## Seminar information archive

Seminar information archive ～12/08｜Today's seminar 12/09 | Future seminars 12/10～

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Braids and hyperbolic 3-manifolds from simple mixing devices (JAPANESE)

**Eiko Kin**(Osaka University)Braids and hyperbolic 3-manifolds from simple mixing devices (JAPANESE)

[ Abstract ]

Taffy pullers are devices for pulling candy. One can build braids from the motion of rods for taffy pullers. According to a beautiful article ``A mathematical history of taffy pullers" by Jean-Luc Thiffeault, all taffy pullers (except the first one) give rise to pseudo-Anosov braids. This means that the devices mix candies effectively. Following a study of Thiffeault, I will discuss which pseudo-Anosov braid is realized by taffy pullers. I will explain an interesting connection between braids coming from taffy pullers. I also discuss the hyperbolic mapping tori obtained from taffy pullers. Intriguingly, the two most common taffy pullers give rise to the complements of the the minimally twisted 4-chain link and 5-chain link which are important examples for the study of cusped hyperbolic 3-manifolds with small volumes.

Reference: A mathematical history of taffy pullers, Jean-Luc Thiffeault, https://arxiv.org/pdf/1608.00152.pdf

Taffy pullers are devices for pulling candy. One can build braids from the motion of rods for taffy pullers. According to a beautiful article ``A mathematical history of taffy pullers" by Jean-Luc Thiffeault, all taffy pullers (except the first one) give rise to pseudo-Anosov braids. This means that the devices mix candies effectively. Following a study of Thiffeault, I will discuss which pseudo-Anosov braid is realized by taffy pullers. I will explain an interesting connection between braids coming from taffy pullers. I also discuss the hyperbolic mapping tori obtained from taffy pullers. Intriguingly, the two most common taffy pullers give rise to the complements of the the minimally twisted 4-chain link and 5-chain link which are important examples for the study of cusped hyperbolic 3-manifolds with small volumes.

Reference: A mathematical history of taffy pullers, Jean-Luc Thiffeault, https://arxiv.org/pdf/1608.00152.pdf

### 2017/06/26

#### Seminar on Geometric Complex Analysis

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

Volume minimization and obstructions to geometric problems

**Akito Futaki**(The University of Tokyo)Volume minimization and obstructions to geometric problems

[ Abstract ]

We discuss on the volume minimization principle for conformally Kaehler Einstein-Maxwell metrics in the similar spirit as the Kaehler-Ricci solitons and Sasaki-Einstein metrics. This talk is base on a joint work with Hajime Ono.

We discuss on the volume minimization principle for conformally Kaehler Einstein-Maxwell metrics in the similar spirit as the Kaehler-Ricci solitons and Sasaki-Einstein metrics. This talk is base on a joint work with Hajime Ono.

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Dimension, comparison, and almost finiteness (English)

**David Kerr**(Texas A & M Univ.)Dimension, comparison, and almost finiteness (English)

### 2017/06/20

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Introduction to the AJ conjecture (ENGLISH)

**Anh Tran**(The University of Texas at Dallas)Introduction to the AJ conjecture (ENGLISH)

[ Abstract ]

The AJ conjecture was proposed by Garoufalidis about 15 years ago. It predicts a strong connection between two important knot invariants derived from very different background, namely the colored Jones function (a quantum invariant) and the A-polynomial (a geometric invariant). The colored Jones function is a sequence of Laurent polynomials which is known to satisfy a linear q-difference equation. The AJ conjecture states that by writing the linear q-difference equation into an operator form and setting q=1, one gets the A-polynomial. In this talk, I will give an introduction to this conjecture.

The AJ conjecture was proposed by Garoufalidis about 15 years ago. It predicts a strong connection between two important knot invariants derived from very different background, namely the colored Jones function (a quantum invariant) and the A-polynomial (a geometric invariant). The colored Jones function is a sequence of Laurent polynomials which is known to satisfy a linear q-difference equation. The AJ conjecture states that by writing the linear q-difference equation into an operator form and setting q=1, one gets the A-polynomial. In this talk, I will give an introduction to this conjecture.

