## Seminar information archive

Seminar information archive ～05/23｜Today's seminar 05/24 | Future seminars 05/25～

#### PDE Real Analysis Seminar

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

Hokkaido University)

A SELECTION CRITERION FOR SOLUTIONS OF A

SYSTEM OF EIKONAL EQUATIONS

(ENGLISH)

**Giovanni Pisante**(Department of MathematicsHokkaido University)

A SELECTION CRITERION FOR SOLUTIONS OF A

SYSTEM OF EIKONAL EQUATIONS

(ENGLISH)

[ Abstract ]

We deal with the system of eikonal equations |ðu/ðx1|=1, |ðu/ðx2|=1 in a planar Lipschitz domain with zero boundary condition. Exploiting the classical pyramidal construction introduced by Cellina, it is easy to prove that there exist infinitely many Lipschitz solutions. Then, the natural problem that has arisen in this framework is to find a way to select and characterize a particular meaningful class of solutions.

We propose a variational method to select the class of solutions which minimize the discontinuity set of the gradient. More precisely we select an optimal weighted measure for the jump set of the second derivatives of a given solution v of the system and we prove the existence of minimizers of the corresponding variational problem.

We deal with the system of eikonal equations |ðu/ðx1|=1, |ðu/ðx2|=1 in a planar Lipschitz domain with zero boundary condition. Exploiting the classical pyramidal construction introduced by Cellina, it is easy to prove that there exist infinitely many Lipschitz solutions. Then, the natural problem that has arisen in this framework is to find a way to select and characterize a particular meaningful class of solutions.

We propose a variational method to select the class of solutions which minimize the discontinuity set of the gradient. More precisely we select an optimal weighted measure for the jump set of the second derivatives of a given solution v of the system and we prove the existence of minimizers of the corresponding variational problem.

#### Numerical Analysis Seminar

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

Existence and non-existence of global solutions for a discrete semilinear heat equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Keisuke Matsuya**(University of Tokyo)Existence and non-existence of global solutions for a discrete semilinear heat equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

#### GCOE Seminars

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

Existence and non-existence of global solutions for a discrete semilinear heat equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**松家 敬介**(東京大学大学院数理科学研究科)Existence and non-existence of global solutions for a discrete semilinear heat equation (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

### 2010/05/25

#### Lie Groups and Representation Theory

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

On endoscopy, packets, and invariants (JAPANESE)

**Kaoru Hiraga**(Kyoto University)On endoscopy, packets, and invariants (JAPANESE)

[ Abstract ]

The theory of endoscopy came out of the Langlands functoriality and the trace formula.

In this talk, I will briefly explain what the endoscopy is, and talk about packet, formal degree and Whittaker normalization of transfer.

I would like to talk about the connection between these topics and the endoscopy.

The theory of endoscopy came out of the Langlands functoriality and the trace formula.

In this talk, I will briefly explain what the endoscopy is, and talk about packet, formal degree and Whittaker normalization of transfer.

I would like to talk about the connection between these topics and the endoscopy.

### 2010/05/24

#### Algebraic Geometry Seminar

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

A counterexample of the birational Torelli problem via Fourier--Mukai transforms (JAPANESE)

**Hokuto Uehara**(Tokyo Metropolitan University)A counterexample of the birational Torelli problem via Fourier--Mukai transforms (JAPANESE)

[ Abstract ]

We study the Fourier--Mukai numbers of rational elliptic surfaces. As

its application, we give an example of a pair of minimal 3-folds $X$

with Kodaira dimensions 1, $h^1(O_X)=h^2(O_X)=0$ such that they are

mutually derived equivalent, deformation equivalent, but not

birationally equivalent. It also supplies a counterexample of the

birational Torelli problem.

We study the Fourier--Mukai numbers of rational elliptic surfaces. As

its application, we give an example of a pair of minimal 3-folds $X$

with Kodaira dimensions 1, $h^1(O_X)=h^2(O_X)=0$ such that they are

mutually derived equivalent, deformation equivalent, but not

birationally equivalent. It also supplies a counterexample of the

birational Torelli problem.

