## Seminar information archive

Seminar information archive ～02/15｜Today's seminar 02/16 | Future seminars 02/17～

### 2009/04/22

#### PDE Real Analysis Seminar

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

From Codazzi-Mainardi to Cauchy-Riemann

**Wilhelm Klingenberg**(University of Durham)From Codazzi-Mainardi to Cauchy-Riemann

[ Abstract ]

In joint work with Brendan Guilfoyle we established an upper bound for the winding number of the principal curvature foliation at any isolated umbilic of a surface in Euclidean three-space. In our talk, we will focus on the analytic core of the problem. Here is a model of the triaxial ellipsoid with its curvature foliation and one umbilic on the right.

In joint work with Brendan Guilfoyle we established an upper bound for the winding number of the principal curvature foliation at any isolated umbilic of a surface in Euclidean three-space. In our talk, we will focus on the analytic core of the problem. Here is a model of the triaxial ellipsoid with its curvature foliation and one umbilic on the right.

#### Geometry Seminar

14:45-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Einstein-Weyl structures on 3-dimensional Severi varieties

Toric non-Abelian Hodge theory

**中田文憲**(東京工業大学理工学研究科) 14:45-16:15Einstein-Weyl structures on 3-dimensional Severi varieties

[ Abstract ]

The space of nodal curves on a projective surface is called a Severi variety.In this talk, we show that any Severi variety of nodal rational curves on a non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space of smooth rational curves on a complex surface, given by N. Hitchin in the context of twistor theory. We will explain some properties of the Einstein-Weyl spaces given by this method, and we will also show some examples of such Einstein-Weyl spaces. (This is a joint work with Nobuhiro Honda.)

The space of nodal curves on a projective surface is called a Severi variety.In this talk, we show that any Severi variety of nodal rational curves on a non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space of smooth rational curves on a complex surface, given by N. Hitchin in the context of twistor theory. We will explain some properties of the Einstein-Weyl spaces given by this method, and we will also show some examples of such Einstein-Weyl spaces. (This is a joint work with Nobuhiro Honda.)

**Tamas Hausel**(Oxford University) 16:30-18:00Toric non-Abelian Hodge theory

[ Abstract ]

First we give an overview of the geometrical and topological aspects of the spaces in the non-Abelian Hodge theory of a curve and their connection with quiver varieties. Then by concentrating on toric hyperkaehler varieties in place of quiver varieties we construct the toric Betti, De Rham and Dolbeault spaces and prove several of the expected properties of the geometry and topology of these varieties. This is joint work with Nick Proudfoot.

First we give an overview of the geometrical and topological aspects of the spaces in the non-Abelian Hodge theory of a curve and their connection with quiver varieties. Then by concentrating on toric hyperkaehler varieties in place of quiver varieties we construct the toric Betti, De Rham and Dolbeault spaces and prove several of the expected properties of the geometry and topology of these varieties. This is joint work with Nick Proudfoot.

#### Seminar on Probability and Statistics

15:00-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Interacting Markov chain Monte Carlo Methods for Solving Nonlinear Measure-Valued Equations

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/02.html

**Arnaud DOUCET**(統計数理研究所)Interacting Markov chain Monte Carlo Methods for Solving Nonlinear Measure-Valued Equations

[ Abstract ]

We present a new class of interacting Markov chain Monte Carlo algorithms for solving numerically discrete-time measure-valued equations. The associated stochastic processes belong to the class of self-interacting Markov chains. In contrast to traditional Markov chains, their time evolution depend on the occupation measure of their past values. This general methodology allows us to provide a natural way to sample from a sequence of target probability measures of increasing complexity. We develop an original theoretical analysis to analyze the behaviour of these iterative algorithms. We establish a variety of convergence results including exponential estimates and a uniform convergence theorem with respect to the number of target distributions. We also illustrate these algorithms in the context of Feynman-Kac distribution flows.

(this is joint work with Professor Pierre Del Moral)

[ Reference URL ]We present a new class of interacting Markov chain Monte Carlo algorithms for solving numerically discrete-time measure-valued equations. The associated stochastic processes belong to the class of self-interacting Markov chains. In contrast to traditional Markov chains, their time evolution depend on the occupation measure of their past values. This general methodology allows us to provide a natural way to sample from a sequence of target probability measures of increasing complexity. We develop an original theoretical analysis to analyze the behaviour of these iterative algorithms. We establish a variety of convergence results including exponential estimates and a uniform convergence theorem with respect to the number of target distributions. We also illustrate these algorithms in the context of Feynman-Kac distribution flows.

(this is joint work with Professor Pierre Del Moral)

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/02.html

### 2009/04/21

#### Tuesday Seminar on Topology

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

Some algebraic aspects of KZ systems

**Ivan Marin**(Univ. Paris VII)Some algebraic aspects of KZ systems

[ Abstract ]

Knizhnik-Zamolodchikov (KZ) systems enables one

to construct representations of (generalized)

braid groups. While this geometric construction is

now very well understood, it still brings to

attention, or helps constructing, new algebraic objects.

In this talk, we will present some of them, including an

infinitesimal version of Iwahori-Hecke algebras and a

generalization of the Krammer representations of the usual

braid groups.