#### Colloquium

15:30-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Some stochastic population models in a random environment (English)

http://www.ummisco.ird.fr/perso/bacaer/

**Nicolas Bacaër**(Institute de Resherrche pour le Developpement, the University of Tokyo)Some stochastic population models in a random environment (English)

[ Abstract ]

Two population models will be considered: an epidemic model [1] and a linear birth-and-death process [2]. The goal is to study the first non-zero eigenvalue, which is related to the speed of convergence towards extinction, using either WKB approximations or probabilistic arguments.

[1] "Le modèle stochastique SIS pour une épidémie dans un environnement aléatoire". Journal of Mathematical Biology (2016)

[2] "Sur les processus linéaires de naissance et de mort sous-critiques dans un environnement aléatoire". Journal of Mathematical Biology (2017)

[ Reference URL ]Two population models will be considered: an epidemic model [1] and a linear birth-and-death process [2]. The goal is to study the first non-zero eigenvalue, which is related to the speed of convergence towards extinction, using either WKB approximations or probabilistic arguments.

[1] "Le modèle stochastique SIS pour une épidémie dans un environnement aléatoire". Journal of Mathematical Biology (2016)

[2] "Sur les processus linéaires de naissance et de mort sous-critiques dans un environnement aléatoire". Journal of Mathematical Biology (2017)

http://www.ummisco.ird.fr/perso/bacaer/

### 2017/06/19

#### Seminar on Geometric Complex Analysis

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

$Q$-prime curvature and Sasakian $\eta$-Einstein manifolds

**Yuya Takeuchi**(The University of Tokyo)$Q$-prime curvature and Sasakian $\eta$-Einstein manifolds

[ Abstract ]

The $Q$-prime curvature is defined for a pseudo-Einstein contact form on a strictly pseudoconvex CR manifold, and its integral, the total $Q$-prime curvature, defines a global CR invariant under some assumptions. In this talk, we will compute the $Q$-prime curvature for Sasakian $\eta$-Einstein manifolds. We will also study the first and the second variation of the total $Q$-prime curvature under deformations of real hypersurfaces at Sasakian $\eta$-Einstein manifolds.

The $Q$-prime curvature is defined for a pseudo-Einstein contact form on a strictly pseudoconvex CR manifold, and its integral, the total $Q$-prime curvature, defines a global CR invariant under some assumptions. In this talk, we will compute the $Q$-prime curvature for Sasakian $\eta$-Einstein manifolds. We will also study the first and the second variation of the total $Q$-prime curvature under deformations of real hypersurfaces at Sasakian $\eta$-Einstein manifolds.

#### Tokyo Probability Seminar

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

Computation of first-order Greeks for barrier options using chain rules for Wiener path integrals (JAPANESE)

**Kensuke Ishitani**(Graduate School of Science and Engineering, Tokyo Metropolitan University)Computation of first-order Greeks for barrier options using chain rules for Wiener path integrals (JAPANESE)

[ Abstract ]

In this presentation, we present a new methodology to compute first-order Greeks for barrier options under the framework of path-dependent payoff functions with European, Lookback, or Asian type and with time-dependent trigger levels. In particular, we develop chain rules for Wiener path integrals between two curves that arise in the computation of first-order Greeks for barrier options. We also illustrate the effectiveness of our method through numerical examples.

In this presentation, we present a new methodology to compute first-order Greeks for barrier options under the framework of path-dependent payoff functions with European, Lookback, or Asian type and with time-dependent trigger levels. In particular, we develop chain rules for Wiener path integrals between two curves that arise in the computation of first-order Greeks for barrier options. We also illustrate the effectiveness of our method through numerical examples.

