## Seminar information archive

Seminar information archive ～07/24｜Today's seminar 07/25 | Future seminars 07/26～

#### GCOE lecture series

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

Holomorphic extensions of highest weight representations to Olshanskii semigroups

**Joachim Hilgert**(Paderborn University)Holomorphic extensions of highest weight representations to Olshanskii semigroups

[ Abstract ]

In this lecture I will present a proof of Olshanskii's Theorem, which says that

for a simple group of Hermitean type unitarizable highest weight

representations can be holomorphically extended to contractive representations

of a complex semigroup containing the group in its boundary.

In this lecture I will present a proof of Olshanskii's Theorem, which says that

for a simple group of Hermitean type unitarizable highest weight

representations can be holomorphically extended to contractive representations

of a complex semigroup containing the group in its boundary.

### 2008/10/24

#### Colloquium

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

On the spectral measure of the sum of elements in a finite von Neumann algebra

**Benoit Collins**(オタワ大学・東京大学大学院数理科学研究科)On the spectral measure of the sum of elements in a finite von Neumann algebra

[ Abstract ]

Given two self-adjoint n×n matrices A and B with prescribed eigenvalues, the set of all possible spectral distributions for A+B has been conjectured by Horn and proved by Knutson, Tao, Klyachko and Totaro.

We address the same question when A and B have prescribed spectral measures but lie in an arbitrary II_1 factor, and we give elements of answers in terms of inequalities between the spectral measures. We explain the relation with the Connes embedding problem.

Given two self-adjoint n×n matrices A and B with prescribed eigenvalues, the set of all possible spectral distributions for A+B has been conjectured by Horn and proved by Knutson, Tao, Klyachko and Totaro.

We address the same question when A and B have prescribed spectral measures but lie in an arbitrary II_1 factor, and we give elements of answers in terms of inequalities between the spectral measures. We explain the relation with the Connes embedding problem.

### 2008/10/23

#### Operator Algebra Seminars

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

Planar algebras and the Haagerup subfactor

**Emily Peters**(UC Berkeley)Planar algebras and the Haagerup subfactor

### 2008/10/22

#### Number Theory Seminar

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

Serre's uniformity in the split Cartan case

**Pierre Parent**(Universite Bordeaux 1)Serre's uniformity in the split Cartan case

[ Abstract ]

We show that, for large enough prime number p, the modular curve

X_{split}(p) has no other point with values in Q than CM points and the rational cusp. This gives a partial answer to an old question of J.-P. Serre concerning the uniform surjectivity of Galois representations associated to torsion points on elliptic curves without complex multiplication.

(Joint work with Yuri Bilu.)

We show that, for large enough prime number p, the modular curve

X_{split}(p) has no other point with values in Q than CM points and the rational cusp. This gives a partial answer to an old question of J.-P. Serre concerning the uniform surjectivity of Galois representations associated to torsion points on elliptic curves without complex multiplication.

(Joint work with Yuri Bilu.)

### 2008/10/21

#### Lie Groups and Representation Theory

17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Invitation to Atlas combinatorics

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**落合啓之**(名古屋大学)Invitation to Atlas combinatorics

[ Abstract ]

半単純リー群のユニタリ表現の分類を手がける Atlas project(J. Adams, D. Vogan らが主催)では、実簡約(real reductive)線形代数群の admissible 表現をパラメトライズし、それに関するいくつかのプログラムが公開されています。ウェブサイトは www.liegroups.org.

現在、そのメインとなるものは Kazhdan-Lusztig-Vogan 多項式です。リー群として複素単純リー群を実リー群と見なしたケースが、通常の Kazhdan-Lusztig 理論に一致し、それを、ある一方向に拡張したのがここで扱われる KLV 理論と考えられます。

この講演では、リー群に関する背景説明などは軽く済ませ、Atlas で公開されているプログラムにおける方言、特に入出力の読み方を通常の言葉に言い換えることで、

プログラムを使ってもらう入り口での障壁を減らしたいと考えています。

ふむ、なかなか、使えるな、自分もインストールしてみようか、と思ってもらえれば、成功です。

なお、サーベイトークなので私のオリジナルな結果は含まれていません。また、計算機を使ってデモをする予定です。京都では計算機と板書の切り替えでばたばたしたので、照準を絞って慌てないように話したいと思います。

[ Reference URL ]半単純リー群のユニタリ表現の分類を手がける Atlas project(J. Adams, D. Vogan らが主催)では、実簡約(real reductive)線形代数群の admissible 表現をパラメトライズし、それに関するいくつかのプログラムが公開されています。ウェブサイトは www.liegroups.org.