Knizhnik-Zamolodchikov (KZ) systems enables one

to construct representations of (generalized)

braid groups. While this geometric construction is

now very well understood, it still brings to

attention, or helps constructing, new algebraic objects.

In this talk, we will present some of them, including an

infinitesimal version of Iwahori-Hecke algebras and a

generalization of the Krammer representations of the usual

braid groups.

### 2009/04/20

#### Seminar on Geometric Complex Analysis

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

Indefinite Kähler surfaces of constant scalar curvature

**鎌田博行**(宮城教育大学)Indefinite Kähler surfaces of constant scalar curvature

### 2009/04/18

#### Infinite Analysis Seminar Tokyo

11:00-14:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Invariant Differential Operators for Non-Compact Lie Groups

TBA

**Vladimir Dobrev**(Institute for Nuclear Reserch and Nuclear Energy, Sofia, Bulgaria) 11:00-12:00Invariant Differential Operators for Non-Compact Lie Groups

[ Abstract ]

We present a canonical procedure for the explicit construction of

invariant differential operators. The exposition is for semi-simple

Lie algebras, but is easily generalized to the supersymmetric and

quantum group settings. Especially important is a narrow class of

algebras, which we call 'conformal Lie algebras', which have very

similar properties to the conformal algebras of n-dimensional

Minkowski space-time. Examples are given in detail, including diagrams of

intertwining operators, or equivalently, multiplets of elementary

representations (generalized Verma modules).

We present a canonical procedure for the explicit construction of

invariant differential operators. The exposition is for semi-simple

Lie algebras, but is easily generalized to the supersymmetric and

quantum group settings. Especially important is a narrow class of

algebras, which we call 'conformal Lie algebras', which have very

similar properties to the conformal algebras of n-dimensional

Minkowski space-time. Examples are given in detail, including diagrams of

intertwining operators, or equivalently, multiplets of elementary

representations (generalized Verma modules).

**笠谷昌弘**(東大数理) 13:30-14:30TBA

[ Abstract ]

TBA

TBA

### 2009/04/16

#### Operator Algebra Seminars

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

Large Deviations in Quantum Spin Chains

**緒方芳子**(東大数理)Large Deviations in Quantum Spin Chains

### 2009/04/15

#### Lectures

15:30-17:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Invariant measures for stochastic partial differential equations: new a priori estimates and applications

**Wilhelm Stannat**(Darmstadt 工科大学)Invariant measures for stochastic partial differential equations: new a priori estimates and applications

#### Seminar on Probability and Statistics

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Estimating the successive Blumenthal-Getoor indices for a discretely observed process

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/00.html

**Jean JACOD**(Universite Paris VI)Estimating the successive Blumenthal-Getoor indices for a discretely observed process

[ Abstract ]

Letting F be a Levy measure whose "tail" $F ([-x, x])$ admits an expansion $\\sigma_{i\\ge 1} a_i/x^\\beta$ as $x \\rightarrow 0$, we call $\\beta_1 > \\beta_2 >...$ the successive Blumenthal-Getoor indices, since $\\beta_1$ is in this case the usual Blumenthal-Getoor index. This notion may be extended to more general semimartingale. We propose here a method to estimate the $\\beta_i$'s and the coefficients $a_i$'s, or rather their extension for semimartingales, when the underlying semimartingale $X$ is observed at discrete times, on fixed time interval. The asymptotic is when the time-lag goes to $0$. It is then possible to construct consistent estimators for $\\beta_i$ and $a_i$ for those $i$'s such that $\\beta_i > \\beta_1 /2$, whereas it is impossible to do so (even when $X$ is a Levy process) for those $i$'s such that $\\beta_i < \\beta_1 /2$. On the other hand, a central limit theorem for $\\beta_1$ is available only when $\\beta_i < \\beta_1 /2$: consequently, when we can actually consistently estimate some $\\beta_i$'s besides $\\beta_1$ , then no central limit theorem can hold, and correlatively the rates of convergence become quite slow (although one know them explicitly): so the results have some theoretical interest in the sense that they set up bounds on what is actually possible to achieve, but the practical applications are probably quite thin.

(joint with Yacine Ait-Sahalia)

[ Reference URL ]Letting F be a Levy measure whose "tail" $F ([-x, x])$ admits an expansion $\\sigma_{i\\ge 1} a_i/x^\\beta$ as $x \\rightarrow 0$, we call $\\beta_1 > \\beta_2 >...$ the successive Blumenthal-Getoor indices, since $\\beta_1$ is in this case the usual Blumenthal-Getoor index. This notion may be extended to more general semimartingale. We propose here a method to estimate the $\\beta_i$'s and the coefficients $a_i$'s, or rather their extension for semimartingales, when the underlying semimartingale $X$ is observed at discrete times, on fixed time interval. The asymptotic is when the time-lag goes to $0$. It is then possible to construct consistent estimators for $\\beta_i$ and $a_i$ for those $i$'s such that $\\beta_i > \\beta_1 /2$, whereas it is impossible to do so (even when $X$ is a Levy process) for those $i$'s such that $\\beta_i < \\beta_1 /2$. On the other hand, a central limit theorem for $\\beta_1$ is available only when $\\beta_i < \\beta_1 /2$: consequently, when we can actually consistently estimate some $\\beta_i$'s besides $\\beta_1$ , then no central limit theorem can hold, and correlatively the rates of convergence become quite slow (although one know them explicitly): so the results have some theoretical interest in the sense that they set up bounds on what is actually possible to achieve, but the practical applications are probably quite thin.

