## Seminar information archive

Seminar information archive ～10/09｜Today's seminar 10/10 | Future seminars 10/11～

### 2007/12/17

#### Seminar on Geometric Complex Analysis

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

種数3の曲線とあるCalabi-Yau threefoldの代数的対応(松本圭司氏との共同研究)

**寺杣友秀**(東京大学)種数3の曲線とあるCalabi-Yau threefoldの代数的対応(松本圭司氏との共同研究)

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Analytic torsion for Calabi-Yau threefolds

**Ken-Ichi Yoshikawa**(The University of Tokyo)Analytic torsion for Calabi-Yau threefolds

[ Abstract ]

In 1994, Bershadky-Cecotti-Ooguri-Vafa conjectured that analytic torsion

gives rise to a function on the moduli space of Calabi-Yau threefolds and

that it coincides with the quantity $F_{1}$ in string theory.

Since the holomorphic part of $F_{1}$ is conjecturally the generating function

of the counting problem of elliptic curves in the mirror Calabi-Yau threefold,

this implies the conjectural equivalence of analytic torsion and the counting

problem of elliptic curves for Calabi-Yau threefolds through mirror symmetry.

After Bershadsky-Cecotti-Ooguri-Vafa, we introduced an invariant of

Calabi-Yau threefolds, which we obtained using analytic torsion and

a Bott-Chern secondary class. In this talk, we will talk about the construction

and some explicit formulae of this analytic torsion invariant.

Some part of this talk is based on the joint work with H. Fang and Z. Lu.

In 1994, Bershadky-Cecotti-Ooguri-Vafa conjectured that analytic torsion

gives rise to a function on the moduli space of Calabi-Yau threefolds and

that it coincides with the quantity $F_{1}$ in string theory.

Since the holomorphic part of $F_{1}$ is conjecturally the generating function

of the counting problem of elliptic curves in the mirror Calabi-Yau threefold,

this implies the conjectural equivalence of analytic torsion and the counting

problem of elliptic curves for Calabi-Yau threefolds through mirror symmetry.

After Bershadsky-Cecotti-Ooguri-Vafa, we introduced an invariant of

Calabi-Yau threefolds, which we obtained using analytic torsion and

a Bott-Chern secondary class. In this talk, we will talk about the construction

and some explicit formulae of this analytic torsion invariant.

Some part of this talk is based on the joint work with H. Fang and Z. Lu.

### 2007/12/13

#### Lectures

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

**Mikael Pichot**(東大数理)Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

#### Applied Analysis

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

Singular limit of a competition-diffusion system

**Danielle Hilhorst**(CNRS / パリ第11大学)Singular limit of a competition-diffusion system

[ Abstract ]

We revisit a competition-diffusion system for the densities of biological populations, and (i) prove the strong convergence in L^2 of the densities of the biological species (joint work with Iida, Mimura and Ninomiya); (ii) derive the singular limit of some reaction terms as the reaction coefficient tends to infinity (joint work with Martin and Mimura).

We revisit a competition-diffusion system for the densities of biological populations, and (i) prove the strong convergence in L^2 of the densities of the biological species (joint work with Iida, Mimura and Ninomiya); (ii) derive the singular limit of some reaction terms as the reaction coefficient tends to infinity (joint work with Martin and Mimura).

### 2007/12/12

#### Seminar on Probability and Statistics

15:20-16:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Inference problems for the telegraph process observed at discrete times

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/12.html

**Stefano IACUS**(Department of Economics, Business and Statistics, University of Milan)Inference problems for the telegraph process observed at discrete times

[ Abstract ]

The telegraph process {X(t), t>0}, has been introduced (see

Goldstein, 1951) as an alternative model to the Brownian motion B(t).

This process describes a motion of a particle on the real line which

alternates its velocity, at Poissonian times, from +v to -v. The

density of the distribution of the position of the particle at time t

solves the hyperbolic differential equation called telegraph equation

and hence the name of the process.

Contrary to B(t) the process X(t) has finite variation and

continuous and differentiable paths. At the same time it is

mathematically challenging to handle. Several variation of this

process have been recently introduced in the context of Finance.

In this talk we will discuss pseudo-likelihood and moment type

estimators of the intensity of the Poisson process, from discrete

time observations of standard telegraph process X(t). We also

discuss the problem of change point estimation for the intensity of

the underlying Poisson process and show the performance of this

estimator on real data.

[ Reference URL ]The telegraph process {X(t), t>0}, has been introduced (see

Goldstein, 1951) as an alternative model to the Brownian motion B(t).

