## Seminar information archive

Seminar information archive ～05/21｜Today's seminar 05/22 | Future seminars 05/23～

### 2010/12/13

#### Seminar on Geometric Complex Analysis

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

An equality estimate for the second main theorem (JAPANESE)

**Katsutoshi Yamanoi**(Tokyo Institute of Technology)An equality estimate for the second main theorem (JAPANESE)

#### Algebraic Geometry Seminar

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

Enumeration of plane curves and labeled floor diagrams (ENGLISH)

**Sergey Fomin**(University of Michigan)Enumeration of plane curves and labeled floor diagrams (ENGLISH)

[ Abstract ]

Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and G. Mikhalkin. Tropical geometry arguments yield combinatorial descriptions of (ordinary and relative) Gromov-Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In the case of the projective plane, these descriptions can be used to obtain new formulas for the corresponding enumerative invariants. In particular, we give a proof of Goettsche's polynomiality conjecture for plane curves, and enumerate plane rational curves of given degree passing through given points and having maximal tangency to a given line. On the combinatorial side, we show that labeled floor diagrams of genus 0 are equinumerous to labeled trees, and therefore counted by the celebrated Cayley's formula. The corresponding bijections lead to interpretations of the Kontsevich numbers (the genus-0 Gromov-Witten invariants of the projective plane) in terms of certain statistics on trees.

This is joint work with Grisha Mikhalkin.

Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and G. Mikhalkin. Tropical geometry arguments yield combinatorial descriptions of (ordinary and relative) Gromov-Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In the case of the projective plane, these descriptions can be used to obtain new formulas for the corresponding enumerative invariants. In particular, we give a proof of Goettsche's polynomiality conjecture for plane curves, and enumerate plane rational curves of given degree passing through given points and having maximal tangency to a given line. On the combinatorial side, we show that labeled floor diagrams of genus 0 are equinumerous to labeled trees, and therefore counted by the celebrated Cayley's formula. The corresponding bijections lead to interpretations of the Kontsevich numbers (the genus-0 Gromov-Witten invariants of the projective plane) in terms of certain statistics on trees.

This is joint work with Grisha Mikhalkin.

### 2010/12/10

#### Colloquium

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

Hamilton-Jacobi equations and crystal growth (JAPANESE)

**Yoshikazu Giga**(The University of Tokyo, Graduate School of Mathematical Sciences)Hamilton-Jacobi equations and crystal growth (JAPANESE)

### 2010/12/09

#### Operator Algebra Seminars

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

Spectral flow associated to KMS states with periodic KMS group action (ENGLISH)

**Ryszard Nest**(Univ. Copenhagen)Spectral flow associated to KMS states with periodic KMS group action (ENGLISH)

### 2010/12/07

#### Tuesday Seminar on Topology

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

Diffeomorphism-invariant geometries and noncommutative geometry (ENGLISH)

**Raphael Ponge**(The University of Tokyo)Diffeomorphism-invariant geometries and noncommutative geometry (ENGLISH)

[ Abstract ]

In many geometric situations we may encounter the action of

a group $G$ on a manifold $M$, e.g., in the context of foliations. If

the action is free and proper, then the quotient $M/G$ is a smooth

manifold. However, in general the quotient $M/G$ need not even be

Hausdorff. Furthermore, it is well-known that a manifold has structure

invariant under the full group of diffeomorphisms except the

differentiable structure itself. Under these conditions how can one do

diffeomorphism-invariant geometry?

Noncommutative geometry provides us with the solution of trading the

ill-behaved space $M/G$ for a non-commutative algebra which

essentially plays the role of the algebra of smooth functions on that

space. The local index formula of Atiyah-Singer ultimately holds in

the setting of noncommutative geometry. Using this framework Connes

and Moscovici then obtained in the 90s a striking reformulation of the

local index formula in diffeomorphism-invariant geometry.

An important part the talk will be devoted to reviewing noncommutative

geometry and Connes-Moscovici's index formula. We will then hint to on-

going projects about reformulating the local index formula in two new

geometric settings: biholomorphism-invariant geometry of strictly

pseudo-convex domains and contactomorphism-invariant geometry.

In many geometric situations we may encounter the action of

a group $G$ on a manifold $M$, e.g., in the context of foliations. If

the action is free and proper, then the quotient $M/G$ is a smooth

manifold. However, in general the quotient $M/G$ need not even be

Hausdorff. Furthermore, it is well-known that a manifold has structure

invariant under the full group of diffeomorphisms except the

differentiable structure itself. Under these conditions how can one do

diffeomorphism-invariant geometry?