(joint with Yacine Ait-Sahalia)

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/00.html

#### Seminar on Probability and Statistics

15:00-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

A survey on realized p-variations for semimartingales

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/00.html

**Jean JACOD**(Universite Paris VI)A survey on realized p-variations for semimartingales

[ Abstract ]

Let $X$ be a process which is observed at the times $i\\Delta_n$ for $i=0,1,\\ldots,$. If $p>0$ the realized $p$-variation over the time interval $[0, t]$ is

V^n(p)_t=\\sum_{i=1}^{[t/\\Delta_n]}|X_{i\\Delta_n}-X_{(i-1)\\Delta_n}|^p.

The behavior of these $p$-variations when $\\Delta_n ightarrow 0$ (and t is fixed) is now well understood, from the point of view of limits in probability (these are basically old results due to Lepingle) and also for the associated central limit theorem.

The aim of this talk is to review those results, as well as a few extensions (multipower variations, truncated variations). We will put some emphasis on the assumptions on $X$ which are needed, depending on the value of $p$.

[ Reference URL ]Let $X$ be a process which is observed at the times $i\\Delta_n$ for $i=0,1,\\ldots,$. If $p>0$ the realized $p$-variation over the time interval $[0, t]$ is

V^n(p)_t=\\sum_{i=1}^{[t/\\Delta_n]}|X_{i\\Delta_n}-X_{(i-1)\\Delta_n}|^p.

The behavior of these $p$-variations when $\\Delta_n ightarrow 0$ (and t is fixed) is now well understood, from the point of view of limits in probability (these are basically old results due to Lepingle) and also for the associated central limit theorem.

The aim of this talk is to review those results, as well as a few extensions (multipower variations, truncated variations). We will put some emphasis on the assumptions on $X$ which are needed, depending on the value of $p$.

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/00.html

### 2009/04/14

#### Lectures

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

Resolution of symplectic orbifolds obtained from reduction

**Klaus Niederkruger**(Ecole normale superieure de Lyon)Resolution of symplectic orbifolds obtained from reduction

[ Abstract ]

We present a method to obtain resolutions of symplectic orbifolds arising from symplectic reduction of a Hamiltonian S1-manifold at a regular value. As an application, we show that all isolated cyclic singularities of a symplectic orbifold admit a resolution and that pre-quantizations of symplectic orbifolds are symplectically fillable by a smooth manifold.

We present a method to obtain resolutions of symplectic orbifolds arising from symplectic reduction of a Hamiltonian S1-manifold at a regular value. As an application, we show that all isolated cyclic singularities of a symplectic orbifold admit a resolution and that pre-quantizations of symplectic orbifolds are symplectically fillable by a smooth manifold.

### 2009/04/13

#### Seminar on Geometric Complex Analysis

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

A new method to generalize the Nevanlinna theory to several complex variables

**千葉優作**(東大数理)A new method to generalize the Nevanlinna theory to several complex variables

### 2009/04/09

#### Operator Algebra Seminars

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

Bimodules, planarity and freeness

**Dietmar Bisch**(Vanderbilt University)Bimodules, planarity and freeness

### 2009/04/08

#### Seminar on Mathematics for various disciplines

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

Growth of an Ice Disk from Supercooled Water: Theory and Space Experiment in Kibo of International Space Station

**横山悦郎**(学習院大学)Growth of an Ice Disk from Supercooled Water: Theory and Space Experiment in Kibo of International Space Station

[ Abstract ]

We present a model of the time evolution of a disk crystal of ice with radius $R$ and thickness $h$ growing from supercooled water and discuss its morphological stability. Disk thickening, {\\it i.e.}, growth along the $c$ axis of ice, is governed by slow molecular rearrangements on the basal faces. Growth of the radius, {\\it i.e.}, growth parallel to the basal plane, is controlled by transport of latent heat. Our analysis is used to understand the symmetry breaking obtained experimentally by Shimada and Furukawa under the one-G condition. We also introduce that the space experiment of the morphological instability on the ice growing in supercooled water, which was carried out on the Japanese Experiment Module "Kibo" of International Space Station from December 2008 and February 2009.

http://kibo.jaxa.jp/experiment/theme/first/ice_crystal_end.html

We show the experimental results under the micro-G condition and discuss the feature on the "Kibo" experoments.

We present a model of the time evolution of a disk crystal of ice with radius $R$ and thickness $h$ growing from supercooled water and discuss its morphological stability. Disk thickening, {\\it i.e.}, growth along the $c$ axis of ice, is governed by slow molecular rearrangements on the basal faces. Growth of the radius, {\\it i.e.}, growth parallel to the basal plane, is controlled by transport of latent heat. Our analysis is used to understand the symmetry breaking obtained experimentally by Shimada and Furukawa under the one-G condition. We also introduce that the space experiment of the morphological instability on the ice growing in supercooled water, which was carried out on the Japanese Experiment Module "Kibo" of International Space Station from December 2008 and February 2009.

http://kibo.jaxa.jp/experiment/theme/first/ice_crystal_end.html

We show the experimental results under the micro-G condition and discuss the feature on the "Kibo" experoments.