### 2010/05/19

#### Seminar on Mathematics for various disciplines

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

Mathematical sciences collaborating with clinical medicine (JAPANESE)

**Hiroshi Suito**(Okayama University)Mathematical sciences collaborating with clinical medicine (JAPANESE)

#### Seminar on Probability and Statistics

15:00-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Estimation of the variance-covariance structure for stochastic processes and applications of YUIMA (JAPANESE)

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2010/01.html

**YOSHIDA, Nakahiro**(University of Tokyo)Estimation of the variance-covariance structure for stochastic processes and applications of YUIMA (JAPANESE)

[ Abstract ]

We discuss limit theorems and asymptotic expansions in estimation of the variance-covariance structure for Ito processes. We will show some numerical examples by YUIMA, a package for statistical analysis and simulation of stochastic differential equations.

[ Reference URL ]We discuss limit theorems and asymptotic expansions in estimation of the variance-covariance structure for Ito processes. We will show some numerical examples by YUIMA, a package for statistical analysis and simulation of stochastic differential equations.

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2010/01.html

### 2010/05/18

#### Tuesday Seminar on Topology

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

On roots of Dehn twists (JAPANESE)

**Naoyuki Monden**(Osaka University)On roots of Dehn twists (JAPANESE)

[ Abstract ]

Let $t_{c}$ be the Dehn twist about a nonseparating simple closed curve

$c$ in a closed orientable surface. If a mapping class $f$ satisfies

$t_{c}=f^{n}$ in mapping class group, we call $f$ a root of $t_{c}$ of

degree $n$. In 2009, Margalit and Schleimer constructed roots of $t_{c}$.

In this talk, I will explain the data set which determine a root of

$t_{c}$ up to conjugacy. Moreover, I will explain the minimal and the

maximal degree.

Let $t_{c}$ be the Dehn twist about a nonseparating simple closed curve

$c$ in a closed orientable surface. If a mapping class $f$ satisfies

$t_{c}=f^{n}$ in mapping class group, we call $f$ a root of $t_{c}$ of

degree $n$. In 2009, Margalit and Schleimer constructed roots of $t_{c}$.

In this talk, I will explain the data set which determine a root of

$t_{c}$ up to conjugacy. Moreover, I will explain the minimal and the

maximal degree.

#### Lie Groups and Representation Theory

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

On the eigenvalues of the Laplacian on locally symmetric hyperbolic spaces (ENGLISH)

**B. Speh**(Cornel University)On the eigenvalues of the Laplacian on locally symmetric hyperbolic spaces (ENGLISH)

[ Abstract ]

A famous Theorem of Selberg says that the non-zero eigenvalues of the Laplacian acting on functions on quotients of the upper half plane H by congruence subgroups of the integral modular group, are bounded away from zero, as the congruence subgroup varies. Analogous questions on Laplacians acting on differential forms of higher degree on locally symmetric spaces (functions may be thought of as differential forms of degree zero) have geometric implications for the cohomology of the locally symmetric space.

Let $X$ be the real hyperbolic n-space and $\\Gamma \\subset $ SO(n, 1) a congruence arithmetic subgroup. Bergeron conjectured that the eigenvalues of the Laplacian acting on the differential forms on $ X / \\Gamma $ are bounded from the below by a constant independent of the congruence subgroup. In the lecture I will show how one can use representation theory to show that this conjecture is true provided that it is true in the middle degree.

This is joint work with T.N. Venkataramana

A famous Theorem of Selberg says that the non-zero eigenvalues of the Laplacian acting on functions on quotients of the upper half plane H by congruence subgroups of the integral modular group, are bounded away from zero, as the congruence subgroup varies. Analogous questions on Laplacians acting on differential forms of higher degree on locally symmetric spaces (functions may be thought of as differential forms of degree zero) have geometric implications for the cohomology of the locally symmetric space.

Let $X$ be the real hyperbolic n-space and $\\Gamma \\subset $ SO(n, 1) a congruence arithmetic subgroup. Bergeron conjectured that the eigenvalues of the Laplacian acting on the differential forms on $ X / \\Gamma $ are bounded from the below by a constant independent of the congruence subgroup. In the lecture I will show how one can use representation theory to show that this conjecture is true provided that it is true in the middle degree.

This is joint work with T.N. Venkataramana

### 2010/05/17

#### Algebraic Geometry Seminar

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

On the GIT stability of Polarized Varieties (JAPANESE)

**Yuji Odaka**(Research Institute for Mathematical Sciences)On the GIT stability of Polarized Varieties (JAPANESE)

[ Abstract ]

Background:

Original GIT-stability notion for polarized variety is

"asymptotic stability", studied by Mumford, Gieseker etc around 1970s.