### 2017/06/14

#### Number Theory Seminar

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

Multiplicity one for the mod p cohomology of Shimura curves (ENGLISH)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~t-saito/title_Hu.pdf

**Yongquan Hu**(Chinese Academy of Sciences, Morningside Center of Mathematics)Multiplicity one for the mod p cohomology of Shimura curves (ENGLISH)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~t-saito/title_Hu.pdf

### 2017/06/13

#### Numerical Analysis Seminar

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Numerical analysis of viscoelastic fluid models (Japanese)

**Hirofumi Notsu**(Kanazawa University)Numerical analysis of viscoelastic fluid models (Japanese)

[ Abstract ]

Numerical methods for viscoelastic fluid models are studied. In viscoelastic fluid models the stress tensor is often written as a sum of the viscous stress tensor depending linearly on the strain rate tensor and the extra stress tensor for the viscoelastic contribution. In order to describe the viscoelastic contribution another equation for the extra stress tensor is required. In the talk we mainly deal with the Oldroyd-B and the Peterlin models among several proposed viscoelastic fluid models, and present error estimates of finite element schemes based on the method of characteristics. The key issue in the estimates is the treatment of the divergence of the extra stress tensor appearing in the equation for the velocity and the pressure.

Numerical methods for viscoelastic fluid models are studied. In viscoelastic fluid models the stress tensor is often written as a sum of the viscous stress tensor depending linearly on the strain rate tensor and the extra stress tensor for the viscoelastic contribution. In order to describe the viscoelastic contribution another equation for the extra stress tensor is required. In the talk we mainly deal with the Oldroyd-B and the Peterlin models among several proposed viscoelastic fluid models, and present error estimates of finite element schemes based on the method of characteristics. The key issue in the estimates is the treatment of the divergence of the extra stress tensor appearing in the equation for the velocity and the pressure.

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Local criteria for non-embeddability of Levi-flat manifolds (JAPANESE)

**Noboru Ogawa**(Tokai University)Local criteria for non-embeddability of Levi-flat manifolds (JAPANESE)

[ Abstract ]

In this talk, we consider the Levi-flat embedding problem. Barrett showed that a smooth Reeb foliation on S^3 cannot be realized as a Levi-flat hypersurface in any complex surfaces. To do this, he focused the relationship between the holonomy along the compact leaf and a system of its defining functions. We will show a new criterion for non-embeddability of Levi-flat manifolds. Our result is a higher dimensional analogue of Barrett's theorem. In particular, this enables us to weaken the compactness assumption. For this purpose, we pose a partial generalization of Ueda theory on the analytic neighborhood structure of complex hypersurfaces. This talk is based on a joint work with Takayuki Koike (Kyoto University).

In this talk, we consider the Levi-flat embedding problem. Barrett showed that a smooth Reeb foliation on S^3 cannot be realized as a Levi-flat hypersurface in any complex surfaces. To do this, he focused the relationship between the holonomy along the compact leaf and a system of its defining functions. We will show a new criterion for non-embeddability of Levi-flat manifolds. Our result is a higher dimensional analogue of Barrett's theorem. In particular, this enables us to weaken the compactness assumption. For this purpose, we pose a partial generalization of Ueda theory on the analytic neighborhood structure of complex hypersurfaces. This talk is based on a joint work with Takayuki Koike (Kyoto University).

### 2017/06/12

#### Seminar on Geometric Complex Analysis

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

On Sp(2)-invariant asymptotically complex hyperbolic Einstein metrics on the 8-ball

**Yoshihiko Matsumoto**(Osaka University)On Sp(2)-invariant asymptotically complex hyperbolic Einstein metrics on the 8-ball

[ Abstract ]

Following a pioneering work of Pedersen, Hitchin studied SU(2)-invariant asymptotically real/complex hyperbolic (often abbreviated as AH/ACH) solution to the Einstein equation on the 4-dimensional unit open ball. We discuss a similar problem on the 8-ball, on which the quaternionic unitary group Sp(2) acts naturally, focusing on ACH solutions rather than AH ones. The Einstein equation is treated as an asymptotic Dirichlet problem, and the Dirichlet data are Sp(2)-invariant “partially integrable” CR structures on the 7-sphere. A remarkable point is that most of such structures are actually non-integrable. I will present how we can practically compute the formal series expansion of the Einstein ACH metric corresponding to a given Dirichlet data, that is, an invariant partially integrable CR structure on the sphere.

