terms of the polynomial representation.

First I will review basic facts about

the double affine Hecke algebra of type $C^¥vee C$

and its polynomial representation.

Next I will intrduce a boundary qKZ equation

and construct its solution in terms of the polynomial representation.

**Yasuhiko Yamada**(Kobe Univ.) 13:00-14:00CFT, Isomonodromy deformations and Nekrasov functions (JAPANESE)

[ Abstract ]

This talk is an introduction to the relation between conformal filed

theories

and super symmetric gauge theories (Alday-Gaiotto-Tachikawa conjecture)

from the point of view of differential equations (in particular

isomonodromy

deformations).

This talk is an introduction to the relation between conformal filed

theories

and super symmetric gauge theories (Alday-Gaiotto-Tachikawa conjecture)

from the point of view of differential equations (in particular

isomonodromy

deformations).

**Katsuhisa Mimachi**(Tokyo Institute of Technology) 14:30-15:30Twisted de Rham theory---resonances and the non-resonance (JAPANESE)

### 2010/09/12

#### Infinite Analysis Seminar Tokyo

10:30-17:00 Room #117 (Graduate School of Math. Sci. Bldg.)

A factorization formula for Macdonald polynomials at roots of unity (JAPANESE)

W algebras and symmetric polynomials (JAPANESE)

Quantizing the difference Painlev¥'e VI equation (JAPANESE)

On a bijective proof of a factorization formula for Macdonald

polynomials at roots of unity (JAPANESE)

**Hideaki Morita**(Muroran Institute of Technology) 10:30-11:30A factorization formula for Macdonald polynomials at roots of unity (JAPANESE)

[ Abstract ]

We consider a combinatorial property of Macdonald polynomials at roots

of unity.

If we made some plethystic substitution to the variables,

Macdonald polynomials are subjected to a certain decomposition rule

when a parameter is specialized at roots of unity.

We review the result and give an outline of the proof.

This talk is based on a joint work with F. Descouens.

We consider a combinatorial property of Macdonald polynomials at roots

of unity.

If we made some plethystic substitution to the variables,

Macdonald polynomials are subjected to a certain decomposition rule

when a parameter is specialized at roots of unity.

We review the result and give an outline of the proof.

This talk is based on a joint work with F. Descouens.

**Junichi Shiraishi**(Tokyo Univ.) 13:00-14:00W algebras and symmetric polynomials (JAPANESE)

[ Abstract ]

It is well known that we have the factorization property of the Macdonald polynomials under the principal specialization $x=(1,t,t^2,t^3,¥cdots)$. We try to better understand this situation in terms of the Ding-Iohara algebra or the deformend $W$-algebra. Some conjectures are presented in the case of $N$-fold tensor representation of the Fock modules.

It is well known that we have the factorization property of the Macdonald polynomials under the principal specialization $x=(1,t,t^2,t^3,¥cdots)$. We try to better understand this situation in terms of the Ding-Iohara algebra or the deformend $W$-algebra. Some conjectures are presented in the case of $N$-fold tensor representation of the Fock modules.

**Koji Hasegawa**(Tohoku Univ.) 14:30-15:30Quantizing the difference Painlev¥'e VI equation (JAPANESE)

[ Abstract ]

I will review two constructions for quantum (=non-commutative) version of

q-difference Painleve VI equation.

I will review two constructions for quantum (=non-commutative) version of

q-difference Painleve VI equation.

**Yasuhide Numata**(Graduate School of Information Science and Technology, Tokyo Univ.) 16:00-17:00On a bijective proof of a factorization formula for Macdonald

polynomials at roots of unity (JAPANESE)

[ Abstract ]

The subject of this talk is a factorization formula for the special

values of modied Macdonald polynomials at roots of unity.

We give a combinatorial proof of the formula, via a result by

Haglund--Haiman--Leohr, for some special classes of partitions,

including two column partitions.

(This talk is based on a joint work with F. Descouens and H. Morita.)

The subject of this talk is a factorization formula for the special

values of modied Macdonald polynomials at roots of unity.

We give a combinatorial proof of the formula, via a result by

Haglund--Haiman--Leohr, for some special classes of partitions,

including two column partitions.

(This talk is based on a joint work with F. Descouens and H. Morita.)

### 2010/09/11

#### Infinite Analysis Seminar Tokyo

13:00-17:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Three-term recurrence relations for a $BC_n$-type basic hypergeometric function and their application (JAPANESE)

TBA (JAPANESE)

AGT conjecture and geometric engineering (JAPANESE)

**Masahiko Ito**(School of Science and Technology for Future Life, Tokyo Denki University) 13:00-14:00Three-term recurrence relations for a $BC_n$-type basic hypergeometric function and their application (JAPANESE)

[ Abstract ]

$BC_n$-type basic hypergeometric series are a certain $q$-analogue

of an integral representation for the Gauss hypergeometric function.