Recently a version appeared, so-called "K-stability", introduced by

Tian(1997) and reformulated by Donaldson(2002), by the way of seeking

the analogue of Kobayashi-Hitchin correspondence, which gives

"differential geometric" interpretation of "stability". These two have

subtle but interesting differences in dimension higher than 1.

Contents:

(1*) Any semistable (in any sense) polarized variety should have only

"semi-log-canonical" singularities. (Partly observed around 1970s)

(2) On the other hand, we proved some stabilities, which corresponds to

"Calabi conjecture", also with admitting mild singularities.

As applications these yield

(3*) Compact moduli spaces with GIT interpretations.

(4) Many counterexamples (as orbifolds) to folklore conjecture:

"K-stability implies asymptotic stability".

(*: Some technical points are yet to be settled.

Some parts for (1)(2) are available on arXiv:0910.1794.)

Background:

Original GIT-stability notion for polarized variety is

"asymptotic stability", studied by Mumford, Gieseker etc around 1970s.

Recently a version appeared, so-called "K-stability", introduced by

Tian(1997) and reformulated by Donaldson(2002), by the way of seeking

the analogue of Kobayashi-Hitchin correspondence, which gives

"differential geometric" interpretation of "stability". These two have

subtle but interesting differences in dimension higher than 1.

Contents:

(1*) Any semistable (in any sense) polarized variety should have only

"semi-log-canonical" singularities. (Partly observed around 1970s)

(2) On the other hand, we proved some stabilities, which corresponds to

"Calabi conjecture", also with admitting mild singularities.

As applications these yield

(3*) Compact moduli spaces with GIT interpretations.

(4) Many counterexamples (as orbifolds) to folklore conjecture:

"K-stability implies asymptotic stability".

(*: Some technical points are yet to be settled.

Some parts for (1)(2) are available on arXiv:0910.1794.)

#### Seminar on Geometric Complex Analysis

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

On the norm defined on the holomorphic maps of compact Riemann surfaces (JAPANESE)

**Masaharu TANABE**(Tokyo Inst. Tech.)On the norm defined on the holomorphic maps of compact Riemann surfaces (JAPANESE)

### 2010/05/15

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Strict positivity of the central values of some Rankin-Selberg

L-functions (JAPANESE)

Calabi-Yau manifolds associated to hypergeometric sheaves and their application

Osaka Pref. Univ. (JAPANESE)

**NARITA, Hiroaki**(Kumamoto University, Fac. of Science) 13:30-14:30Strict positivity of the central values of some Rankin-Selberg

L-functions (JAPANESE)

[ Abstract ]

We consider the Arakawa lift which is an automprphic form on an inner twist of $GSp(2)$. We construct examples the case when the central values of the $L$-functions of Rankin-Selberg type with degree 8 Euler factors take positive values. ....

We consider the Arakawa lift which is an automprphic form on an inner twist of $GSp(2)$. We construct examples the case when the central values of the $L$-functions of Rankin-Selberg type with degree 8 Euler factors take positive values. ....

**YAMAUCHI, Takuya**(Osaka Pref. Univ. ) 15:00-16:00Calabi-Yau manifolds associated to hypergeometric sheaves and their application

Osaka Pref. Univ. (JAPANESE)

[ Abstract ]

Let U be the P_1 munus 3 points, and form hypergeometric sheaves on U, by iterative convolutions of certain local sysytem of rank 1 on U. We construct certain families of Calabi-Yau manifolds whose cohomology groups of middle degree are these hypergeometric sheaves. We discuss the potential-modularity of these varieties and unit root formula. This is a joint work with Michio Tsuzuki. (trans. by the organizer of the seminar)

Let U be the P_1 munus 3 points, and form hypergeometric sheaves on U, by iterative convolutions of certain local sysytem of rank 1 on U. We construct certain families of Calabi-Yau manifolds whose cohomology groups of middle degree are these hypergeometric sheaves. We discuss the potential-modularity of these varieties and unit root formula. This is a joint work with Michio Tsuzuki. (trans. by the organizer of the seminar)

### 2010/05/12

#### Lectures

15:30-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Independence of families of $\\ell$-adic representations and uniform constructibility (ENGLISH)

**Luc Illusie**(東京大学/Paris南大学)Independence of families of $\\ell$-adic representations and uniform constructibility (ENGLISH)

[ Abstract ]