現在、そのメインとなるものは Kazhdan-Lusztig-Vogan 多項式です。リー群として複素単純リー群を実リー群と見なしたケースが、通常の Kazhdan-Lusztig 理論に一致し、それを、ある一方向に拡張したのがここで扱われる KLV 理論と考えられます。

この講演では、リー群に関する背景説明などは軽く済ませ、Atlas で公開されているプログラムにおける方言、特に入出力の読み方を通常の言葉に言い換えることで、

プログラムを使ってもらう入り口での障壁を減らしたいと考えています。

ふむ、なかなか、使えるな、自分もインストールしてみようか、と思ってもらえれば、成功です。

なお、サーベイトークなので私のオリジナルな結果は含まれていません。また、計算機を使ってデモをする予定です。京都では計算機と板書の切り替えでばたばたしたので、照準を絞って慌てないように話したいと思います。

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

#### Tuesday Seminar on Topology

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

On embeddings of 3-manifolds in 6-manifolds

**森山 哲裕**(東京大学大学院数理科学研究科)On embeddings of 3-manifolds in 6-manifolds

[ Abstract ]

In this talk, we give a simple axiomatic definition of an invariant of

smooth embeddings of 3-manifolds in 6-manifolds.

The axiom is expressed in terms of some cobordisms of pairs of manifolds of

dimensions 6 and 3 (equipped with some cohomology class of the complement) and

the signature of 4-manifolds.

We then show that our invariant gives a unified framework for some classical

invariants in low-dimensions (Haefliger invariant, Milnor's triple

linking number, Rokhlin invariant, Casson invariant,

Takase's invariant, Skopenkov's invariants).

In this talk, we give a simple axiomatic definition of an invariant of

smooth embeddings of 3-manifolds in 6-manifolds.

The axiom is expressed in terms of some cobordisms of pairs of manifolds of

dimensions 6 and 3 (equipped with some cohomology class of the complement) and

the signature of 4-manifolds.

We then show that our invariant gives a unified framework for some classical

invariants in low-dimensions (Haefliger invariant, Milnor's triple

linking number, Rokhlin invariant, Casson invariant,

Takase's invariant, Skopenkov's invariants).

### 2008/10/20

#### Seminar on Geometric Complex Analysis

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

パンルヴェ方程式と複素力学系

**岩崎 克則**(九大数理)パンルヴェ方程式と複素力学系

### 2008/10/17

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #128 (Graduate School of Math. Sci. Bldg.)

岩根研究所に於ける画像処理技術の紹介Ⅱ; CV技術による空間認識と対象物認識、及び人工知能

**岩根 和郎**(岩根研究所)岩根研究所に於ける画像処理技術の紹介Ⅱ; CV技術による空間認識と対象物認識、及び人工知能

#### Lectures

15:00-16:00 Room #570 (Graduate School of Math. Sci. Bldg.)

GCOEレクチャー"Holomorphic extensions of unitary representations" その4 "Applications and open problems"

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

**Joachim Hilgert**(Paderborn University)GCOEレクチャー"Holomorphic extensions of unitary representations" その4 "Applications and open problems"

[ Abstract ]

In this lecture we present further applications of the given extension results and describe some open problems. In particular, we will mention estimates for automorphic forms (Krotz-Stanton), random matrices (Huckleberry-Puttmann-Zirnbauer), unitarizability of highest weight representation with non-scalar lowest K-type, and infinite dimensional groups.

