## Seminar information archive

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

### 2011/11/17

#### GCOE lecture series

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Recovery of weakly coupled system from partial Cauchy data (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Recovery of weakly coupled system from partial Cauchy data (ENGLISH)

[ Abstract ]

We consider the inverse problem for recovery of coefficients of weakly coupled system of elliptic equations in a bounded 2D domain.

We consider the inverse problem for recovery of coefficients of weakly coupled system of elliptic equations in a bounded 2D domain.

#### Operator Algebra Seminars

16:30-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Hecke pairs in ergodic measured equivalence relations (JAPANESE)

**Takehiko Yamanouchi**(Tokyo Gakugei University)Hecke pairs in ergodic measured equivalence relations (JAPANESE)

### 2011/11/16

#### Seminar on Geometric Complex Analysis

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

Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties (ENGLISH)

**Franc Forstneric**(University of Ljubljana)Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties (ENGLISH)

[ Abstract ]

A disc functional on a complex space, $X$, is a function P that assign a real number $P(f)$ (possibly minus infinity) to every analytic disc $f$ in $X$. An examples is the Poisson functional $P_u$ of an upper semicontinuous function $u$ on $X$: in that case $P_u(f)$ is the average of u over the boundary curve of the disc $f$. Other natural examples include the Lelong and the Riesz functionals. The envelope of a disc functional $P$ is a function on $X$ associating to every point $x$ of $X$ the infimum of the values $P(f)$ over all analytic discs $f$ in $X$ satisfying $f(0)=x$. The main point of interest is that the envelopes of many natural disc functionals are plurisubharmonic functions solving certain extremal problems. In the classical case when $X=\mathbf{C}^n$ this was first discovered by E. Poletsky in the early 1990's. In this talk I will discuss recent results on plurisubharmonicity of envelopes of all the classical disc functional mentioned above on locally irreducible complex spaces. In the second part of the talk I will give formulas expressing the classical Siciak-Zaharyuta maximal function of an open set in an affine algebraic variety as the envelope of certain disc functionals. We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in $\mathbf{C}^n$ by Lempert and by Larusson and Sigurdsson.

A disc functional on a complex space, $X$, is a function P that assign a real number $P(f)$ (possibly minus infinity) to every analytic disc $f$ in $X$. An examples is the Poisson functional $P_u$ of an upper semicontinuous function $u$ on $X$: in that case $P_u(f)$ is the average of u over the boundary curve of the disc $f$. Other natural examples include the Lelong and the Riesz functionals. The envelope of a disc functional $P$ is a function on $X$ associating to every point $x$ of $X$ the infimum of the values $P(f)$ over all analytic discs $f$ in $X$ satisfying $f(0)=x$. The main point of interest is that the envelopes of many natural disc functionals are plurisubharmonic functions solving certain extremal problems. In the classical case when $X=\mathbf{C}^n$ this was first discovered by E. Poletsky in the early 1990's. In this talk I will discuss recent results on plurisubharmonicity of envelopes of all the classical disc functional mentioned above on locally irreducible complex spaces. In the second part of the talk I will give formulas expressing the classical Siciak-Zaharyuta maximal function of an open set in an affine algebraic variety as the envelope of certain disc functionals. We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in $\mathbf{C}^n$ by Lempert and by Larusson and Sigurdsson.

#### GCOE Seminars

10:00-11:00 Room #270 (Graduate School of Math. Sci. Bldg.)

All you never really wanted to know about QRT, but were foolhardy enough to ask (ENGLISH)

**Alfred Ramani**(Ecole Polytechnique)All you never really wanted to know about QRT, but were foolhardy enough to ask (ENGLISH)

[ Abstract ]

We discuss various extensions of the famous QRT second order, first degree, integrable mapping. We show how these extensions can be combined. A discussion of integrable correspondences related to these extended QRT mappings is also presented.

We discuss various extensions of the famous QRT second order, first degree, integrable mapping. We show how these extensions can be combined. A discussion of integrable correspondences related to these extended QRT mappings is also presented.

### 2011/11/15

#### Tuesday Seminar on Topology

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

Singular codimension-one foliations

and twisted open books in dimension 3.

(joint work with G. Meigniez)

(ENGLISH)

**Francois Laudenbach**(Univ. de Nantes)Singular codimension-one foliations

and twisted open books in dimension 3.