### 2009/03/25

#### GCOE lecture series

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

The Strominger-Yau-Zaslow conjecture and mirror symmetry via degenerations II

**Mark Gross**(University of California, San Diego)The Strominger-Yau-Zaslow conjecture and mirror symmetry via degenerations II

[ Abstract ]

The second half of the lecture.

The second half of the lecture.

### 2009/03/24

#### GCOE lecture series

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

The Strominger-Yau-Zaslow conjecture and mirror symmetry via degenerations I

**Mark Gross**(University of California, San Diego)The Strominger-Yau-Zaslow conjecture and mirror symmetry via degenerations I

[ Abstract ]

I will discuss the SYZ conjecture which attempts to explain mirror symmetry via the existence of dual torus fibrations on mirror pairs of Calabi-Yau manifolds. After reviewing some older work on this subject, I will explain how it leads to an algebro-geometric version of this conjecture and will discuss recent work with Bernd Siebert. This recent work gives a mirror construction along with far more detailed information about the B-model side of mirror symmetry, leading to new mirror symmetry predictions.

I will discuss the SYZ conjecture which attempts to explain mirror symmetry via the existence of dual torus fibrations on mirror pairs of Calabi-Yau manifolds. After reviewing some older work on this subject, I will explain how it leads to an algebro-geometric version of this conjecture and will discuss recent work with Bernd Siebert. This recent work gives a mirror construction along with far more detailed information about the B-model side of mirror symmetry, leading to new mirror symmetry predictions.

### 2009/03/21

#### Infinite Analysis Seminar Tokyo

11:00-14:30 Room #117 (Graduate School of Math. Sci. Bldg.)

On classes of transformations for bilinear sum of

(basic) hypergeometric series and multivariate generalizations.

On explicit formulas for Whittaker functions on real semisimple Lie groups

**梶原 康史**(神戸理) 11:00-12:00On classes of transformations for bilinear sum of

(basic) hypergeometric series and multivariate generalizations.

[ Abstract ]

In this talk, I will present classes of bilinear transformation

formulas for basic hypergeometric series and Milne's multivariate

basic hypergeometric series associated with the root system of

type $A$. Our construction is similar to one of elementary

proof of Sears-Whipple transformation formula for terminating

balanced ${}_4 \\phi_3$ series while we use multiple Euler

transformation formula with different dimensions which has

obtained in our previous work.

In this talk, I will present classes of bilinear transformation

formulas for basic hypergeometric series and Milne's multivariate

basic hypergeometric series associated with the root system of

type $A$. Our construction is similar to one of elementary

proof of Sears-Whipple transformation formula for terminating

balanced ${}_4 \\phi_3$ series while we use multiple Euler

transformation formula with different dimensions which has

obtained in our previous work.

**石井 卓**(成蹊大理工) 13:30-14:30On explicit formulas for Whittaker functions on real semisimple Lie groups

[ Abstract ]

will report explicit formulas

for Whittaker functions related to principal series

reprensetations on real semisimple Lie groups $G$ of

classical type.

Our explicit formulas are recursive formulas with

respect to the real rank of $G$, and in some lower rank

cases they are related to generalized

hypergeometric series $ {}_3F_2(1) $ and $ {}_4F_3(1) $.

will report explicit formulas

for Whittaker functions related to principal series

reprensetations on real semisimple Lie groups $G$ of

classical type.

Our explicit formulas are recursive formulas with

respect to the real rank of $G$, and in some lower rank

cases they are related to generalized

hypergeometric series $ {}_3F_2(1) $ and $ {}_4F_3(1) $.

### 2009/03/17

#### GCOE lecture series

10:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Dirac Cohomology

Enright-Varadarajan modules and harmonic spinors

Harish-Chandra modules

Special unipotent representations of real reductive groups

**Roger Zierau**(Oklahoma State University) 11:00-12:00Dirac Cohomology

**Salah Mehdi**(Metz University) 13:30-14:30Enright-Varadarajan modules and harmonic spinors

**Bernhard Krötz**

(Max Planck Institute) 15:00-16:00Harish-Chandra modules

**Peter Trapa**(Utah) 16:30-17:30Special unipotent representations of real reductive groups

### 2009/03/16

#### GCOE lecture series

10:00-16:20 Room #123 (Graduate School of Math. Sci. Bldg.)

Harish-Chandra modules

http://www.ms.u-tokyo.ac.jp/~toshi/index.files/springschooltokyo200903.html#kroetz

Special unipotent representations of real reductive groups

http://www.ms.u-tokyo.ac.jp/~toshi/index.files/springschooltokyo200903.html#trapa

Dirac Cohomology

http://www.ms.u-tokyo.ac.jp/~toshi/index.files/springschooltokyo200903.html#zierau

Enright-Varadarajan modules and harmonic spinors

http://www.ms.u-tokyo.ac.jp/~toshi/index.files/springschooltokyo200903.html#mehdi

**Bernhard Krötz**

(Max Planck Institute) 10:00-11:00Harish-Chandra modules

[ Abstract ]

We plan to give a course on the various types of topological globalizations of Harish-Chandra modules. It is intended to cover the following topics:

1.Topological representation theory on various types of locally convex vector spaces.