[ Reference URL ]In this lecture we present further applications of the given extension results and describe some open problems. In particular, we will mention estimates for automorphic forms (Krotz-Stanton), random matrices (Huckleberry-Puttmann-Zirnbauer), unitarizability of highest weight representation with non-scalar lowest K-type, and infinite dimensional groups.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

#### Algebraic Geometry Seminar

13:00-14:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Construction of surfaces of general type with pg=0 via

Q-Gorenstein smoothing

**Yongnam Lee**(Sogang U.)Construction of surfaces of general type with pg=0 via

Q-Gorenstein smoothing

#### GCOE lecture series

15:00-16:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Holomorphic extensions of unitary representations その4 Applications and open problems

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

**Joachim Hilgert**(Paderborn University)Holomorphic extensions of unitary representations その4 Applications and open problems

[ Abstract ]

In this lecture we present further applications of the given extension results and describe some open problems. In particular, we will mention estimates for automorphic forms (Krötz-Stanton), random matrices (Huckleberry-Püttmann-Zirnbauer), unitarizability of highest weight representation with non-scalar lowest K-type, and infinite dimensional groups.

[ Reference URL ]In this lecture we present further applications of the given extension results and describe some open problems. In particular, we will mention estimates for automorphic forms (Krötz-Stanton), random matrices (Huckleberry-Püttmann-Zirnbauer), unitarizability of highest weight representation with non-scalar lowest K-type, and infinite dimensional groups.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

### 2008/10/16

#### Operator Algebra Seminars

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

The $D_{2n}$ planar algebras

**Scott Morrison**(UC Santa Barbara)The $D_{2n}$ planar algebras

#### Lectures

15:00-16:00 Room #570 (Graduate School of Math. Sci. Bldg.)

GCOEレクチャー"Holomorphic extensions of unitary representations" その3 "Highest weight representations"

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

**Joachim Hilgert**(Paderborn University)GCOEレクチャー"Holomorphic extensions of unitary representations" その3 "Highest weight representations"

[ Abstract ]

In this lecture we explain the extension results in a little more detail and explain how they lead to geometric realizations of singular highest weight representations on nilpotent coadjoint orbits.

[ Reference URL ]In this lecture we explain the extension results in a little more detail and explain how they lead to geometric realizations of singular highest weight representations on nilpotent coadjoint orbits.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

#### Applied Analysis

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Two-dimensional harmonic map heat flow versus four-dimensional Yang-Mills heat flow

**Joseph F. Grotowski**(University of Queensland)Two-dimensional harmonic map heat flow versus four-dimensional Yang-Mills heat flow

[ Abstract ]

Harmonic map heat flow and Yang-Mills heat flow are the gradient flows associated to particular energy functionals. In the considered dimension, (i.e. dimension two for the harmonic map heat flow, dimension four for the Yang-Mills heat flow), the associated energy functional is (locally) conformally invariant, that is, the dimension is critical. This leads to a number of interesting phenomena when considering both the functionals and the associated flows. In this talk we discuss qualitative similarities and differences between the flows.

Harmonic map heat flow and Yang-Mills heat flow are the gradient flows associated to particular energy functionals. In the considered dimension, (i.e. dimension two for the harmonic map heat flow, dimension four for the Yang-Mills heat flow), the associated energy functional is (locally) conformally invariant, that is, the dimension is critical. This leads to a number of interesting phenomena when considering both the functionals and the associated flows. In this talk we discuss qualitative similarities and differences between the flows.

#### GCOE lecture series

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

Holomorphic extensions of unitary representations その3 Highest weight representations

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

**Joachim Hilgert**(Paderborn University)Holomorphic extensions of unitary representations その3 Highest weight representations

[ Abstract ]

In this lecture we explain the extension results in a little more detail and explain how they lead to geometric realizations of singular highest weight representations on nilpotent coadjoint orbits.