Let $k$ be a number field, $\\overline{k}$ an algebraic closure of $k$, $\\Gamma_k = \\mathrm{Gal}(\\overline{k}/k)$. A family of continuous homomorphisms $\\rho_{\\ell} : \\Gamma_k \\rightarrow G_{\\ell}$, indexed by prime numbers $\\ell$, where $G_{\\ell}$ is a locally compact $\\ell$-adic Lie group, is said to be independent if $\\rho(\\Gamma_k) = \\prod \\rho_{\\ell}(\\Gamma_k)$, where $\\rho = (\\rho_{\\ell}) : \\Gamma_k \\rightarrow \\prod G_{\\ell}$. Serre gave a criterion for such a family to become independent after a finite extension of $k$. We will explain Serre's criterion and show that it applies to families coming from the $\\ell$-adic cohomology (or cohomology with compact support) of schemes separated and of finite type over $k$. This application uses a variant of Deligne's generic constructibility theorem with uniformity in $\\ell$.

Let $k$ be a number field, $\\overline{k}$ an algebraic closure of $k$, $\\Gamma_k = \\mathrm{Gal}(\\overline{k}/k)$. A family of continuous homomorphisms $\\rho_{\\ell} : \\Gamma_k \\rightarrow G_{\\ell}$, indexed by prime numbers $\\ell$, where $G_{\\ell}$ is a locally compact $\\ell$-adic Lie group, is said to be independent if $\\rho(\\Gamma_k) = \\prod \\rho_{\\ell}(\\Gamma_k)$, where $\\rho = (\\rho_{\\ell}) : \\Gamma_k \\rightarrow \\prod G_{\\ell}$. Serre gave a criterion for such a family to become independent after a finite extension of $k$. We will explain Serre's criterion and show that it applies to families coming from the $\\ell$-adic cohomology (or cohomology with compact support) of schemes separated and of finite type over $k$. This application uses a variant of Deligne's generic constructibility theorem with uniformity in $\\ell$.

#### Number Theory Seminar

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Differences between

Galois representations in outer-automorphisms

of the fundamental groups and those in automorphisms, implied by

topology of moduli spaces (ENGLISH)

**Makoto Matsumoto**(University of Tokyo)Differences between

Galois representations in outer-automorphisms

of the fundamental groups and those in automorphisms, implied by

topology of moduli spaces (ENGLISH)

[ Abstract ]

Fix a prime l. Let C be a proper smooth geometrically connected curve over a number ﬁeld K, and x be its closed point. Let Π denote the pro-l completion of the geometric fundamental group of C with geometric base point over x. We have two non-abelian Galois representations:

ρA : Galk(x) → Aut(Π),ρO : GalK → Out(Π).

Our question is: in the natural inclusion Ker(ρA) ⊂ Ker(ρO) ∩ Galk(x), whether the equality holds or not. Theorem: Assume that g ≥ 3, l divides 2g -2. Then, there are inﬁnitely many pairs (C, K) with the following property. If l does not divide the extension degree [k(x): K], then Ker(ρA) = (Ker(ρO) ∩ Galk(x)) holds.

This is in contrast to the case of the projective line minus three points and its canonical tangential base points, where the equality holds (a result of Deligne and Ihara).

There are two ingredients in the proof: (1) Galois representations often contain the image of the geometric monodromy (namely, the mapping class group) [M-Tamagawa 2000] (2) A topological result [S. Morita 98] [Hain-Reed 2000] on the cohomological obstruction of lifting the outer action of the mapping class group to automorphisms.

(This lecture is held as `Arithmetic Geometry Seminar Tokyo-Paris' and it is transmitted to IHES by the internet.)

Fix a prime l. Let C be a proper smooth geometrically connected curve over a number ﬁeld K, and x be its closed point. Let Π denote the pro-l completion of the geometric fundamental group of C with geometric base point over x. We have two non-abelian Galois representations:

ρA : Galk(x) → Aut(Π),ρO : GalK → Out(Π).

Our question is: in the natural inclusion Ker(ρA) ⊂ Ker(ρO) ∩ Galk(x), whether the equality holds or not. Theorem: Assume that g ≥ 3, l divides 2g -2. Then, there are inﬁnitely many pairs (C, K) with the following property. If l does not divide the extension degree [k(x): K], then Ker(ρA) = (Ker(ρO) ∩ Galk(x)) holds.

This is in contrast to the case of the projective line minus three points and its canonical tangential base points, where the equality holds (a result of Deligne and Ihara).