Noncommutative geometry provides us with the solution of trading the

ill-behaved space $M/G$ for a non-commutative algebra which

essentially plays the role of the algebra of smooth functions on that

space. The local index formula of Atiyah-Singer ultimately holds in

the setting of noncommutative geometry. Using this framework Connes

and Moscovici then obtained in the 90s a striking reformulation of the

local index formula in diffeomorphism-invariant geometry.

An important part the talk will be devoted to reviewing noncommutative

geometry and Connes-Moscovici's index formula. We will then hint to on-

going projects about reformulating the local index formula in two new

geometric settings: biholomorphism-invariant geometry of strictly

pseudo-convex domains and contactomorphism-invariant geometry.

#### Numerical Analysis Seminar

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

Numerical verification of existence for solutions to Dirichlet

boundary value problems of semilinear elliptic equations

(JAPANESE)

[ Reference URL ]

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

**Akitoshi Takayasu**(Waseda University)Numerical verification of existence for solutions to Dirichlet

boundary value problems of semilinear elliptic equations

(JAPANESE)

[ Reference URL ]

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

### 2010/12/06

#### Seminar on Geometric Complex Analysis

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

Chow semistability of polarized toric manifolds (JAPANESE)

**Hajime Ono**(Tokyo Univ of Science)Chow semistability of polarized toric manifolds (JAPANESE)

### 2010/12/04

#### Classical Analysis

09:30-10:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Orbit decomposition of multiple flag varieties and representations of of quiver (JAPANESE)

**Toshihiko Matsuki**(Kyoto University)Orbit decomposition of multiple flag varieties and representations of of quiver (JAPANESE)

#### Classical Analysis

10:40-11:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Integral transformations on the Heun equation and its applications (JAPANESE)

**Kouichi Takemura**(Chuo University)Integral transformations on the Heun equation and its applications (JAPANESE)

#### Classical Analysis

13:00-14:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Weyl group symmetries of double confluent Heun equations (JAPANESE)

**Kazuki Hiroe**(University of Tokyo)Weyl group symmetries of double confluent Heun equations (JAPANESE)

#### Classical Analysis

14:10-15:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Affine root systems, monodromy preserving deformation, and hypergeometric functions (JAPANESE)

**Takao Suzuki**(Kobe University)Affine root systems, monodromy preserving deformation, and hypergeometric functions (JAPANESE)

#### Classical Analysis

15:30-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On the uniformization equations which have singularities along discriminant of complex reflection groups of rank three (JAPANESE)

**Jiro Sekiguchi**(Tokyo University of Agriculture and Technology)On the uniformization equations which have singularities along discriminant of complex reflection groups of rank three (JAPANESE)

### 2010/12/03

#### GCOE Seminars

11:00-12:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Discrete Integrability and Consistency-Around-the-Cube (CAC) (ENGLISH)

**Jarmo Hietarinta**(University of Turku)Discrete Integrability and Consistency-Around-the-Cube (CAC) (ENGLISH)

[ Abstract ]

For integrable lattice equations we can still apply many integrability criteria that are regularly used for continuous systems, but there are also some that are specific for discrete systems. One particularly successful discrete integrability criterion is the multidimensional consistency, or CAC. We review the classic results of Nijhoff and of Adler-Bobenko-Suris and then present some extensions.

For integrable lattice equations we can still apply many integrability criteria that are regularly used for continuous systems, but there are also some that are specific for discrete systems. One particularly successful discrete integrability criterion is the multidimensional consistency, or CAC. We review the classic results of Nijhoff and of Adler-Bobenko-Suris and then present some extensions.

#### GCOE Seminars

13:30-14:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Geometric asymptotics of the first Painleve equation (ENGLISH)

**Nalini Joshi**(University of Sydney)Geometric asymptotics of the first Painleve equation (ENGLISH)

[ Abstract ]

I will report on my recent collaboration with Hans Duistermaat on the geometry of the space of initial values of the first Painleve equation, which was first constructed by Okamoto. We show that highly accurate information about solutions can be found by utilizing the regularized and compactified space of initial values in Boutroux's coordinates. I will also describe numerical explorations based on this work obtained in collaboration with Holger Dullin.