[ Reference URL ]In this lecture we explain the extension results in a little more detail and explain how they lead to geometric realizations of singular highest weight representations on nilpotent coadjoint orbits.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

### 2008/10/15

#### PDE Real Analysis Seminar

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

Asymptotic behavior of solutions for BCF model describing crystal surface growth

**八木厚志**(大阪大学)Asymptotic behavior of solutions for BCF model describing crystal surface growth

[ Abstract ]

This talk is concerned with the initial-boundary value problem for a nonlinear parabolic equation which was presented Johnson et al. for describing the process of growth of a crystal surface on the

basis of the BCF theory. We will investigate asymptotic behavior of solutions by construct exponential attractors and a Lyapunov function and by examining a structure of the $\\omega$ limit set.

This talk is concerned with the initial-boundary value problem for a nonlinear parabolic equation which was presented Johnson et al. for describing the process of growth of a crystal surface on the

basis of the BCF theory. We will investigate asymptotic behavior of solutions by construct exponential attractors and a Lyapunov function and by examining a structure of the $\\omega$ limit set.

#### Lectures

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

連続講演 "Thin 3D Navier-Stokes equations" (3次元薄層領域上のナビエストークス方程式) その2 The role of the 2D limit problem

**George Sell**(ミネソタ大学)連続講演 "Thin 3D Navier-Stokes equations" (3次元薄層領域上のナビエストークス方程式) その2 The role of the 2D limit problem

[ Abstract ]

In both lectures we will examine a new topic of the thin

3D Navier-Stokes equations with Navier boundary conditions.

In the first lecture we will treat the ultimate boundedness

of strong solutions and the related theory of global

attractors.

In the second lecture, which will include a brief summary

of the first lecture, we will examine the role played by the

2D Limit Problem. These issues are a special challenge for

analysis because the 2D Limit Problem is NOT imbedded the

3D problem.

These lectures are based on joint work with Genevieve Raugel,Dragos Iftimie, and Luan Hoang.

In both lectures we will examine a new topic of the thin

3D Navier-Stokes equations with Navier boundary conditions.

In the first lecture we will treat the ultimate boundedness

of strong solutions and the related theory of global

attractors.

In the second lecture, which will include a brief summary

of the first lecture, we will examine the role played by the

2D Limit Problem. These issues are a special challenge for

analysis because the 2D Limit Problem is NOT imbedded the

3D problem.

These lectures are based on joint work with Genevieve Raugel,Dragos Iftimie, and Luan Hoang.

#### Lectures

15:00-16:00 Room #570 (Graduate School of Math. Sci. Bldg.)

GCOEレクチャー"Holomorphic extensions of unitary representations" その2 "Geometric Background"

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

**Joachim Hilgert**(Paderborn University)GCOEレクチャー"Holomorphic extensions of unitary representations" その2 "Geometric Background"

[ Abstract ]

In this lecture we will explain the complex geometry needed to understand the phenomena described in the first lecture. The key words here are Olshanski semigroups, invariant cones in Lie algebras, Akhiezer-Gindikin domain, and coadjoint orbits of convex type.

[ Reference URL ]In this lecture we will explain the complex geometry needed to understand the phenomena described in the first lecture. The key words here are Olshanski semigroups, invariant cones in Lie algebras, Akhiezer-Gindikin domain, and coadjoint orbits of convex type.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

#### GCOE lecture series

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

Holomorphic extensions of unitary representations その2 Geometric background

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2008.html#20081014hilgert

**Joachim Hilgert**(Paderborn University)Holomorphic extensions of unitary representations その2 Geometric background

[ Abstract ]

In this lecture we will explain the complex geometry needed to understand the phenomena described in the first lecture. The key words here are Olshanski semigroups, invariant cones in Lie algebras, Akhiezer-Gindikin domain, and coadjoint orbits of convex type.