2.Basic algebraic theory of Harish-Chandra modules

3. Unique globalization versus lower bounds for matrix coefficients

4. Dirac type sequences for representations

5. Deformation theory of Harish-Chandra modules

The new material presented was obtained in collaboration with Joseph Bernstein and Henrik Schlichtkrull. A first reference is the recent preprint "Smooth Frechet Globalizations of Harish-Chandra Modules" by J. Bernstein and myself, downloadable at arXiv:0812.1684v1.

[ Reference URL ]We plan to give a course on the various types of topological globalizations of Harish-Chandra modules. It is intended to cover the following topics:

1.Topological representation theory on various types of locally convex vector spaces.

2.Basic algebraic theory of Harish-Chandra modules

3. Unique globalization versus lower bounds for matrix coefficients

4. Dirac type sequences for representations

5. Deformation theory of Harish-Chandra modules

The new material presented was obtained in collaboration with Joseph Bernstein and Henrik Schlichtkrull. A first reference is the recent preprint "Smooth Frechet Globalizations of Harish-Chandra Modules" by J. Bernstein and myself, downloadable at arXiv:0812.1684v1.

http://www.ms.u-tokyo.ac.jp/~toshi/index.files/springschooltokyo200903.html#kroetz

**Peter Trapa**(Utah) 11:15-12:15Special unipotent representations of real reductive groups

[ Abstract ]

These lectures are aimed at beginning graduate students interested in the representation theory of real Lie groups. A familiarity with the theory of compact Lie groups and the basics of Harish-Chandra modules will be assumed. The goal of the lecture series is to give an exposition (with many examples) of the algebraic and geometric theory of special unipotent representations. These representations are of considerable interest; in particular, they are predicted to be the building blocks of all representation which can contribute to spaces of automorphic forms. They admit many beautiful characterizations, but their construction and unitarizability still remain mysterious.

The following topics are planned:

1.Algebraic definition of special unipotent representations and examples.

2.Localization and duality for Harish-Chandra modules.

3. Geometric definition of special unipotent representations.

[ Reference URL ]These lectures are aimed at beginning graduate students interested in the representation theory of real Lie groups. A familiarity with the theory of compact Lie groups and the basics of Harish-Chandra modules will be assumed. The goal of the lecture series is to give an exposition (with many examples) of the algebraic and geometric theory of special unipotent representations. These representations are of considerable interest; in particular, they are predicted to be the building blocks of all representation which can contribute to spaces of automorphic forms. They admit many beautiful characterizations, but their construction and unitarizability still remain mysterious.

The following topics are planned:

1.Algebraic definition of special unipotent representations and examples.

2.Localization and duality for Harish-Chandra modules.

3. Geometric definition of special unipotent representations.

http://www.ms.u-tokyo.ac.jp/~toshi/index.files/springschooltokyo200903.html#trapa

**Roger Zierau**(Oklahoma State University) 13:30-14:30Dirac Cohomology

[ Abstract ]

Dirac operators have played an important role in representation theory. An early example is the construction of discrete series representations as spaces of L2 harmonic spinors on symmetric spaces G/K. More recently a very natural Dirac operator has been discovered by Kostant; it is referred to as the cubic Dirac operator. There are algebraic and geometric versions. Suppose G/H is a reductive homogeneous space and $\\mathfrak g = \\mathfrak h + \\mathfrak q$. Let S\\mathfrak q be the restriction of the spin representation of SO(\\mathfrak q) to H ⊂ SO(\\mathfrak q). The algebraic cubic Dirac operator is an H-homomorphism \\mathcal D: V \\otimes S\\mathfrak q → V \\otimes S\\mathfrak q, where V is an $\\mathfrak g$-module. The geometric geometric version is a differential operator acting on smooth sections of vector bundles of spinors on G/H. The algebraic cubic Dirac operator leads to a notion of Dirac cohomology, generalizing $\\mathfrak n$-cohomology. The lectures will roughly contain the following.

1.Construction of the spin representations of \\widetilde{SO}(n).

2.The algebraic cubic Dirac operator \\mathcal D: V \\otimes S\\mathfrak q → V \\otimes S\\mathfrak q will be defined and some properties, including a formula for the square, will be given.

3. Of special interest is the case when H=K, a maximal compact subgroup of G and V is a unitarizable $(\\mathfrak g,K)$-module. This case will be discussed.

4.The Dirac cohomology of a finite dimensional representation will be computed. We will see how this is related to $\\mathfrak n$-cohomology of V.

5. The relationship between the algebraic and geometric cubic Dirac operators will be described. A couple of open questions will then be discussed.

The lectures will be fairly elementary.

[ Reference URL ]Dirac operators have played an important role in representation theory. An early example is the construction of discrete series representations as spaces of L2 harmonic spinors on symmetric spaces G/K. More recently a very natural Dirac operator has been discovered by Kostant; it is referred to as the cubic Dirac operator. There are algebraic and geometric versions. Suppose G/H is a reductive homogeneous space and $\\mathfrak g = \\mathfrak h + \\mathfrak q$. Let S\\mathfrak q be the restriction of the spin representation of SO(\\mathfrak q) to H ⊂ SO(\\mathfrak q). The algebraic cubic Dirac operator is an H-homomorphism \\mathcal D: V \\otimes S\\mathfrak q → V \\otimes S\\mathfrak q, where V is an $\\mathfrak g$-module. The geometric geometric version is a differential operator acting on smooth sections of vector bundles of spinors on G/H. The algebraic cubic Dirac operator leads to a notion of Dirac cohomology, generalizing $\\mathfrak n$-cohomology. The lectures will roughly contain the following.