(joint work with G. Meigniez)

(ENGLISH)

[ Abstract ]

The allowed singularities are those of functions.

According to A. Haefliger (1958),

such structures on manifolds, called $\\Gamma_1$-structures,

are objects of a cohomological

theory with a classifying space $B\\Gamma_1$.

The problem of cancelling the singularities

(or regularization problem)

arise naturally.

For a closed manifold, it was solved by W.Thurston in a famous paper

(1976), with a proof relying on Mather's isomorphism (1971):

Diff$^\\infty(\\mathbb R)$ as a discrete group has the same homology

as the based loop space

$\\Omega B\\Gamma_1^+$.

For further extension to contact geometry, it is necessary

to solve the regularization problem

without using Mather's isomorphism.

That is what we have done in dimension 3. Our result is the following.

{\\it Every $\\Gamma_1$-structure $\\xi$ on a 3-manifold $M$ whose

normal bundle

embeds into the tangent bundle to $M$ is $\\Gamma_1$-homotopic

to a regular foliation

carried by a (possibily twisted) open book.}

The proof is elementary and relies on the dynamics of a (twisted)

pseudo-gradient of $\\xi$.

All the objects will be defined in the talk, in particular the notion

of twisted open book which is a central object in the reported paper.

The allowed singularities are those of functions.

According to A. Haefliger (1958),

such structures on manifolds, called $\\Gamma_1$-structures,

are objects of a cohomological

theory with a classifying space $B\\Gamma_1$.

The problem of cancelling the singularities

(or regularization problem)

arise naturally.

For a closed manifold, it was solved by W.Thurston in a famous paper

(1976), with a proof relying on Mather's isomorphism (1971):

Diff$^\\infty(\\mathbb R)$ as a discrete group has the same homology

as the based loop space

$\\Omega B\\Gamma_1^+$.

For further extension to contact geometry, it is necessary

to solve the regularization problem

without using Mather's isomorphism.

That is what we have done in dimension 3. Our result is the following.

{\\it Every $\\Gamma_1$-structure $\\xi$ on a 3-manifold $M$ whose

normal bundle

embeds into the tangent bundle to $M$ is $\\Gamma_1$-homotopic

to a regular foliation

carried by a (possibily twisted) open book.}

The proof is elementary and relies on the dynamics of a (twisted)

pseudo-gradient of $\\xi$.

All the objects will be defined in the talk, in particular the notion

of twisted open book which is a central object in the reported paper.

#### Lie Groups and Representation Theory

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

Categorification of cluster algebras arising from unipotent subgroups of non-simply laced Lie groups (ENGLISH)

**Laurant Demonet**(Nagoya University)Categorification of cluster algebras arising from unipotent subgroups of non-simply laced Lie groups (ENGLISH)

[ Abstract ]

We introduce an abstract framework to categorify some antisymetrizable cluster algebras by using actions of finite groups on stably 2-Calabi-Yau exact categories. We introduce the notion of the equivariant category and, with similar technics as in [K], [CK], [GLS1], [GLS2], [DK], [FK], [P], we construct some examples of such categorifications. For example, if we let Z/2Z act on the category of representations of the preprojective algebra of type A2n-1 via the only non trivial action on the diagram, we obtain the cluster structure on the coordinate ring of the maximal unipotent subgroup of the semi-simple Lie group of type Bn [D]. Hence, we can get relations between the cluster algebras categorified by some exact subcategories of these two categories. We also prove by the same methods as in [FK] a conjecture of Fomin and Zelevinsky stating that the cluster monomials are linearly independent.

References

[CK] P. Caldero, B. Keller, From triangulated categories to cluster algebras, Invent. Math. 172 (2008), no. 1, 169--211.

[DK] R. Dehy, B. Keller, On the combinatorics of rigid objects in 2-Calabi-Yau categories, arXiv: 0709.0882.

[D] L. Demonet, Cluster algebras and preprojective algebras: the non simply-laced case, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 379--384.

[FK] C. Fu, B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, arXiv: 0710.3152.

[GLS1] C. Geiss, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589--632.

[GLS2] C. Geiss, B. Leclerc, J. Schröer, Cluster algebra structures and semicanoncial bases for unipotent groups, arXiv: math/0703039.

[K] B. Keller, Categorification of acyclic cluster algebras: an introduction, arXiv: 0801.3103.

[P] Y. Palu, Cluster characters for triangulated 2-Calabi--Yau categories, arXiv: math/0703540.