Following a pioneering work of Pedersen, Hitchin studied SU(2)-invariant asymptotically real/complex hyperbolic (often abbreviated as AH/ACH) solution to the Einstein equation on the 4-dimensional unit open ball. We discuss a similar problem on the 8-ball, on which the quaternionic unitary group Sp(2) acts naturally, focusing on ACH solutions rather than AH ones. The Einstein equation is treated as an asymptotic Dirichlet problem, and the Dirichlet data are Sp(2)-invariant “partially integrable” CR structures on the 7-sphere. A remarkable point is that most of such structures are actually non-integrable. I will present how we can practically compute the formal series expansion of the Einstein ACH metric corresponding to a given Dirichlet data, that is, an invariant partially integrable CR structure on the sphere.

#### Algebraic Geometry Seminar

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Rational and irrational singular quartic threefolds (English)

**Ivan Cheltsov**(The University of Edinburgh)Rational and irrational singular quartic threefolds (English)

[ Abstract ]

Burkhardt and Igusa quartics admit a faithful action of the symmetric group of degree 6.

There are other quartic threefolds with this property. All of them are singular.

Beauville proved that all but four of them are irrational. Burkhardt and Igusa quartics are known to be rational.

Two constructions of Todd imply the rationality of the remaining two quartic threefolds.

In this talk, I will give an alternative proof of both these (irrationality and rationality) results.

This proof is based on explicit small resolutions of the so-called Coble fourfold.

This fourfold is the double cover of the four-dimensional projective space branched over Igusa quartic.

This is a joint work with Sasha Kuznetsov and Costya Shramov.

Burkhardt and Igusa quartics admit a faithful action of the symmetric group of degree 6.

There are other quartic threefolds with this property. All of them are singular.

Beauville proved that all but four of them are irrational. Burkhardt and Igusa quartics are known to be rational.

Two constructions of Todd imply the rationality of the remaining two quartic threefolds.

In this talk, I will give an alternative proof of both these (irrationality and rationality) results.

This proof is based on explicit small resolutions of the so-called Coble fourfold.

This fourfold is the double cover of the four-dimensional projective space branched over Igusa quartic.

This is a joint work with Sasha Kuznetsov and Costya Shramov.

### 2017/06/06

#### Algebraic Geometry Seminar

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

Fano varieties: K-stability and boundedness (English)

https://sites.google.com/site/chenjiangmath/

**Chen Jiang**(IPMU)Fano varieties: K-stability and boundedness (English)

[ Abstract ]

There are two interesting problems for Fano varieties, K-stability and boundedness.

Significant progress has been made for both problems recently.

In this talk, I will show the boundedness of K-semistable Fano varieties with anti-canonical degree bounded from below, by using methods from birational geometry.

[ Reference URL ]There are two interesting problems for Fano varieties, K-stability and boundedness.

Significant progress has been made for both problems recently.

In this talk, I will show the boundedness of K-semistable Fano varieties with anti-canonical degree bounded from below, by using methods from birational geometry.

https://sites.google.com/site/chenjiangmath/

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

A formula for the action of Dehn twists on the HOMFLY-PT type skein algebra and its application (JAPANESE)

**Shunsuke Tsuji**(The University of Tokyo)A formula for the action of Dehn twists on the HOMFLY-PT type skein algebra and its application (JAPANESE)

[ Abstract ]

We give an explicit formula for the action of the Dehn twist along a simple closed curve of a surface on the completed HOMFLY-PT type skein modules of the surface in terms of the action of the completed HOMFLY-PT type skein algebra of the surface. As an application, using this formula, we construct an invariant for an integral homology 3-sphere which is an element of Q[ρ] [[h]].

We give an explicit formula for the action of the Dehn twist along a simple closed curve of a surface on the completed HOMFLY-PT type skein modules of the surface in terms of the action of the completed HOMFLY-PT type skein algebra of the surface. As an application, using this formula, we construct an invariant for an integral homology 3-sphere which is an element of Q[ρ] [[h]].