1.Construction of the spin representations of \\widetilde{SO}(n).

2.The algebraic cubic Dirac operator \\mathcal D: V \\otimes S\\mathfrak q → V \\otimes S\\mathfrak q will be defined and some properties, including a formula for the square, will be given.

3. Of special interest is the case when H=K, a maximal compact subgroup of G and V is a unitarizable $(\\mathfrak g,K)$-module. This case will be discussed.

4.The Dirac cohomology of a finite dimensional representation will be computed. We will see how this is related to $\\mathfrak n$-cohomology of V.

5. The relationship between the algebraic and geometric cubic Dirac operators will be described. A couple of open questions will then be discussed.

The lectures will be fairly elementary.

http://www.ms.u-tokyo.ac.jp/~toshi/index.files/springschooltokyo200903.html#zierau

**Salah Mehdi**(Metz University) 15:20-16:20Enright-Varadarajan modules and harmonic spinors

[ Abstract ]

The aim of these lectures is twofold. First we would like to describe the construction of the Enright-Varadarajan modules which provide a nice algebraic characterization of discrete series representations. This construction uses several important tools of representations theory.

Then we shall use the Enright-Varadarajan modules to define a product for harmonic spinors on homogeneous spaces.

[ Reference URL ]The aim of these lectures is twofold. First we would like to describe the construction of the Enright-Varadarajan modules which provide a nice algebraic characterization of discrete series representations. This construction uses several important tools of representations theory.

Then we shall use the Enright-Varadarajan modules to define a product for harmonic spinors on homogeneous spaces.

http://www.ms.u-tokyo.ac.jp/~toshi/index.files/springschooltokyo200903.html#mehdi

### 2009/03/14

#### GCOE lecture series

09:00-14:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Dirac cohomology

Enright-Varadarajan modules and harmonic spinors

Harish-Chandra modules

Special unipotent representations of real reductive groups

**Roger Zierau**(Oklahoma State University) 09:00-10:00Dirac cohomology

**Salah Mehdi**(Metz University) 10:15-11:15Enright-Varadarajan modules and harmonic spinors

**Bernhard Krötz**(Max Planck Institute) 11:45-12:45Harish-Chandra modules

**Peter Trapa**(Utah University) 13:00-14:00Special unipotent representations of real reductive groups

### 2009/03/13

#### GCOE lecture series

09:30-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Enright-Varadarajan modules and harmonic spinors

Special unipotent representations of real reductive groups

Harish-Chandra modules

Dirac Cohomology

**Salah Mehdi**(Metz) 09:30-10:30Enright-Varadarajan modules and harmonic spinors

[ Abstract ]

The aim of these lectures is twofold. First we would like to describe the construction of the Enright-Varadarajan modules which provide a nice algebraic characterization of discrete series representations. This construction uses several important tools of representations theory. Then we shall use the Enright-Varadarajan modules to define a product for harmonic spinors on homogeneous spaces.

The aim of these lectures is twofold. First we would like to describe the construction of the Enright-Varadarajan modules which provide a nice algebraic characterization of discrete series representations. This construction uses several important tools of representations theory. Then we shall use the Enright-Varadarajan modules to define a product for harmonic spinors on homogeneous spaces.

**Peter Trapa**(Utah) 11:00-12:00Special unipotent representations of real reductive groups

**Bernhard Krötz**

(Max Planck Institute) 13:30-14:30Harish-Chandra modules

**Roger Zierau**(Oklahoma State University) 15:00-16:00Dirac Cohomology

### 2009/03/12

#### Colloquium

15:00-17:30 Room #050 (Graduate School of Math. Sci. Bldg.)

数値解析:得られた成果と残された課題

正標数の世界に40年

**菊地文雄**(東京大学大学院数理科学研究科) 15:00-16:00数値解析:得られた成果と残された課題

[ Abstract ]

有限要素法を中心とする偏微分方程式の数値計算と数値解析に従事して長い歳月を経た。その間に偏微分方程式としては、Poisson方程式、弾性論のCauchy-Navierの方程式、非圧縮流体のStokes方程式、平板の曲げに対する重調和方程式やReissner-Mindlinの方程式、電磁気学のMaxwell方程式、プラズマ平衡のGrad-Shafranov方程式などを扱ってきたが、得られた成果もかなりある反面、残された課題も多いと思う。定年退職にあたり、少々整理と総括をしておきたい。

有限要素法を中心とする偏微分方程式の数値計算と数値解析に従事して長い歳月を経た。その間に偏微分方程式としては、Poisson方程式、弾性論のCauchy-Navierの方程式、非圧縮流体のStokes方程式、平板の曲げに対する重調和方程式やReissner-Mindlinの方程式、電磁気学のMaxwell方程式、プラズマ平衡のGrad-Shafranov方程式などを扱ってきたが、得られた成果もかなりある反面、残された課題も多いと思う。定年退職にあたり、少々整理と総括をしておきたい。