We introduce an abstract framework to categorify some antisymetrizable cluster algebras by using actions of finite groups on stably 2-Calabi-Yau exact categories. We introduce the notion of the equivariant category and, with similar technics as in [K], [CK], [GLS1], [GLS2], [DK], [FK], [P], we construct some examples of such categorifications. For example, if we let Z/2Z act on the category of representations of the preprojective algebra of type A2n-1 via the only non trivial action on the diagram, we obtain the cluster structure on the coordinate ring of the maximal unipotent subgroup of the semi-simple Lie group of type Bn [D]. Hence, we can get relations between the cluster algebras categorified by some exact subcategories of these two categories. We also prove by the same methods as in [FK] a conjecture of Fomin and Zelevinsky stating that the cluster monomials are linearly independent.

References

[CK] P. Caldero, B. Keller, From triangulated categories to cluster algebras, Invent. Math. 172 (2008), no. 1, 169--211.

[DK] R. Dehy, B. Keller, On the combinatorics of rigid objects in 2-Calabi-Yau categories, arXiv: 0709.0882.

[D] L. Demonet, Cluster algebras and preprojective algebras: the non simply-laced case, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 379--384.

[FK] C. Fu, B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, arXiv: 0710.3152.

[GLS1] C. Geiss, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589--632.

[GLS2] C. Geiss, B. Leclerc, J. Schröer, Cluster algebra structures and semicanoncial bases for unipotent groups, arXiv: math/0703039.

[K] B. Keller, Categorification of acyclic cluster algebras: an introduction, arXiv: 0801.3103.

[P] Y. Palu, Cluster characters for triangulated 2-Calabi--Yau categories, arXiv: math/0703540.

### 2011/11/14

#### GCOE lecture series

17:00-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Recovery of weakly coupled system from partial Cauchy data (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Recovery of weakly coupled system from partial Cauchy data (ENGLISH)

[ Abstract ]

We consider the inverse problem for recovery of coefficients of weakly coupled system of elliptic equations in a bounded 2D domain.

We consider the inverse problem for recovery of coefficients of weakly coupled system of elliptic equations in a bounded 2D domain.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On projective manifolds swept out by cubic varieties (JAPANESE)

**Kiwamu Watanabe**(University of Tokyo)On projective manifolds swept out by cubic varieties (JAPANESE)

[ Abstract ]

The structures of embedded complex projective manifolds swept out by varieties with preassigned properties have been studied by several authors. In this talk, we study structures of embedded projective manifolds swept out by cubic varieties.

The structures of embedded complex projective manifolds swept out by varieties with preassigned properties have been studied by several authors. In this talk, we study structures of embedded projective manifolds swept out by cubic varieties.

### 2011/11/10

#### GCOE lecture series

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)

[ Abstract ]

We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).

We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).

#### Applied Analysis

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

Mathematical Modeling of Cellular Electrodiffusion and Osmosis (JAPANESE)

**Yoichiro Mori**(University of Minnesota )Mathematical Modeling of Cellular Electrodiffusion and Osmosis (JAPANESE)

#### Applied Analysis

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

Schoenflies spheres in Sturm attractors (ENGLISH)

**Bernold Fiedler**(Free University of Berlin)Schoenflies spheres in Sturm attractors (ENGLISH)

[ Abstract ]

In gradient systems on compact manifolds the boundary of the unstable manifold of an equilibrium need not be homeomorphic to a sphere, or to any compact manifold.

For scalar parabolic equations in one space dimension, however, we can exlude complications like Reidemeister torsion and the Alexander horned sphere. Instead the boundary is a Schoenflies embedded sphere. This is due to Sturm nodal properties related to the Matano lap number.

In gradient systems on compact manifolds the boundary of the unstable manifold of an equilibrium need not be homeomorphic to a sphere, or to any compact manifold.

For scalar parabolic equations in one space dimension, however, we can exlude complications like Reidemeister torsion and the Alexander horned sphere. Instead the boundary is a Schoenflies embedded sphere. This is due to Sturm nodal properties related to the Matano lap number.