This process describes a motion of a particle on the real line which

alternates its velocity, at Poissonian times, from +v to -v. The

density of the distribution of the position of the particle at time t

solves the hyperbolic differential equation called telegraph equation

and hence the name of the process.

Contrary to B(t) the process X(t) has finite variation and

continuous and differentiable paths. At the same time it is

mathematically challenging to handle. Several variation of this

process have been recently introduced in the context of Finance.

In this talk we will discuss pseudo-likelihood and moment type

estimators of the intensity of the Poisson process, from discrete

time observations of standard telegraph process X(t). We also

discuss the problem of change point estimation for the intensity of

the underlying Poisson process and show the performance of this

estimator on real data.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/12.html

### 2007/12/11

#### Tuesday Seminar on Topology

16:30-18:40 Room #056 (Graduate School of Math. Sci. Bldg.)

A Singular Version of The Poincar\\'e-Hopf Theorem

Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology

**Xavier G\'omez-Mont**(CIMAT, Mexico) 16:30-17:30A Singular Version of The Poincar\\'e-Hopf Theorem

[ Abstract ]

The Poincar\\'e-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.

A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains “hidden” topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.

One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincar\\'e-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.

I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincar\\'e-Hopf Theorem.

I will also explain how some of these ideas can be extended to complete intersections.

The Poincar\\'e-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.

A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains “hidden” topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.

One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincar\\'e-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.

I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincar\\'e-Hopf Theorem.

I will also explain how some of these ideas can be extended to complete intersections.

**Miguel A. Xicotencatl**(CINVESTAV, Mexico) 17:40-18:40Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology

[ Abstract ]

(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)

At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.

(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)

At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.

#### Algebraic Geometry Seminar

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

Homological methods in non-commutative geometry, part 7

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homological methods in non-commutative geometry, part 7

#### Lie Groups and Representation Theory

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

Characterization of some smooth vectors for irreducible representations of exponential solvable Lie groups

[ Reference URL ]

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

**井上順子**(鳥取大学)Characterization of some smooth vectors for irreducible representations of exponential solvable Lie groups

[ Reference URL ]

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

### 2007/12/10

#### Seminar on Geometric Complex Analysis

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

岩澤予想の幾何学的類似の量子化(予想される結果)

**杉山健一**(千葉大学)岩澤予想の幾何学的類似の量子化(予想される結果)

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Deligne conjecture and the Drinfeld double.

**Dmitry Kaledin**(Steklov Institute and The University of Tokyo)Deligne conjecture and the Drinfeld double.

[ Abstract ]

Deligne conjecture describes the structure which exists on

the Hochschild cohomology $HH(A)$ of an associative algebra

$A$. Several proofs exists, but they all combinatorial to a certain

extent. I will present another proof which is more categorical in

nature (in particular, the input data are not the algebra $A$, but

rather, the tensor category of $A$-bimodules). Combinatorics is

still there, but now it looks more natural -- in particular, the

action of the Gerstenhaber operad, which is know to consist of

homology of pure braid groups, is induced by the action of the braid

groups themselves on the so-called "Drinfeld double" of the category

$A$-bimod.

If time permits, I will also discuss what additional structures

appear in the Calabi-Yau case, and what one needs to impose to

insure Hodge-to-de Rham degeneration.

Deligne conjecture describes the structure which exists on

the Hochschild cohomology $HH(A)$ of an associative algebra

$A$. Several proofs exists, but they all combinatorial to a certain

extent. I will present another proof which is more categorical in

nature (in particular, the input data are not the algebra $A$, but

rather, the tensor category of $A$-bimodules). Combinatorics is

still there, but now it looks more natural -- in particular, the

action of the Gerstenhaber operad, which is know to consist of

homology of pure braid groups, is induced by the action of the braid

groups themselves on the so-called "Drinfeld double" of the category

$A$-bimod.

If time permits, I will also discuss what additional structures

appear in the Calabi-Yau case, and what one needs to impose to

insure Hodge-to-de Rham degeneration.

### 2007/12/06

#### Lectures

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

**Mikael Pichot**(東大数理)Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

#### Applied Analysis

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

藤田型方程式における時間大域解の挙動について

**柳田 英二**(東北大学大学院理学研究科)藤田型方程式における時間大域解の挙動について

[ Abstract ]

この講演では,藤田型の半線形放物型偏微分方程式に関する M. Fila, J. King, P. Polacik, M. Winkler らとの共同研究による成果についてその概要を紹介する.全空間上の藤田型方程式については,これまで様々な挙動を示す時間大域解の存在が示されている.そこで大域解の時間的挙動と初期値の空間的挙動の関係を詳細に調べることにより,大域解をいくつかに分類し,その挙動がそれぞれ異なるメカニズムに支配されていることを明らかにする.時間が許せば,最近の進展や関連する話題についても触れる予定である.