There are two ingredients in the proof: (1) Galois representations often contain the image of the geometric monodromy (namely, the mapping class group) [M-Tamagawa 2000] (2) A topological result [S. Morita 98] [Hain-Reed 2000] on the cohomological obstruction of lifting the outer action of the mapping class group to automorphisms.

(This lecture is held as `Arithmetic Geometry Seminar Tokyo-Paris' and it is transmitted to IHES by the internet.)

#### PDE Real Analysis Seminar

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

The University of Tokyo)

Exact controllability for incompressible fluids (ENGLISH)

**Jean-Pierre Puel**(Graduate School of Mathematical SciencesThe University of Tokyo)

Exact controllability for incompressible fluids (ENGLISH)

[ Abstract ]

After a short presentation of J.-M. Coron's results for Euler equation, we will give the good notions of controllability for Navier-Stokes equations, namely the exact controllability to trajectories.

We will outline the strategy for obtaining local results, based on a fixed point argument following the study of null controllability for the linearized problem. This is equivalent to an observability inequality for the adjoint system, which requires a global Carleman estimate for linearized Navier-Stokes equations. We will explain this estimate and the different steps for obtaining it along the lines of the articles by E.Fernadez-Cara, S.Guerrero, O.Imanuvilov and J.-P.Puel (JMPA, 2004) and M.Gonzalez-Burgos, S.Guerrero and J.-P.Puel (CPAA, 2009).

We will end up with some important open problems.

After a short presentation of J.-M. Coron's results for Euler equation, we will give the good notions of controllability for Navier-Stokes equations, namely the exact controllability to trajectories.

We will outline the strategy for obtaining local results, based on a fixed point argument following the study of null controllability for the linearized problem. This is equivalent to an observability inequality for the adjoint system, which requires a global Carleman estimate for linearized Navier-Stokes equations. We will explain this estimate and the different steps for obtaining it along the lines of the articles by E.Fernadez-Cara, S.Guerrero, O.Imanuvilov and J.-P.Puel (JMPA, 2004) and M.Gonzalez-Burgos, S.Guerrero and J.-P.Puel (CPAA, 2009).

We will end up with some important open problems.

#### Numerical Analysis Seminar

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

Accurate factorizations of ill-conditioned matrices and applications (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Takeshi Ogita**(Tokyo Woman's Christian University)Accurate factorizations of ill-conditioned matrices and applications (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

#### GCOE Seminars

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

悪条件行列の高精度な分解法とその応用 (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**荻田 武史**(東京女子大学現代教養学部)悪条件行列の高精度な分解法とその応用 (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

### 2010/05/11

#### Tuesday Seminar on Topology

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

The logarithms of Dehn twists (JAPANESE)

**Nariya Kawazumi**(The University of Tokyo)The logarithms of Dehn twists (JAPANESE)

[ Abstract ]

We establish an explicit formula for the action of (non-separating and

separating) Dehn twists on the complete group ring of the fundamental group of a

surface. It generalizes the classical transvection formula on the first homology.

The proof is involved with a homological interpretation of the Goldman

Lie algebra. This talk is based on a jointwork with Yusuke Kuno (Hiroshima U./JSPS).

We establish an explicit formula for the action of (non-separating and

separating) Dehn twists on the complete group ring of the fundamental group of a

surface. It generalizes the classical transvection formula on the first homology.

The proof is involved with a homological interpretation of the Goldman

Lie algebra. This talk is based on a jointwork with Yusuke Kuno (Hiroshima U./JSPS).

#### Lie Groups and Representation Theory

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

On a finite $W$-algebra module structure on the space of

continuous Whittaker vectors for an irreducible Harish-Chandra module (ENGLISH)

**Hisayosi Matumoto**(the University of Tokyo)On a finite $W$-algebra module structure on the space of

continuous Whittaker vectors for an irreducible Harish-Chandra module (ENGLISH)

[ Abstract ]

Let $G$ be a real reductive Lie group. The space of continuous Whittaker vectors for an irreducible Harish-Chandra module has a structure of a module over a finite $W$-algebra. We have seen such modules are irreducible for groups of type A. However, there is a counterexample to the naive conjecture. We discuss a refined version of the conjecture and further examples in this talk.

Let $G$ be a real reductive Lie group. The space of continuous Whittaker vectors for an irreducible Harish-Chandra module has a structure of a module over a finite $W$-algebra. We have seen such modules are irreducible for groups of type A. However, there is a counterexample to the naive conjecture. We discuss a refined version of the conjecture and further examples in this talk.