### 2011/11/09

#### Number Theory Seminar

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On extension and restriction of overconvergent isocrystals (ENGLISH)

**Atsushi Shiho**(University of Tokyo)On extension and restriction of overconvergent isocrystals (ENGLISH)

[ Abstract ]

First we explain two theorems concerning (log) extension of overconvergent isocrystals. One is a p-adic analogue of the theorem of logarithmic extension of regular integrable connections, and the other is a p-adic analogue of Zariski-Nagata purity. Next we explain a theorem which says that we can check certain property of overconvergent isocrystals by restricting them to curves.

(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)

First we explain two theorems concerning (log) extension of overconvergent isocrystals. One is a p-adic analogue of the theorem of logarithmic extension of regular integrable connections, and the other is a p-adic analogue of Zariski-Nagata purity. Next we explain a theorem which says that we can check certain property of overconvergent isocrystals by restricting them to curves.

(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)

### 2011/11/08

#### Tuesday Seminar on Topology

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

Fiberwise bordism groups and related topics (JAPANESE)

**Shoji Yokura**(Kagoshima University )Fiberwise bordism groups and related topics (JAPANESE)

[ Abstract ]

We have recently introduced the notion of fiberwise bordism. In this talk, after a quick review of some of the classical (co)bordism theories, we will explain motivations of considering fiberwise bordism and some results and connections with other known works, such as M. Kreck's bordism groups of orientation preserving diffeomorphisms and Emerson-Meyer's bivariant K-theory etc. An essential motivation is our recent work towards constructing a bivariant-theoretic analogue (in the sense of Fulton-MacPherson) of Levine-Morel's or Levine-Pandharipande's algebraic cobordism.

We have recently introduced the notion of fiberwise bordism. In this talk, after a quick review of some of the classical (co)bordism theories, we will explain motivations of considering fiberwise bordism and some results and connections with other known works, such as M. Kreck's bordism groups of orientation preserving diffeomorphisms and Emerson-Meyer's bivariant K-theory etc. An essential motivation is our recent work towards constructing a bivariant-theoretic analogue (in the sense of Fulton-MacPherson) of Levine-Morel's or Levine-Pandharipande's algebraic cobordism.

#### Tuesday Seminar of Analysis

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

The University of Tokyo (JSPS Research Fellow))

Stochastic Power-Law Fluid equations: Existence and Uniqueness of weak solutions (joint work with Nobuo Yoshida) (JAPANESE)

**Yutaka Terasawa**(Graduate School of Mathematical Sciences,The University of Tokyo (JSPS Research Fellow))

Stochastic Power-Law Fluid equations: Existence and Uniqueness of weak solutions (joint work with Nobuo Yoshida) (JAPANESE)

#### Seminar on Mathematics for various disciplines

16:30-17:30 Room #052 (Graduate School of Math. Sci. Bldg.)

Interactive Data Visualization challenges, approaches and examples (ENGLISH)

**Ralph Bruckschen**(ベルリン工科大学、MATHEON)Interactive Data Visualization challenges, approaches and examples (ENGLISH)

[ Abstract ]

Data visualization is probably the most important method to analyze scientific datasets. In the time of petaflop supercomputers and high resolution sensors, the visualization of such datasets became a challenge because of the sheer magnitude. Using the latest technology I will describe some of the challenges and approaches to visualize large and massive datasets. The main bottle necks will be explained, as some algorithms and data structures to widen them. Finally I will show some examples of data visualization using a CAVE environment and virtual prototyping from the 3D Labor at the Technical University of Berlin.

Data visualization is probably the most important method to analyze scientific datasets. In the time of petaflop supercomputers and high resolution sensors, the visualization of such datasets became a challenge because of the sheer magnitude. Using the latest technology I will describe some of the challenges and approaches to visualize large and massive datasets. The main bottle necks will be explained, as some algorithms and data structures to widen them. Finally I will show some examples of data visualization using a CAVE environment and virtual prototyping from the 3D Labor at the Technical University of Berlin.

### 2011/11/07

#### GCOE lecture series

17:00-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)

[ Abstract ]

We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).

We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Okounkov bodies and Seshadri constants (JAPANESE)

**Atsushi Ito**(University of Tokyo)Okounkov bodies and Seshadri constants (JAPANESE)

[ Abstract ]

Okounkov bodies, which are convex bodies associated to big line bundles, have rich information of the line bundles. On the other hand, Seshadri constants are invariants which measure the positivities of line bundles. In this talk, I will explain a relation between Okounkov bodies and Seshadri constants.