この講演では,藤田型の半線形放物型偏微分方程式に関する M. Fila, J. King, P. Polacik, M. Winkler らとの共同研究による成果についてその概要を紹介する.全空間上の藤田型方程式については,これまで様々な挙動を示す時間大域解の存在が示されている.そこで大域解の時間的挙動と初期値の空間的挙動の関係を詳細に調べることにより,大域解をいくつかに分類し,その挙動がそれぞれ異なるメカニズムに支配されていることを明らかにする.時間が許せば,最近の進展や関連する話題についても触れる予定である.

### 2007/12/05

#### Number Theory Seminar

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

Classification of two dimensional trianguline representations of p-adic fields

**中村健太郎**(東京大学大学院数理科学研究科)Classification of two dimensional trianguline representations of p-adic fields

[ Abstract ]

Trianguline representation is a class of p-adic Galois representations of p-adic fields. This was defined by P.Colmez by using ($\\varphi, \\Gamma$)-modules over Robba ring. In his study of p-adic local Langlands correspondence of GL_2(Q_p), he completely classified two dimensional trianguline representations of Q_p. On the other hand, L.Berger recently defined the category of B-pairs and established the equivalence between the category of B-pairs and the category of ($\\varphi,\\Gamma$)-modules over Robba ring. In this talk, we extend the Colmez's result by using B-pairs. We completely classify two dimensional trianguline representations of K for any finite extension of Q_p. We also talk about a relation between two dimensional trianguline representations and principal series or special series of GL_2(K).

Trianguline representation is a class of p-adic Galois representations of p-adic fields. This was defined by P.Colmez by using ($\\varphi, \\Gamma$)-modules over Robba ring. In his study of p-adic local Langlands correspondence of GL_2(Q_p), he completely classified two dimensional trianguline representations of Q_p. On the other hand, L.Berger recently defined the category of B-pairs and established the equivalence between the category of B-pairs and the category of ($\\varphi,\\Gamma$)-modules over Robba ring. In this talk, we extend the Colmez's result by using B-pairs. We completely classify two dimensional trianguline representations of K for any finite extension of Q_p. We also talk about a relation between two dimensional trianguline representations and principal series or special series of GL_2(K).

#### Seminar on Probability and Statistics

16:20-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

A Decision-Theoretic Approach to Estimation from Wishart matrices on Symmetric Cones

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/11.html

**今野 良彦**(日本女子大学理学部)A Decision-Theoretic Approach to Estimation from Wishart matrices on Symmetric Cones

[ Abstract ]

James and Stein(1961) have considered the problem of estimating the mean matrix of Wishart distributions under so-called Stein's loss function and obtained a minimax estimator with a constant risk. Later Stein(1977) has given an unbiased risk estimate for a class of orthogonally invariant estimators, from which he obtained orthogonally invariant minimax estimators which are uniformly better than the best triangular-invariant estimator in James and Stein(1961). The works mentioned above lead to the following natural question: Is it possible for any estimators to improve upon the maximum likelihood estimator for the mean matrix of the complex or quaternion Wishart distributions? This talk shows that we can obtain improved estimators for the mean matrix under these models in a unified manner. The method involves an abstract theory of finite-dimensional Euclidean simple Jordan algebra

[ Reference URL ]James and Stein(1961) have considered the problem of estimating the mean matrix of Wishart distributions under so-called Stein's loss function and obtained a minimax estimator with a constant risk. Later Stein(1977) has given an unbiased risk estimate for a class of orthogonally invariant estimators, from which he obtained orthogonally invariant minimax estimators which are uniformly better than the best triangular-invariant estimator in James and Stein(1961). The works mentioned above lead to the following natural question: Is it possible for any estimators to improve upon the maximum likelihood estimator for the mean matrix of the complex or quaternion Wishart distributions? This talk shows that we can obtain improved estimators for the mean matrix under these models in a unified manner. The method involves an abstract theory of finite-dimensional Euclidean simple Jordan algebra

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/11.html

### 2007/12/04

#### Seminar on Mathematics for various disciplines

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

Quasilinear hyperbolic equations with hysteresis

http://coe.math.sci.hokudai.ac.jp/sympo/various/index.html

Carleman estimates for second order operators with two large parameters

http://coe.math.sci.hokudai.ac.jp/sympo/various/index.html

**Pavel Krejci**(Weierstrass Institute for Applied Analysis and Stochastics) 15:00-16:00Quasilinear hyperbolic equations with hysteresis

[ Abstract ]

We consider a wave propagation problem in a rate independent elastoplastic material described by a counterclockwise convex hysteresis operator. Unlike in viscoelasticity, the speed of propagation is bounded above by the speed of the corresponding elastic waves. The smoothening dissipative effect is due to the convexity of the hysteresis branches. We present some recent results on the long time behavior of solutions under various boundary conditions, including the stability of time periodic solutions under periodic forcing.