[ Reference URL ]In this lecture we will explain the complex geometry needed to understand the phenomena described in the first lecture. The key words here are Olshanski semigroups, invariant cones in Lie algebras, Akhiezer-Gindikin domain, and coadjoint orbits of convex type.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2008.html#20081014hilgert

### 2008/10/14

#### Tuesday Seminar on Topology

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

Pontrjagin-Thom maps and the Deligne-Mumford compactification

**Jeffrey Herschel Giansiracusa**(Oxford University)Pontrjagin-Thom maps and the Deligne-Mumford compactification

[ Abstract ]

An embedding f: M -> N produces, via a construction of Pontrjagin-Thom, a map from N to the Thom space of the normal bundle over M. If f is an arbitrary map then one instead gets a map from N to the infinite loop space of the Thom spectrum of the normal bundle of f. We extend this Pontrjagin-Thom construction of wrong-way maps to differentiable stacks and use it to produce interesting maps from the Deligne-Mumford compactification of the moduli space of curves to certain infinite loop spaces. We show that these maps are surjective on mod p homology in a range of degrees. We thus produce large new families of torsion cohomology classes on the Deligne-Mumford compactification.

An embedding f: M -> N produces, via a construction of Pontrjagin-Thom, a map from N to the Thom space of the normal bundle over M. If f is an arbitrary map then one instead gets a map from N to the infinite loop space of the Thom spectrum of the normal bundle of f. We extend this Pontrjagin-Thom construction of wrong-way maps to differentiable stacks and use it to produce interesting maps from the Deligne-Mumford compactification of the moduli space of curves to certain infinite loop spaces. We show that these maps are surjective on mod p homology in a range of degrees. We thus produce large new families of torsion cohomology classes on the Deligne-Mumford compactification.

#### Lectures

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

連続講演 "Thin 3D Navier-Stokes equations" (3次元薄層領域上のナビエストークス方程式) その1

Ultimate boundedness of solutions with large data and global attractors

**George Sell**(ミネソタ大学)連続講演 "Thin 3D Navier-Stokes equations" (3次元薄層領域上のナビエストークス方程式) その1

Ultimate boundedness of solutions with large data and global attractors

[ Abstract ]

In both lectures we will examine a new topic of the thin 3D Navier-Stokes equations with Navier boundary conditions.

In the first lecture we will treat the ultimate boundedness of strong solutions and the related theory of global attractors.

In the second lecture, which will include a brief summary of the first lecture, we will examine the role played by the 2D Limit Problem. These issues are a special challenge for analysis because the 2D Limit Problem is NOT imbedded the 3D problem.

These lectures are based on joint work with Genevieve Raugel, Dragos Iftimie, and Luan Hoang.

In both lectures we will examine a new topic of the thin 3D Navier-Stokes equations with Navier boundary conditions.

In the first lecture we will treat the ultimate boundedness of strong solutions and the related theory of global attractors.

In the second lecture, which will include a brief summary of the first lecture, we will examine the role played by the 2D Limit Problem. These issues are a special challenge for analysis because the 2D Limit Problem is NOT imbedded the 3D problem.

These lectures are based on joint work with Genevieve Raugel, Dragos Iftimie, and Luan Hoang.

#### Tuesday Seminar of Analysis

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Thin 3D Navier-Stokes equations: Ultimate boundedness of solutions with large data and global attractors

**George Sell**(ミネソタ大学)Thin 3D Navier-Stokes equations: Ultimate boundedness of solutions with large data and global attractors

[ Abstract ]

グローバルCOE連続講演会と共催です.詳細はそちらをご覧ください.

グローバルCOE連続講演会と共催です.詳細はそちらをご覧ください.

#### Lectures

15:00-16:00 Room #570 (Graduate School of Math. Sci. Bldg.)

GCOEレクチャー"Holomorphic extensions of unitary representations" その1 "Overview and Examples"

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

**Joachim Hilgert**(Paderborn University)GCOEレクチャー"Holomorphic extensions of unitary representations" その1 "Overview and Examples"

[ Abstract ]

In this lecture we present the Gelfand-Gindikin program of decomposing $L^2$-spaces into families of irreducible representations using complex geometry. We then briefly outline results due to Olshanski, Hilgert-Olafsson-Orsted, Hilgert-Neeb-Orsted, Krotz-Stanton and others in this direction. In particular, we will explain holomorphic extensions of holomorphic discrete series representations and their relation to Hardy and weighted Bergman spaces.