They are defined as multiple $q$-series satisfying Weyl group symmetry of type $C_n$,

and they are a multi-sum generalization of the basic hypergeometric series

in a class of what is called (very-)well-poised. In my talk I will explain

an explicit expression for the $q$-difference system of rank $n+1$

satisfied by a $BC_n$-type basic hypergeometric series with 6+1 parameters

as first order simultaneous $q$-difference equations with a concrete basis.

For this purpose I introduce two types of symmetric Laurent polynomials

which I call the $BC$-type interpolation polynomials. The polynomials satisfy

three-term relations like a contiguous relation for the Gauss hypergeometric

function. As an application, I will show another proof for the product formula

of the $q$-integral introduced by Gustafson.

$BC_n$-type basic hypergeometric series are a certain $q$-analogue

of an integral representation for the Gauss hypergeometric function.

They are defined as multiple $q$-series satisfying Weyl group symmetry of type $C_n$,

and they are a multi-sum generalization of the basic hypergeometric series

in a class of what is called (very-)well-poised. In my talk I will explain

an explicit expression for the $q$-difference system of rank $n+1$

satisfied by a $BC_n$-type basic hypergeometric series with 6+1 parameters

as first order simultaneous $q$-difference equations with a concrete basis.

For this purpose I introduce two types of symmetric Laurent polynomials

which I call the $BC$-type interpolation polynomials. The polynomials satisfy

three-term relations like a contiguous relation for the Gauss hypergeometric

function. As an application, I will show another proof for the product formula

of the $q$-integral introduced by Gustafson.

**Masatoshi Noumi**(Kobe Univ.) 14:30-15:30TBA (JAPANESE)

**Masato Taki**(YITP Kyoto Univ.) 16:00-17:00AGT conjecture and geometric engineering (JAPANESE)

### 2010/09/09

#### Lectures

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

Mini-course on Buildings (3/3) (ENGLISH)

**Bernhard Mühlherr**(Justus-Liebig-Universität Gießen)Mini-course on Buildings (3/3) (ENGLISH)

[ Abstract ]

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

The third lecture will be then devoted to classification results,

mainly the classification of spherical buildings. However, I will try to say some words on the classification of affine buildings and twin buildings as well.

This is Part 3 of a 3-part lecture.

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

The third lecture will be then devoted to classification results,

mainly the classification of spherical buildings. However, I will try to say some words on the classification of affine buildings and twin buildings as well.

This is Part 3 of a 3-part lecture.

### 2010/09/06

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Non-reduced components of the Noether-Lefschetz locus (ENGLISH)

**Prof. Remke Kloosterman**(Humboldt University, Berlin)Non-reduced components of the Noether-Lefschetz locus (ENGLISH)

[ Abstract ]

Let $M_d$ be the moduli space of complex smooth degree $d$ surfaces in $\\mathbb{P}3$. Let $NL_d \\subset M_d$ be the subset corresponding to surfaces with Picard number at least 2. It is known that $NL_r$ is Zariski-constructable, and each irreducible component of $NL_r$ has a natural scheme structure. In this talk we describe the largest non-reduced components of $NL_r$. This extends work of Maclean and Otwinowska.

This is joint work with my PhD student Ananyo Dan.

Let $M_d$ be the moduli space of complex smooth degree $d$ surfaces in $\\mathbb{P}3$. Let $NL_d \\subset M_d$ be the subset corresponding to surfaces with Picard number at least 2. It is known that $NL_r$ is Zariski-constructable, and each irreducible component of $NL_r$ has a natural scheme structure. In this talk we describe the largest non-reduced components of $NL_r$. This extends work of Maclean and Otwinowska.

This is joint work with my PhD student Ananyo Dan.

### 2010/09/04

#### Lectures

09:30-11:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Mini-course on Buildings (1/3) (ENGLISH)

**Bernhard Mühlherr**(Justus-Liebig-Universität Gießen)Mini-course on Buildings (1/3) (ENGLISH)

[ Abstract ]

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

In my first lecture I will begin by introducing generalized polygons, namely rank two spherical buildings, and discussing several aspects of them which will be generalized later, and then move on to defining Coxeter complexes and giving the classical definition of buildings as simplicial complexes. I will try to include as many examples as possible.

This is Part 1 of a 3-part lecture. The second lecture will follow after a ten-minute break.

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

In my first lecture I will begin by introducing generalized polygons, namely rank two spherical buildings, and discussing several aspects of them which will be generalized later, and then move on to defining Coxeter complexes and giving the classical definition of buildings as simplicial complexes. I will try to include as many examples as possible.

This is Part 1 of a 3-part lecture. The second lecture will follow after a ten-minute break.

#### Lectures

11:10-12:40 Room #126 (Graduate School of Math. Sci. Bldg.)

Mini-course on Buildings (2/3) (ENGLISH)

**Bernhard Mühlherr**(Justus-Liebig-Universität Gießen)Mini-course on Buildings (2/3) (ENGLISH)

[ Abstract ]

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

In my second lecture I will start with chamber systems and coset

geometries, introducing some special properties of chamber systems in order to give another definition of a building. This definition is less standard but it will give some results on presentations of groups acting on buildings for free. In particular it will enable me to present a sketch of a proof of the Curtis-Tits theorem for Chevalley groups and to formulate Tits' extension theorem.