**桂 利行**(東京大学大学院数理科学研究科) 16:30-17:30正標数の世界に40年

[ Abstract ]

正標数における代数幾何学には、標数0の場合とは異なる特有の現象がある。1950年代には、これらは病理的現象として捉えられ、研究している人の数も少なかった。現在では、特有の現象を扱うための手段がかなり整備され、正標数の様々な対象に対して興味ある現象が解析されている。代数多様体の単有理性、野性的ファイバーの問題、正標数特有のサイクルの構造等、これまで正標数の世界で行ってきた研究を中心に思い出を交えてお話ししたい。

正標数における代数幾何学には、標数0の場合とは異なる特有の現象がある。1950年代には、これらは病理的現象として捉えられ、研究している人の数も少なかった。現在では、特有の現象を扱うための手段がかなり整備され、正標数の様々な対象に対して興味ある現象が解析されている。代数多様体の単有理性、野性的ファイバーの問題、正標数特有のサイクルの構造等、これまで正標数の世界で行ってきた研究を中心に思い出を交えてお話ししたい。

#### GCOE lecture series

09:30-14:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Dirac Cohomology

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

Harish-Chandra modules

Special unipotent representations of real reductive groups

**Roger Zierau**(Oklahoma State University) 09:30-10:30Dirac Cohomology

[ Abstract ]

Dirac operators have played an important role in representation theory. An early example is the construction of discrete series representations as spaces of L2 harmonic spinors on symmetric spaces G/K. More recently a very natural Dirac operator has been discovered by Kostant; it is referred to as the cubic Dirac operator. There are algebraic and geometric versions. Suppose G/H is a reductive homogeneous space and $\\mathfrak g = \\mathfrak h + \\mathfrak q$. Let S\\mathfrak q be the restriction of the spin representation of SO(\\mathfrak q) to H ⊂ SO(\\mathfrak q). The algebraic cubic Dirac operator is an H-homomorphism \\mathcal D: V \\otimes S\\mathfrak q → V \\otimes S\\mathfrak q, where V is an $\\mathfrak g$-module. The geometric geometric version is a differential operator acting on smooth sections of vector bundles of spinors on G/H. The algebraic cubic Dirac operator leads to a notion of Dirac cohomology, generalizing $\\mathfrak n$-cohomology. The lectures will roughly contain the following.

1.Construction of the spin representations of \\widetilde{SO}(n).

2.The algebraic cubic Dirac operator \\mathcal D: V \\otimes S\\mathfrak q → V \\otimes S\\mathfrak q will be defined and some properties, including a formula for the square, will be given.

3. Of special interest is the case when H=K, a maximal compact subgroup of G and V is a unitarizable $(\\mathfrak g,K)$-module. This case will be discussed.

4.The Dirac cohomology of a finite dimensional representation will be computed. We will see how this is related to $\\mathfrak n$-cohomology of V.

5. The relationship between the algebraic and geometric cubic Dirac operators will be described. A couple of open questions will then be discussed.

The lectures will be fairly elementary.

[ Reference URL ]Dirac operators have played an important role in representation theory. An early example is the construction of discrete series representations as spaces of L2 harmonic spinors on symmetric spaces G/K. More recently a very natural Dirac operator has been discovered by Kostant; it is referred to as the cubic Dirac operator. There are algebraic and geometric versions. Suppose G/H is a reductive homogeneous space and $\\mathfrak g = \\mathfrak h + \\mathfrak q$. Let S\\mathfrak q be the restriction of the spin representation of SO(\\mathfrak q) to H ⊂ SO(\\mathfrak q). The algebraic cubic Dirac operator is an H-homomorphism \\mathcal D: V \\otimes S\\mathfrak q → V \\otimes S\\mathfrak q, where V is an $\\mathfrak g$-module. The geometric geometric version is a differential operator acting on smooth sections of vector bundles of spinors on G/H. The algebraic cubic Dirac operator leads to a notion of Dirac cohomology, generalizing $\\mathfrak n$-cohomology. The lectures will roughly contain the following.

1.Construction of the spin representations of \\widetilde{SO}(n).

2.The algebraic cubic Dirac operator \\mathcal D: V \\otimes S\\mathfrak q → V \\otimes S\\mathfrak q will be defined and some properties, including a formula for the square, will be given.

3. Of special interest is the case when H=K, a maximal compact subgroup of G and V is a unitarizable $(\\mathfrak g,K)$-module. This case will be discussed.

4.The Dirac cohomology of a finite dimensional representation will be computed. We will see how this is related to $\\mathfrak n$-cohomology of V.

5. The relationship between the algebraic and geometric cubic Dirac operators will be described. A couple of open questions will then be discussed.

The lectures will be fairly elementary.

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

**Bernhard Krötz**(Max Planck) 11:00-12:00Harish-Chandra modules

[ Abstract ]

We plan to give a course on the various types of topological globalizations of Harish-Chandra modules. It is intended to cover the following topics:

1.Topological representation theory on various types of locally convex vector spaces.

2.Basic algebraic theory of Harish-Chandra modules

3. Unique globalization versus lower bounds for matrix coefficients

4. Dirac type sequences for representations

5. Deformation theory of Harish-Chandra modules

The new material presented was obtained in collaboration with Joseph Bernstein and Henrik Schlichtkrull. A first reference is the recent preprint "Smooth Frechet Globalizations of Harish-Chandra Modules" by J. Bernstein and myself, downloadable at arXiv:0812.1684v1.