### 2017/06/01

#### Classical Analysis

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

Homological and monodromy representations of framed braid groups

(JAPANESE)

**Akishi Ikeda**(IPMU, University of Tokyo)Homological and monodromy representations of framed braid groups

(JAPANESE)

### 2017/05/31

#### Number Theory Seminar

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stark Systems over Gorenstein Rings (JAPANESE)

**Ryotaro Sakamoto**(University of Tokyo)Stark Systems over Gorenstein Rings (JAPANESE)

### 2017/05/30

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On a conjecture of Dunfield, Friedl and Jackson for hyperbolic knots (JAPANESE)

**Takayuki Morifuji**(Keio University)On a conjecture of Dunfield, Friedl and Jackson for hyperbolic knots (JAPANESE)

[ Abstract ]

The hyperbolic torsion polynomial is defined to be the twisted Alexander polynomial associated to the holonomy representation of a hyperbolic knot. Dunfield, Friedl and Jackson conjecture that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. In this talk we will survey recent results on the conjecture and explain its generalization to hyperbolic links.

The hyperbolic torsion polynomial is defined to be the twisted Alexander polynomial associated to the holonomy representation of a hyperbolic knot. Dunfield, Friedl and Jackson conjecture that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. In this talk we will survey recent results on the conjecture and explain its generalization to hyperbolic links.

#### Infinite Analysis Seminar Tokyo

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

$Q$-functions associated to the root system of type $C$ (JAPANESE)

**Soichi Okada**(Graduate School of Mathematics, Nagoya University)$Q$-functions associated to the root system of type $C$ (JAPANESE)

[ Abstract ]

Schur $Q$-functions are a family of symmetric functions introduced

by Schur in his study of projective representations of symmetric

groups. They are obtained by putting $t=-1$ in the Hall-Littlewood

functions associated to the root system of type $A$. (Schur

functions are the $t=0$ specialization.) This talk concerns

symplectic $Q$-functions, which are obtained by putting $t=-1$

in the Hall-Littlewood functions associated to the root system

of type $C$. We discuss several Pfaffian identities as well

as a combinatorial description for them. Also we present some

positivity conjectures.

Schur $Q$-functions are a family of symmetric functions introduced

by Schur in his study of projective representations of symmetric

groups. They are obtained by putting $t=-1$ in the Hall-Littlewood

functions associated to the root system of type $A$. (Schur

functions are the $t=0$ specialization.) This talk concerns

symplectic $Q$-functions, which are obtained by putting $t=-1$

in the Hall-Littlewood functions associated to the root system

of type $C$. We discuss several Pfaffian identities as well

as a combinatorial description for them. Also we present some

positivity conjectures.

#### Algebraic Geometry Seminar

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

Contractible affine threefolds in smooth Fano threefolds (English or Japanese)

**Masaru Nagaoka**(The University of Tokyo)Contractible affine threefolds in smooth Fano threefolds (English or Japanese)

[ Abstract ]

By the contribution of M. Furushima, N. Nakayama, Th. Peternell and M.

Schneider, it is completed to classify all projective compactifications

of the affine $3$-space $\mathbb{A}^3$ with Picard number one.

As a similar question, T. Kishimoto raised the problem to classify all

triplets $(V, U, D_1 \cup D_2)$ which consist of smooth Fano threefolds

$V$ of Picard number two, contractible affine threefolds $U$ as open

subsets of $V$, and the complements $D_1 \cup D_2 =V \setminus U$.

He also solved this problem when the log canonical divisors $K_V+D_1+D_2

$ are not nef.

In this talk, I will discuss the triplets $(V, U, D_1 \cup D_2)$ whose

log canonical divisors are linearly equivalent to zero.

I will also explain how to determine all Fano threefolds $V$ which

appear in such triplets.

By the contribution of M. Furushima, N. Nakayama, Th. Peternell and M.

Schneider, it is completed to classify all projective compactifications

of the affine $3$-space $\mathbb{A}^3$ with Picard number one.