### 2010/05/10

#### Algebraic Geometry Seminar

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

Toric degenerations of Grassmannians and mirror symmetry (JAPANESE)

**Makoto Miura**

(The University of Tokyo)Toric degenerations of Grassmannians and mirror symmetry (JAPANESE)

[ Abstract ]

I will talk about toric degenerarions of Grassmannians and

an application to the mirror constructions for complete intersection

Calabi-Yau manifolds in Grassmannians.

In particular, if we focus on toric degenerations by term orderings on

polynomial rings,

we have to choose a term ordering for which the coordinate ring has an

uniformly homogeneous sagbi basis.

We discuss this condition for some examples of ordinary Grassmannians

and a spinor variety.

I will talk about toric degenerarions of Grassmannians and

an application to the mirror constructions for complete intersection

Calabi-Yau manifolds in Grassmannians.

In particular, if we focus on toric degenerations by term orderings on

polynomial rings,

we have to choose a term ordering for which the coordinate ring has an

uniformly homogeneous sagbi basis.

We discuss this condition for some examples of ordinary Grassmannians

and a spinor variety.

#### Seminar on Geometric Complex Analysis

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

Asymptotics of ACH-Einstein metrics, and an invariant tensor of partially-integrable almost CR manifolds (JAPANESE)

**Yoshihiko MATSUMOTO**(Univ. of Tokyo)Asymptotics of ACH-Einstein metrics, and an invariant tensor of partially-integrable almost CR manifolds (JAPANESE)

[ Abstract ]

To investigate strictly pseudoconvex partially-integrable almost CR manifolds as boundaries at infinity of noncompact complete Riemannian spaces, we study the Einstein equation for ACH metrics. At the jet level (of a certain order that depends only on the dimension) along the boundary, a solution uniquely exists up to the action of boundary-preserving diffeomorphisms. If we further consider higher-order solutions, without logarithmic singularities, in general we encounter an obstruction for construction, which is a local invariant tensor of the boundary. Some properties of that invariant tensor are also mentioned.

To investigate strictly pseudoconvex partially-integrable almost CR manifolds as boundaries at infinity of noncompact complete Riemannian spaces, we study the Einstein equation for ACH metrics. At the jet level (of a certain order that depends only on the dimension) along the boundary, a solution uniquely exists up to the action of boundary-preserving diffeomorphisms. If we further consider higher-order solutions, without logarithmic singularities, in general we encounter an obstruction for construction, which is a local invariant tensor of the boundary. Some properties of that invariant tensor are also mentioned.

### 2010/05/07

#### Lectures

16:00-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Independence of families of $\\ell$-adic representations and uniform constructibility

**Luc Illusie**(東京大学/Paris南大学)Independence of families of $\\ell$-adic representations and uniform constructibility

[ Abstract ]

Let $k$ be a number field, $\\overline{k}$ an algebraic closure of $k$, $\\Gamma_k = \\mathrm{Gal}(\\overline{k}/k)$. A family of continuous homomorphisms $\\rho_{\\ell} : \\Gamma_k \\rightarrow G_{\\ell}$, indexed by prime numbers $\\ell$, where $G_{\\ell}$ is a locally compact $\\ell$-adic Lie group, is said to be independent if $\\rho(\\Gamma_k) = \\prod \\rho_{\\ell}(\\Gamma_k)$, where $\\rho = (\\rho_{\\ell}) : \\Gamma_k \\rightarrow \\prod G_{\\ell}$. Serre gave a criterion for such a family to become independent after a finite extension of $k$. We will explain Serre's criterion and show that it applies to families coming from the $\\ell$-adic cohomology (or cohomology with compact support) of schemes separated and of finite type over $k$. This application uses a variant of Deligne's generic constructibility theorem with uniformity in $\\ell$.

Let $k$ be a number field, $\\overline{k}$ an algebraic closure of $k$, $\\Gamma_k = \\mathrm{Gal}(\\overline{k}/k)$. A family of continuous homomorphisms $\\rho_{\\ell} : \\Gamma_k \\rightarrow G_{\\ell}$, indexed by prime numbers $\\ell$, where $G_{\\ell}$ is a locally compact $\\ell$-adic Lie group, is said to be independent if $\\rho(\\Gamma_k) = \\prod \\rho_{\\ell}(\\Gamma_k)$, where $\\rho = (\\rho_{\\ell}) : \\Gamma_k \\rightarrow \\prod G_{\\ell}$. Serre gave a criterion for such a family to become independent after a finite extension of $k$. We will explain Serre's criterion and show that it applies to families coming from the $\\ell$-adic cohomology (or cohomology with compact support) of schemes separated and of finite type over $k$. This application uses a variant of Deligne's generic constructibility theorem with uniformity in $\\ell$.