[ Reference URL ]In this lecture we present the Gelfand-Gindikin program of decomposing $L^2$-spaces into families of irreducible representations using complex geometry. We then briefly outline results due to Olshanski, Hilgert-Olafsson-Orsted, Hilgert-Neeb-Orsted, Krotz-Stanton and others in this direction. In particular, we will explain holomorphic extensions of holomorphic discrete series representations and their relation to Hardy and weighted Bergman spaces.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

#### Lie Groups and Representation Theory

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

The Dirichlet-to-Neumann map as a pseudodifferential

operator

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**Jan Moellers**(Paderborn University)The Dirichlet-to-Neumann map as a pseudodifferential

operator

[ Abstract ]

Both Dirichlet and Neumann boundary conditions for the Laplace equation are of fundamental importance in Mathematics and Physics. Given a compact connected Riemannian manifold $M$ with boundary $\\partial M$ the Dirichlet-to-Neumann operator $\\Lambda_g$ maps Dirichlet boundary data $f$ to the corresponding Neumann boundary data $\\Lambda_g f =(\\partial_\\nu u)|_{\\partial M}$ where $u$ denotes the unique solution to the Dirichlet problem $\\laplace_g u=0$ in $M$, $u|_{\\partial M} = f$.

The main statement is that this operator is a first order elliptic pseudodifferential operator on the boundary $\\partial M$.

We will first give a brief overview of how to define the Dirichlet-to-Neumann operator as a map $\\Lambda_g:H^{1/2}(\\partial M)\\longrightarrow H^{-1/2}(\\partial M)$ between Sobolev spaces. In order to show that it is actually a pseudodifferential operator we introduce tangential pseudodifferential operators. This allows us to derive a

microlocal factorization of the Laplacian near boundary points. Together with a regularity statement for the heat equation this will finally give the main result.

[ Reference URL ]Both Dirichlet and Neumann boundary conditions for the Laplace equation are of fundamental importance in Mathematics and Physics. Given a compact connected Riemannian manifold $M$ with boundary $\\partial M$ the Dirichlet-to-Neumann operator $\\Lambda_g$ maps Dirichlet boundary data $f$ to the corresponding Neumann boundary data $\\Lambda_g f =(\\partial_\\nu u)|_{\\partial M}$ where $u$ denotes the unique solution to the Dirichlet problem $\\laplace_g u=0$ in $M$, $u|_{\\partial M} = f$.

The main statement is that this operator is a first order elliptic pseudodifferential operator on the boundary $\\partial M$.

We will first give a brief overview of how to define the Dirichlet-to-Neumann operator as a map $\\Lambda_g:H^{1/2}(\\partial M)\\longrightarrow H^{-1/2}(\\partial M)$ between Sobolev spaces. In order to show that it is actually a pseudodifferential operator we introduce tangential pseudodifferential operators. This allows us to derive a

microlocal factorization of the Laplacian near boundary points. Together with a regularity statement for the heat equation this will finally give the main result.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

#### GCOE lecture series

15:00-16:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Holomorphic extensions of unitary representations" その1 "Overview and Examples"

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2008.html#20081014hilgert

**Joachim Hilgert**(Paderborn University)Holomorphic extensions of unitary representations" その1 "Overview and Examples"

[ Abstract ]

In this lecture we present the Gelfand-Gindikin program of decomposing $L^2$-spaces into families of irreducible representations using complex geometry. We then briefly outline results due to Olshanski, Hilgert-Olafsson-Orsted, Hilgert-Neeb-Orsted, Krotz-Stanton and others in this direction. In particular, we will explain holomorphic extensions of holomorphic discrete series representations and their relation to Hardy and weighted Bergman spaces.

[ Reference URL ]In this lecture we present the Gelfand-Gindikin program of decomposing $L^2$-spaces into families of irreducible representations using complex geometry. We then briefly outline results due to Olshanski, Hilgert-Olafsson-Orsted, Hilgert-Neeb-Orsted, Krotz-Stanton and others in this direction. In particular, we will explain holomorphic extensions of holomorphic discrete series representations and their relation to Hardy and weighted Bergman spaces.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2008.html#20081014hilgert

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