I will report on my recent collaboration with Hans Duistermaat on the geometry of the space of initial values of the first Painleve equation, which was first constructed by Okamoto. We show that highly accurate information about solutions can be found by utilizing the regularized and compactified space of initial values in Boutroux's coordinates. I will also describe numerical explorations based on this work obtained in collaboration with Holger Dullin.

#### Classical Analysis

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

The third Painlev¥'e equation and quiver varieties (JAPANESE)

**Daisuke Yamakawa**(Kobe University)The third Painlev¥'e equation and quiver varieties (JAPANESE)

### 2010/12/01

#### Number Theory Seminar

16:30-18:45 Room #056 (Graduate School of Math. Sci. Bldg.)

On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves (JAPANESE)

Galois theory for schemes (ENGLISH)

**Yuichiro Hoshi**(RIMS, Kyoto University) 16:30-17:30On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves (JAPANESE)

[ Abstract ]

In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.

For a hyperbolic curve X over a number field, are the following three conditions equivalent?

(A) For any prime number l, X is quasi-l-monodromically full.

(B) There exists a prime number l such that X is l-monodromically full.

(C) X is l-monodromically full for all but finitely many prime numbers l.

The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.

For an elliptic curve E over a number field, the following four conditions are equivalent:

(0) E does not admit complex multiplication.

(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.

(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.

(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.

In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies condition (C) but does not satisfy condition (A).

In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.

For a hyperbolic curve X over a number field, are the following three conditions equivalent?

(A) For any prime number l, X is quasi-l-monodromically full.

(B) There exists a prime number l such that X is l-monodromically full.

(C) X is l-monodromically full for all but finitely many prime numbers l.

The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.

For an elliptic curve E over a number field, the following four conditions are equivalent:

(0) E does not admit complex multiplication.

(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.

(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.

(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.

In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies condition (C) but does not satisfy condition (A).

**Marco Garuti**(University of Padova) 17:45-18:45Galois theory for schemes (ENGLISH)

[ Abstract ]

We discuss some aspects of finite group scheme actions: the Galois correspondence and the notion of Galois closure.

We discuss some aspects of finite group scheme actions: the Galois correspondence and the notion of Galois closure.

### 2010/11/30

#### Numerical Analysis Seminar

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

Finite volume element method for singular solutions of elliptic PDEs

(JAPANESE)

[ Reference URL ]

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

**Yasunori Aoki**(University of Waterloo/NII)Finite volume element method for singular solutions of elliptic PDEs

(JAPANESE)

[ Reference URL ]

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

#### Tuesday Seminar on Topology

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

Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds (JAPANESE)

**Nobuhiro Nakamura**(The University of Tokyo)Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds (JAPANESE)

[ Abstract ]

We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds.

The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients.

The second one is a local coefficient version of Furuta's 10/8-inequality.

As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds.

The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients.

The second one is a local coefficient version of Furuta's 10/8-inequality.

As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

#### Operator Algebra Seminars

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

Noncommutative geometry and Rankin-Cohen brackets (ENGLISH)

**Yi-Jun Yao**(Fudan Univ.)Noncommutative geometry and Rankin-Cohen brackets (ENGLISH)

### 2010/11/29

#### Kavli IPMU Komaba Seminar

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

Borcherds products in monstrous moonshine. (ENGLISH)

**Scott Carnahan**(IPMU)Borcherds products in monstrous moonshine. (ENGLISH)

[ Abstract ]

During the 1980s, Koike, Norton, and Zagier independently found an

infinite product expansion for the difference of two modular j-functions

on a product of half planes. Borcherds showed that this product identity

is the Weyl denominator formula for an infinite dimensional Lie algebra

that has an action of the monster simple group by automorphisms, and used

this action to prove the monstrous moonshine conjectures.

I will describe a more general construction that yields an infinite

product identity and an infinite dimensional Lie algebra for each element

of the monster group. The above objects then arise as the special cases

assigned to the identity element. Time permitting, I will attempt to

describe a connection to conformal field theory.

During the 1980s, Koike, Norton, and Zagier independently found an

infinite product expansion for the difference of two modular j-functions

on a product of half planes. Borcherds showed that this product identity

is the Weyl denominator formula for an infinite dimensional Lie algebra

that has an action of the monster simple group by automorphisms, and used

this action to prove the monstrous moonshine conjectures.

I will describe a more general construction that yields an infinite

product identity and an infinite dimensional Lie algebra for each element

of the monster group. The above objects then arise as the special cases

assigned to the identity element. Time permitting, I will attempt to

describe a connection to conformal field theory.