[ Reference URL ]We consider a wave propagation problem in a rate independent elastoplastic material described by a counterclockwise convex hysteresis operator. Unlike in viscoelasticity, the speed of propagation is bounded above by the speed of the corresponding elastic waves. The smoothening dissipative effect is due to the convexity of the hysteresis branches. We present some recent results on the long time behavior of solutions under various boundary conditions, including the stability of time periodic solutions under periodic forcing.

http://coe.math.sci.hokudai.ac.jp/sympo/various/index.html

**Victor Isakov**(Wichita State University) 16:15-17:15Carleman estimates for second order operators with two large parameters

[ Abstract ]

We obtain new Carleman type estimates for general second order linear partial differential operators. These estimates hold for the weight functions under pseudoconvexity conditions relating the operator and weight function. We discuss these conditions. We give applications to uniqueness and stability of the continuation and inverse problems for elasticity system with residual stress without smallness assumptions on residual stress. This is a joint work with Nanhee Kim.

[ Reference URL ]We obtain new Carleman type estimates for general second order linear partial differential operators. These estimates hold for the weight functions under pseudoconvexity conditions relating the operator and weight function. We discuss these conditions. We give applications to uniqueness and stability of the continuation and inverse problems for elasticity system with residual stress without smallness assumptions on residual stress. This is a joint work with Nanhee Kim.

http://coe.math.sci.hokudai.ac.jp/sympo/various/index.html

#### Tuesday Seminar on Topology

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

Morse theory for abelian hyperkahler quotients

**今野 宏**(東京大学大学院数理科学研究科)Morse theory for abelian hyperkahler quotients

[ Abstract ]

In 1980's Kirwan computed Betti numbers of symplectic quotients by using Morse theory. In this talk, we develop this method to hyperkahler quotients by abelian Lie groups. In this method, many computations are much more simplified in the case of hyperkahler quotients than the case of symplectic quotients. As a result we compute not only the Betti numbers, but also the cohomology rings of abelian hyperkahler quotients.

In 1980's Kirwan computed Betti numbers of symplectic quotients by using Morse theory. In this talk, we develop this method to hyperkahler quotients by abelian Lie groups. In this method, many computations are much more simplified in the case of hyperkahler quotients than the case of symplectic quotients. As a result we compute not only the Betti numbers, but also the cohomology rings of abelian hyperkahler quotients.

### 2007/12/03

#### Seminar on Geometric Complex Analysis

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

等質有界領域の対称性条件、性質の良い有界領域実現について

**甲斐千舟**(九州大学)等質有界領域の対称性条件、性質の良い有界領域実現について

### 2007/11/29

#### Lectures

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

**Mikael Pichot**(東大数理)Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

### 2007/11/27

#### Algebraic Geometry Seminar

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

Categorical resolutions of singularities

**Alexander Kuznetsov**(Steklov Inst)Categorical resolutions of singularities

[ Abstract ]

I will give a definition of a categorical resolution of singularities and explain how such resolutions can be constructed.

I will give a definition of a categorical resolution of singularities and explain how such resolutions can be constructed.

#### Tuesday Seminar of Analysis

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

Heaviside's theory of signal transmission on submarine cables

**小松 彦三郎**(東大数理(名誉教授))Heaviside's theory of signal transmission on submarine cables

#### Tuesday Seminar on Topology

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

A quandle cocycle invariant for handlebody-links

**石井 敦**(京都大学数理解析研究所)A quandle cocycle invariant for handlebody-links

[ Abstract ]

[joint work with Masahide Iwakiri (Osaka City University)]

A handlebody-link is a disjoint union of circles and a

finite trivalent graph embedded in a closed 3-manifold.

We consider it up to isotopies and IH-moves.

Then it represents an ambient isotopy class of

handlebodies embedded in the closed 3-manifold.

In this talk, I explain how a quandle cocycle invariant

is defined for handlebody-links.

[joint work with Masahide Iwakiri (Osaka City University)]

A handlebody-link is a disjoint union of circles and a

finite trivalent graph embedded in a closed 3-manifold.