#### Colloquium

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

Why to study controllability problems and the mathematical tools involved (ENGLISH)

**Jean-Pierre Puel**(The University of Tokyo, Universite de Versailles Saint-Quentin)Why to study controllability problems and the mathematical tools involved (ENGLISH)

[ Abstract ]

We will give some examples of controllability problems and the underlying applications to practical situations. This includes vibrations of membranes or plates, motion of incompressible fluids or quantum systems occuring in quantum chemistry or in quantum logic information theory. These examples correspond to different types of partial differential equations for which specific analysis has to be done. Of course, at the moment, very few results are known and the domain is widely open. We will describe very briefly the mathematical tools used for each type of PDE, in particular microlocal analysis, global Carleman estimates or some specific real analysis estimates.These methods appear to be also useful to study some inverse problems and, if time permits, we will give a few elements on some examples.

We will give some examples of controllability problems and the underlying applications to practical situations. This includes vibrations of membranes or plates, motion of incompressible fluids or quantum systems occuring in quantum chemistry or in quantum logic information theory. These examples correspond to different types of partial differential equations for which specific analysis has to be done. Of course, at the moment, very few results are known and the domain is widely open. We will describe very briefly the mathematical tools used for each type of PDE, in particular microlocal analysis, global Carleman estimates or some specific real analysis estimates.These methods appear to be also useful to study some inverse problems and, if time permits, we will give a few elements on some examples.

### 2010/05/06

#### Operator Algebra Seminars

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

Connes-Landi Deformation of Spectral Triples (ENGLISH)

**Makoto Yamashita**(Univ. Tokyo)Connes-Landi Deformation of Spectral Triples (ENGLISH)

[ Abstract ]

We describe a way to deform spectral triples with a 2-torus action and a real deformation parameter, motivated by deformation of manifolds after Connes-Landi. Such deformations are shown to have naturally isomorphic K-theoretic invariants independent of the deformation parameter.

We describe a way to deform spectral triples with a 2-torus action and a real deformation parameter, motivated by deformation of manifolds after Connes-Landi. Such deformations are shown to have naturally isomorphic K-theoretic invariants independent of the deformation parameter.

### 2010/04/28

#### PDE Real Analysis Seminar

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

)

GLOBAL AND SINGULAR SOLUTIONS TO SOME

HYDRODYNAMIC EVOLUTION EQUATIONS

**Marcus Wunsch**(Kyoto University)

GLOBAL AND SINGULAR SOLUTIONS TO SOME

HYDRODYNAMIC EVOLUTION EQUATIONS

[ Abstract ]

The two-component Hunter-Saxton system is a recently derived system of evolution equations modeling, e.g., the nonlinear dynamics of nondissipative dark matter and the propagation of orientation waves in nematic liquid crystals. It is imbedded into a parameterized family of systems called the generalized Hunter-Saxton (2HS) system [2] reducing, if one component is omitted, to the generalized Proudman-Johnson(gPJ) equation [1] modeling three-dimensional vortex dynamics.

After demonstrating, by means of Kato's semigroup theory, the local-in-time existence of classical solutions, the blow-up scenarios for the 2HS system and the gPJ equation are described. The explicit construction of weak dissipative solutions for both models is discussed in detail.

Finally, global existence in time of these weak solutions is proved.

The two-component Hunter-Saxton system is a recently derived system of evolution equations modeling, e.g., the nonlinear dynamics of nondissipative dark matter and the propagation of orientation waves in nematic liquid crystals. It is imbedded into a parameterized family of systems called the generalized Hunter-Saxton (2HS) system [2] reducing, if one component is omitted, to the generalized Proudman-Johnson(gPJ) equation [1] modeling three-dimensional vortex dynamics.

After demonstrating, by means of Kato's semigroup theory, the local-in-time existence of classical solutions, the blow-up scenarios for the 2HS system and the gPJ equation are described. The explicit construction of weak dissipative solutions for both models is discussed in detail.

Finally, global existence in time of these weak solutions is proved.

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