#### Algebraic Geometry Seminar

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

K3 surfaces and log del Pezzo surfaces of index three (JAPANESE)

**Hisanori Ohashi**(Nagoya Univ. )K3 surfaces and log del Pezzo surfaces of index three (JAPANESE)

[ Abstract ]

Alexeev and Nikulin have classified log del Pezzo surfaces of index 1 and 2 by using the classification of non-symplectic involutions on K3 surfaces. We want to discuss the generalization of this result to the index 3 cases. In this case we are also able to construct log del Pezzos $Z$ from K3 surfaces $X$, but the converse is not necessarily true. The condition on $Z$ is exactly the "multiple smooth divisor property", which we will define. Our theorem is the classification of log del Pezzo surfaces of index 3 with this property.

The idea of the proof is similar to that of Alexeev and Nikulin, but the methods are different because of the existence of singularities: although the singularity is mild, the description of nef cone by reflection groups cannot be used. Instead

we construct and analyze good elliptic fibrations on K3 surfaces $X$ and use it to obtain the classification. It includes a partial but geometric generalization of the classification of non-symplectic automorphisms of order three, recently done by Artebani, Sarti and Taki.

Alexeev and Nikulin have classified log del Pezzo surfaces of index 1 and 2 by using the classification of non-symplectic involutions on K3 surfaces. We want to discuss the generalization of this result to the index 3 cases. In this case we are also able to construct log del Pezzos $Z$ from K3 surfaces $X$, but the converse is not necessarily true. The condition on $Z$ is exactly the "multiple smooth divisor property", which we will define. Our theorem is the classification of log del Pezzo surfaces of index 3 with this property.

The idea of the proof is similar to that of Alexeev and Nikulin, but the methods are different because of the existence of singularities: although the singularity is mild, the description of nef cone by reflection groups cannot be used. Instead

we construct and analyze good elliptic fibrations on K3 surfaces $X$ and use it to obtain the classification. It includes a partial but geometric generalization of the classification of non-symplectic automorphisms of order three, recently done by Artebani, Sarti and Taki.

### 2010/11/26

#### Kavli IPMU Komaba Seminar

14:40-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Hamiltonian torus actions on orbifolds and orbifold-GKM theorem (joint

work with T. Holm) (JAPANESE)

**Tomoo Matsumura**(Cornell University)Hamiltonian torus actions on orbifolds and orbifold-GKM theorem (joint

work with T. Holm) (JAPANESE)

[ Abstract ]

When a symplectic manifold M carries a Hamiltonian torus R action, the

injectivity theorem states that the R-equivariant cohomology of M is a

subring of the one of the fixed points and the GKM theorem allows us

to compute this subring by only using the data of 1-dimensional

orbits. The results in the first part of this talk are a

generalization of this technique to Hamiltonian R actions on orbifolds

and an application to the computation of the equivariant cohomology of

toric orbifolds. In the second part, we will introduce the equivariant

Chen-Ruan cohomology ring which is a symplectic invariant of the

action on the orbifold and explain the injectivity/GKM theorem for this ring.

When a symplectic manifold M carries a Hamiltonian torus R action, the

injectivity theorem states that the R-equivariant cohomology of M is a

subring of the one of the fixed points and the GKM theorem allows us

to compute this subring by only using the data of 1-dimensional

orbits. The results in the first part of this talk are a

generalization of this technique to Hamiltonian R actions on orbifolds

and an application to the computation of the equivariant cohomology of

toric orbifolds. In the second part, we will introduce the equivariant

Chen-Ruan cohomology ring which is a symplectic invariant of the

action on the orbifold and explain the injectivity/GKM theorem for this ring.

### 2010/11/25

#### Operator Algebra Seminars

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

Classification of actions of Kac algebras (JAPANESE)

**Reiji Tomatsu**(Tokyo Univ. Science)Classification of actions of Kac algebras (JAPANESE)

### 2010/11/18

#### Operator Algebra Seminars

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

Perturbation of dual operator algebras and similarity (ENGLISH)

**Jean Roydor**(Univ. Tokyo)Perturbation of dual operator algebras and similarity (ENGLISH)

### 2010/11/17

#### Number Theory Seminar

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

Fast lattice reduction for F_2-linear pseudorandom number generators (JAPANESE)

**Shin Harase**(University of Tokyo)Fast lattice reduction for F_2-linear pseudorandom number generators (JAPANESE)

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