Okounkov bodies, which are convex bodies associated to big line bundles, have rich information of the line bundles. On the other hand, Seshadri constants are invariants which measure the positivities of line bundles. In this talk, I will explain a relation between Okounkov bodies and Seshadri constants.

#### Seminar on Geometric Complex Analysis

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

Oka's extra-zero problem and related topics (JAPANESE)

**Junjiro Nocuchi**(University of Tokyo)Oka's extra-zero problem and related topics (JAPANESE)

[ Abstract ]

The main part of this talk is a joint work with my colleagues, M. Abe and S. Hamano. After the solution of Cousin II problem by K. Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. Some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was obtained. We will give a complete solution of this problem with examples and to discuss some new questions. An example on a toric variety of which idea is based on K. Stein's paper in 1941 has some special interest and will be discussed. I would like also to discuss some analytic intersections form the viewpoint of Nevanlinna theory.

The main part of this talk is a joint work with my colleagues, M. Abe and S. Hamano. After the solution of Cousin II problem by K. Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. Some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was obtained. We will give a complete solution of this problem with examples and to discuss some new questions. An example on a toric variety of which idea is based on K. Stein's paper in 1941 has some special interest and will be discussed. I would like also to discuss some analytic intersections form the viewpoint of Nevanlinna theory.

### 2011/11/04

#### Colloquium

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

Cross ratio, and all that

(JAPANESE)

**KANAI Masahiko**(Graduate School of Mathematical Sciences, University of Tokyo)Cross ratio, and all that

(JAPANESE)

[ Abstract ]

Although the origin of cross ratio goes back to ancient Greek mathematics, new discoveries about it has been made even in the past few decades. It seems that our understanding as to cross ratio is still limited. I am going to show you the present status and my own concerns, as well.

Although the origin of cross ratio goes back to ancient Greek mathematics, new discoveries about it has been made even in the past few decades. It seems that our understanding as to cross ratio is still limited. I am going to show you the present status and my own concerns, as well.

### 2011/11/02

#### Number Theory Seminar

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

Hypergeometric series and arithmetic-geometric mean over 2-adic fields (JAPANESE)

**Kensaku Kinjo**(University of Tokyo)Hypergeometric series and arithmetic-geometric mean over 2-adic fields (JAPANESE)

[ Abstract ]

Dwork proved that the Gaussian hypergeometric function on p-adic numbers

can be extended to a function which takes values of the unit roots of

ordinary elliptic curves over a finite field of characteristic p>2.

We present an analogous theory in the case p=2.

As an application, we give a relation between the canonical lift

and the unit root of an elliptic curve over a finite field of

characteristic 2

by using the 2-adic arithmetic-geometric mean.

Dwork proved that the Gaussian hypergeometric function on p-adic numbers

can be extended to a function which takes values of the unit roots of

ordinary elliptic curves over a finite field of characteristic p>2.

We present an analogous theory in the case p=2.

As an application, we give a relation between the canonical lift

and the unit root of an elliptic curve over a finite field of

characteristic 2

by using the 2-adic arithmetic-geometric mean.

### 2011/11/01

#### Tuesday Seminar on Topology

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

Motivic Milnor fibers and Jordan normal forms of monodromies (JAPANESE)

**Kiyoshi Takeuchi**(University of Tsukuba)Motivic Milnor fibers and Jordan normal forms of monodromies (JAPANESE)

[ Abstract ]

We introduce a method to calculate the equivariant

Hodge-Deligne numbers of toric hypersurfaces.

Then we apply it to motivic Milnor

fibers introduced by Denef-Loeser and study the Jordan

normal forms of the local and global monodromies

of polynomials maps in various situations.

Especially we focus our attention on monodromies

at infinity studied by many people. The results will be

explicitly described by the ``convexity" of

the Newton polyhedra of polynomials. This is a joint work

with Y. Matsui and A. Esterov.

We introduce a method to calculate the equivariant

Hodge-Deligne numbers of toric hypersurfaces.

Then we apply it to motivic Milnor

fibers introduced by Denef-Loeser and study the Jordan

normal forms of the local and global monodromies

of polynomials maps in various situations.

Especially we focus our attention on monodromies

at infinity studied by many people. The results will be

explicitly described by the ``convexity" of

the Newton polyhedra of polynomials. This is a joint work

with Y. Matsui and A. Esterov.