This is Part 2 of a 3-part lecture. Part 1 takes place ealier on the same day. Part 3 will take place on Thursday, September 9.

The goal of this course is to provide an overview on the theory of buildings which was developed by Jacques Tits.

In my second lecture I will start with chamber systems and coset

geometries, introducing some special properties of chamber systems in order to give another definition of a building. This definition is less standard but it will give some results on presentations of groups acting on buildings for free. In particular it will enable me to present a sketch of a proof of the Curtis-Tits theorem for Chevalley groups and to formulate Tits' extension theorem.

This is Part 2 of a 3-part lecture. Part 1 takes place ealier on the same day. Part 3 will take place on Thursday, September 9.

### 2010/09/03

#### Lectures

14:30-15:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Carleman estimates and boundary problems. (JAPANESE)

**Luc Robbiano**(University of Versailles)Carleman estimates and boundary problems. (JAPANESE)

[ Abstract ]

In this presentation, based on joint works with Jerome LeRousseau and Matthieu Leautaud, we consider boundary problems for elliptic/parabolic operators. We prove Carleman estimates in such cases. One of the interest for such an estimate is the implied controllability of (semi-linear) heat equations.

One of the main aspects of the proof is a microlocal decomposition separating high and low tangential frequencies.

If time permits, we will present how such an approach can be used to prove Carleman estimates in the case of non smooth coefficients at an interface, possibly with an additional diffusion process along the interface.

In this presentation, based on joint works with Jerome LeRousseau and Matthieu Leautaud, we consider boundary problems for elliptic/parabolic operators. We prove Carleman estimates in such cases. One of the interest for such an estimate is the implied controllability of (semi-linear) heat equations.

One of the main aspects of the proof is a microlocal decomposition separating high and low tangential frequencies.

If time permits, we will present how such an approach can be used to prove Carleman estimates in the case of non smooth coefficients at an interface, possibly with an additional diffusion process along the interface.

### 2010/09/01

#### Lie Groups and Representation Theory

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

Groups of Kac-Moody type (ENGLISH)

**Bernhard M\"uhlherr**(Justus-Liebig-Universit\"at Giessen)Groups of Kac-Moody type (ENGLISH)

[ Abstract ]

Groups of Kac-Moody type are natural generalizations of Kac-Moody groups over fields in the sense that they have an RGD-system. This is a system of subgroups indexed by the roots of a root system and satisfying certain commutation relations.

Roughly speaking, there is a one-to-one correspondence between groups of Kac-Moody type and Moufang twin buildings. This correspondence was used in the last decade to prove several group theoretic results on RGD-systems and in particular on Kac-

Moody groups over fields.

In my talk I will explain RGD-systems and how they provide twin

buildings in a natural way. I will then present some of the group theoretic applications mentioned above and describe how twin buildings are used as a main tool in their proof.

Groups of Kac-Moody type are natural generalizations of Kac-Moody groups over fields in the sense that they have an RGD-system. This is a system of subgroups indexed by the roots of a root system and satisfying certain commutation relations.

Roughly speaking, there is a one-to-one correspondence between groups of Kac-Moody type and Moufang twin buildings. This correspondence was used in the last decade to prove several group theoretic results on RGD-systems and in particular on Kac-

Moody groups over fields.

In my talk I will explain RGD-systems and how they provide twin

buildings in a natural way. I will then present some of the group theoretic applications mentioned above and describe how twin buildings are used as a main tool in their proof.

#### thesis presentations

16:30-17:45 Room #123 (Graduate School of Math. Sci. Bldg.)

On the moduli spaces of finite flat models of Galois representations (JAPANESE)

**Naoki IMAI**(Graduate School of Mathematical Sciences the University of Tokyo )On the moduli spaces of finite flat models of Galois representations (JAPANESE)

### 2010/08/06

#### Lectures

15:30-17:45 Room #370 (Graduate School of Math. Sci. Bldg.)

A Spectral Method for Space--

Time Fractional Diffusion Equation (ENGLISH)

A multidimensional Borg-Levinson theorem (ENGLISH)

**Leevan Ling**(Hong Kong Baptist University) 15:30-16:30A Spectral Method for Space--

Time Fractional Diffusion Equation (ENGLISH)

**Mourad Choulli**(University of Metz) 16:45-17:45A multidimensional Borg-Levinson theorem (ENGLISH)

#### GCOE Seminars

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

Motion by mean curvature and Allen-Cahn equations (ENGLISH)

**Matthieu Alfaro**(University Montpellier 2)Motion by mean curvature and Allen-Cahn equations (ENGLISH)

[ Abstract ]

After introducing the classical and the generalized motion by mean curvature, we give some connections with the singular limit of Allen-Cahn equations in both framework. New results and estimates shall be provided.

After introducing the classical and the generalized motion by mean curvature, we give some connections with the singular limit of Allen-Cahn equations in both framework. New results and estimates shall be provided.

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