We consider it up to isotopies and IH-moves.

Then it represents an ambient isotopy class of

handlebodies embedded in the closed 3-manifold.

In this talk, I explain how a quandle cocycle invariant

is defined for handlebody-links.

### 2007/11/26

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Kontsevich quantization of Poisson manifolds and Duflo isomorphism.

**Mich\"ael Pevzner**(Universit\'e de Reims and the University of Tokyo)Kontsevich quantization of Poisson manifolds and Duflo isomorphism.

[ Abstract ]

Abstract: Since the fundamental results by Chevalley, Harish-Chandra and Dixmier one knows that the set of invariant polynomials on the dual of a Lie algebra of a particular type (solvable, simple or nilpotent) is isomorphic, as an algebra, to the center of the enveloping algebra. This fact was generalized to an arbitrary finite-dimensional real Lie algebra by M. Duflo in late 1970's. His proof was based on the Kirillov's orbits method that parametrizes infinitesimal characters of unitary irreducible representations of the corresponding Lie group in terms of co-adjoint orbits.

The Kontsevich' Formality theorem implies not only the existence of the Duflo map but shows that it is canonical. We shall describe this construction and indicate how does this construction extend to the whole Poisson cohomology of an arbitrary finite-dimensional real Lie algebra.

Abstract: Since the fundamental results by Chevalley, Harish-Chandra and Dixmier one knows that the set of invariant polynomials on the dual of a Lie algebra of a particular type (solvable, simple or nilpotent) is isomorphic, as an algebra, to the center of the enveloping algebra. This fact was generalized to an arbitrary finite-dimensional real Lie algebra by M. Duflo in late 1970's. His proof was based on the Kirillov's orbits method that parametrizes infinitesimal characters of unitary irreducible representations of the corresponding Lie group in terms of co-adjoint orbits.

The Kontsevich' Formality theorem implies not only the existence of the Duflo map but shows that it is canonical. We shall describe this construction and indicate how does this construction extend to the whole Poisson cohomology of an arbitrary finite-dimensional real Lie algebra.

#### Seminar on Geometric Complex Analysis

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

Analysis related to probability theory based on p-adic hierarchical structure

**金子宏**(東京理科大学)Analysis related to probability theory based on p-adic hierarchical structure

### 2007/11/22

#### Applied Analysis

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

Critical frequencyをもつ非線形シュレディンガー方程式のマルチピーク解

**佐藤 洋平**(早稲田大学・基幹理工学部・数学科)Critical frequencyをもつ非線形シュレディンガー方程式のマルチピーク解

[ Abstract ]

非線形シュレディンガー方程式

$$ -\\epsilon2 \\Delta u +V(x)u= u^p, u>0 \\ \\hbox{in} \\R^N,

u\\in H1(\\R^N)$$

において、$\\epsilon \\to 0$ としたときに V(x) の k個の極小点にピークが集中していくマルチピーク解 $u_\\epsilon$ について考える。

ここで、p はsuperlinear, subcriticalの条件を満たし, ポテンシャル関数 V(x) は非負の有界な関数で $\\liminf_{|x|\\to \\infty}V(x)>0$ を満たすとする。

もし V(x) の各極小点に集中するピークがあるとしたら、そのピークの形状や大きさはその極小値が正であるか、0であるかによって大きく異なることが知られている。

この講演では V(x) の各極小値が正であるか 0 であるかにかかわらず、各 k個の極小点にピークが集中するマルチピーク解 $u_\\epsilon$ を構成する。

非線形シュレディンガー方程式

$$ -\\epsilon2 \\Delta u +V(x)u= u^p, u>0 \\ \\hbox{in} \\R^N,

u\\in H1(\\R^N)$$

において、$\\epsilon \\to 0$ としたときに V(x) の k個の極小点にピークが集中していくマルチピーク解 $u_\\epsilon$ について考える。

ここで、p はsuperlinear, subcriticalの条件を満たし, ポテンシャル関数 V(x) は非負の有界な関数で $\\liminf_{|x|\\to \\infty}V(x)>0$ を満たすとする。

もし V(x) の各極小点に集中するピークがあるとしたら、そのピークの形状や大きさはその極小値が正であるか、0であるかによって大きく異なることが知られている。

この講演では V(x) の各極小値が正であるか 0 であるかにかかわらず、各 k個の極小点にピークが集中するマルチピーク解 $u_\\epsilon$ を構成する。

#### Operator Algebra Seminars

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

Spatial property of the canonical map associated to von Neumann algebras

**張欽**(東大数理)Spatial property of the canonical map associated to von Neumann algebras

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