### 2011/10/31

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Minimal model theory for log surfaces (JAPANESE)

**Osamu Fujino**(Kyoto University)Minimal model theory for log surfaces (JAPANESE)

[ Abstract ]

We discuss the log minimal model theory for log sur- faces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.

We discuss the log minimal model theory for log sur- faces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.

#### PDE Real Analysis Seminar

13:30-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Stationary Weak Solutions of the Navier-Stokes Equations Past an Obstacle (ENGLISH)

**Horst Heck**

(Technische Universität Darmstadt)Stationary Weak Solutions of the Navier-Stokes Equations Past an Obstacle (ENGLISH)

[ Abstract ]

Consider the stationary Navier-Stokes equations in an exterior smooth domain $\\Omega$. If the flow condition $u_\\infty$ for $u$ at infinity is non-zero and the external force $f\\in \\dot H^{-1}_2(\\Omega)$ is given Leray constructed a weak solution $u\\in \\dot H^1_2(\\Omega)$, the homogeneous Sobolev space, with $u-u_\\infty\\in L^6(\\Omega)$.

We show that if in addition $f\\in \\dot H^{-1}_q(\\Omega)$ for some $q\\in (4/3,4)$ then the weak solution has the property $u-u_\\infty\\in L^{4q/(4-q)}(\\Omega)$.

This additional integrability implies that $u$ satisfies the energy identity. Further consequences are uniqueness results for small $u_\\infty$ and $f$ and continuous dependence of the solution with respect to $u_\\infty$.

The presented results are joint work with Hyunseok Kim and Hideo Kozono.

Consider the stationary Navier-Stokes equations in an exterior smooth domain $\\Omega$. If the flow condition $u_\\infty$ for $u$ at infinity is non-zero and the external force $f\\in \\dot H^{-1}_2(\\Omega)$ is given Leray constructed a weak solution $u\\in \\dot H^1_2(\\Omega)$, the homogeneous Sobolev space, with $u-u_\\infty\\in L^6(\\Omega)$.

We show that if in addition $f\\in \\dot H^{-1}_q(\\Omega)$ for some $q\\in (4/3,4)$ then the weak solution has the property $u-u_\\infty\\in L^{4q/(4-q)}(\\Omega)$.

This additional integrability implies that $u$ satisfies the energy identity. Further consequences are uniqueness results for small $u_\\infty$ and $f$ and continuous dependence of the solution with respect to $u_\\infty$.

The presented results are joint work with Hyunseok Kim and Hideo Kozono.

#### Seminar on Geometric Complex Analysis

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

Classification of Moishezon twistor spaces on 4CP^2 (JAPANESE)

**Nobuhiro Honda**(Tohoku Univeristy)Classification of Moishezon twistor spaces on 4CP^2 (JAPANESE)

### 2011/10/29

#### Infinite Analysis Seminar Tokyo

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

CKP Hierarchy, Bosonic Tau Function, Bosonization Formulae and Orthogonal Polynomials both in Odd and Even Variables

(based on a joint work with Johan van de Leur and Takahiro Shiota) (ENGLISH)

Kernel function identities associated with van Diejen's $q$-difference operators

and transformation formulas for multiple $q$-hypergeometric series (JAPANESE)

**Alexander Orlov**(Nonlinear Wave Processes Laboratory, Oceanology Institute (Moscow)) 11:00-12:00CKP Hierarchy, Bosonic Tau Function, Bosonization Formulae and Orthogonal Polynomials both in Odd and Even Variables

(based on a joint work with Johan van de Leur and Takahiro Shiota) (ENGLISH)

[ Abstract ]

We develop the theory of CKP hierarchy introduced in the papers of Kyoto school where the CKP tau function is written as a vacuum expectation value in terms of free bosons. We show that a sort of odd currents naturaly appear. We consider bosonization formulae which relate bosonic Fock vectors to polynomials in even and odd Grassmannian variables, where both sets play a role of higher times.

We develop the theory of CKP hierarchy introduced in the papers of Kyoto school where the CKP tau function is written as a vacuum expectation value in terms of free bosons. We show that a sort of odd currents naturaly appear. We consider bosonization formulae which relate bosonic Fock vectors to polynomials in even and odd Grassmannian variables, where both sets play a role of higher times.

**Yasuho Masuda**(Kobe Univ. ) 13:30-14:30Kernel function identities associated with van Diejen's $q$-difference operators

and transformation formulas for multiple $q$-hypergeometric series (JAPANESE)

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