We plan to give a course on the various types of topological globalizations of Harish-Chandra modules. It is intended to cover the following topics:

1.Topological representation theory on various types of locally convex vector spaces.

2.Basic algebraic theory of Harish-Chandra modules

3. Unique globalization versus lower bounds for matrix coefficients

4. Dirac type sequences for representations

5. Deformation theory of Harish-Chandra modules

The new material presented was obtained in collaboration with Joseph Bernstein and Henrik Schlichtkrull. A first reference is the recent preprint "Smooth Frechet Globalizations of Harish-Chandra Modules" by J. Bernstein and myself, downloadable at arXiv:0812.1684v1.

**Peter Trapa**(Utah大学) 13:30-14:30Special unipotent representations of real reductive groups

[ Abstract ]

These lectures are aimed at beginning graduate students interested in the representation theory of real Lie groups. A familiarity with the theory of compact Lie groups and the basics of Harish-Chandra modules will be assumed. The goal of the lecture series is to give an exposition (with many examples) of the algebraic and geometric theory of special unipotent representations. These representations are of considerable interest; in particular, they are predicted to be the building blocks of all representation which can contribute to spaces of automorphic forms. They admit many beautiful characterizations, but their construction and unitarizability still remain mysterious.

The following topics are planned:

1.Algebraic definition of special unipotent representations and examples.

2.Localization and duality for Harish-Chandra modules.

3. Geometric definition of special unipotent representations.

These lectures are aimed at beginning graduate students interested in the representation theory of real Lie groups. A familiarity with the theory of compact Lie groups and the basics of Harish-Chandra modules will be assumed. The goal of the lecture series is to give an exposition (with many examples) of the algebraic and geometric theory of special unipotent representations. These representations are of considerable interest; in particular, they are predicted to be the building blocks of all representation which can contribute to spaces of automorphic forms. They admit many beautiful characterizations, but their construction and unitarizability still remain mysterious.

The following topics are planned:

1.Algebraic definition of special unipotent representations and examples.

2.Localization and duality for Harish-Chandra modules.

3. Geometric definition of special unipotent representations.

### 2009/03/05

#### Tuesday Seminar on Topology

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

Extending surface automorphisms over 4-space

**Shicheng Wang**(Peking University)Extending surface automorphisms over 4-space

[ Abstract ]

Let $e: M^p\\to R^{p+2}$ be a co-dimensional 2 smooth embedding

from a closed orientable manifold to the Euclidean space and $E_e$ be the subgroup of ${\\cal M}_M$, the mapping class group

of $M$, whose elements extend over $R^{p+2}$ as self-diffeomorphisms. Then there is a spin structure

on $M$ derived from the embedding which is preserved by each $\\tau \\in E_e$.

Some applications: (1) the index $[{\\cal M}_{F_g}:E_e]\\geq 2^{2g-1}+2^{g-1}$ for any embedding $e:F_g\\to R^4$, where $F_g$

is the surface of genus $g$. (2) $[{\\cal M}_{T^p}:E_e]\\geq 2^p-1$ for any unknotted embedding

$e:T^p\\to R^{p+2}$, where $T^p$ is the $p$-dimensional torus. Those two lower bounds are known to be sharp.

This is a joint work of Ding-Liu-Wang-Yao.

Let $e: M^p\\to R^{p+2}$ be a co-dimensional 2 smooth embedding

from a closed orientable manifold to the Euclidean space and $E_e$ be the subgroup of ${\\cal M}_M$, the mapping class group

of $M$, whose elements extend over $R^{p+2}$ as self-diffeomorphisms. Then there is a spin structure

on $M$ derived from the embedding which is preserved by each $\\tau \\in E_e$.

Some applications: (1) the index $[{\\cal M}_{F_g}:E_e]\\geq 2^{2g-1}+2^{g-1}$ for any embedding $e:F_g\\to R^4$, where $F_g$

is the surface of genus $g$. (2) $[{\\cal M}_{T^p}:E_e]\\geq 2^p-1$ for any unknotted embedding

$e:T^p\\to R^{p+2}$, where $T^p$ is the $p$-dimensional torus. Those two lower bounds are known to be sharp.

This is a joint work of Ding-Liu-Wang-Yao.

#### GCOE Seminars

10:15-11:15 Room #270 (Graduate School of Math. Sci. Bldg.)

Carleman type estimates with two large parameters and applications to elasticity theory woth residual stress

**V. Isakov**(Wichita State Univ.)Carleman type estimates with two large parameters and applications to elasticity theory woth residual stress

[ Abstract ]

We give Carleman estimates with two large parameters for general second order partial differential operators with real-valued coefficients.

We outline proofs based on differential quadratic forms and Fourier analysis. As an application, we give Carleman estimates for (anisotropic)elasticity system with residual stress and discuss applications to control theory and inverse problems.

We give Carleman estimates with two large parameters for general second order partial differential operators with real-valued coefficients.

We outline proofs based on differential quadratic forms and Fourier analysis. As an application, we give Carleman estimates for (anisotropic)elasticity system with residual stress and discuss applications to control theory and inverse problems.

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