As a similar question, T. Kishimoto raised the problem to classify all

triplets $(V, U, D_1 \cup D_2)$ which consist of smooth Fano threefolds

$V$ of Picard number two, contractible affine threefolds $U$ as open

subsets of $V$, and the complements $D_1 \cup D_2 =V \setminus U$.

He also solved this problem when the log canonical divisors $K_V+D_1+D_2

$ are not nef.

In this talk, I will discuss the triplets $(V, U, D_1 \cup D_2)$ whose

log canonical divisors are linearly equivalent to zero.

I will also explain how to determine all Fano threefolds $V$ which

appear in such triplets.

### 2017/05/29

#### Seminar on Geometric Complex Analysis

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

LCK structures on compact solvmanifolds

**Hiroshi Sawai**(National Institute of Technology, Numazu College)LCK structures on compact solvmanifolds

[ Abstract ]

A locally conformal Kähler (in short LCK) manifold is said to be Vaisman if Lee form is parallel with respect to Levi-Civita connection. In this talk, we prove that a Vaisman structure on a compact solvmanifolds depends only on the form of the fundamental 2-form, and it do not depends on a complex structure. As an application, we give the structure theorem for Vaisman (completely solvable) solvmanifolds and LCK nilmanifolds. Moreover, we show the existence of LCK solvmanifolds without Vaisman structures.

A locally conformal Kähler (in short LCK) manifold is said to be Vaisman if Lee form is parallel with respect to Levi-Civita connection. In this talk, we prove that a Vaisman structure on a compact solvmanifolds depends only on the form of the fundamental 2-form, and it do not depends on a complex structure. As an application, we give the structure theorem for Vaisman (completely solvable) solvmanifolds and LCK nilmanifolds. Moreover, we show the existence of LCK solvmanifolds without Vaisman structures.

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

On fundamental groups of tensor product II$_1$ factors (English)

**Yusuke Isono**(RIMS, Kyoto Univ.)On fundamental groups of tensor product II$_1$ factors (English)

#### Tokyo Probability Seminar

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

The cardinality of infinite geodesics originating from zero in First Passage Percolation (JAPANESE)

**Shuta Nakajima**(Research Institute for Mathematical Sciences, Kyoto University)The cardinality of infinite geodesics originating from zero in First Passage Percolation (JAPANESE)

### 2017/05/26

#### Colloquium

15:30-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

ループ空間上のスペクトルギャップの漸近挙動について (JAPANESE)

**Shigeki Aida**(Graduate School of Mathematical Sciences, The University of Tokyo)ループ空間上のスペクトルギャップの漸近挙動について (JAPANESE)

### 2017/05/23

#### Tuesday Seminar on Topology

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

Johnson homomorphisms, stable and unstable (ENGLISH)

**Richard Hain**(Duke University)Johnson homomorphisms, stable and unstable (ENGLISH)

[ Abstract ]

In this talk I will recall how motivic structures (Hodge and/or Galois) on the relative completions of mapping class groups yield non-trivial information about Johnson homomorphisms. I will explain how these motivic structures can be pasted, and why I believe that the genus 1 case is of fundamental importance in studying the higher genus (stable) case.

In this talk I will recall how motivic structures (Hodge and/or Galois) on the relative completions of mapping class groups yield non-trivial information about Johnson homomorphisms. I will explain how these motivic structures can be pasted, and why I believe that the genus 1 case is of fundamental importance in studying the higher genus (stable) case.

#### Algebraic Geometry Seminar

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

Perverse coherent sheaves on blow-ups at codimension two loci (English)

**Naoki Koseki**(The University of Tokyo)Perverse coherent sheaves on blow-ups at codimension two loci (English)

[ Abstract ]

I would like to talk about my recent work in progress.

Let us consider the blow-up X of Y along a subvariety C.

Then the following natural question arises:

What is the relation between moduli space of sheaves on Y

and that of X?

H.Nakajima and K.Yoshioka answered the above question

in the case when Y is a surface and C is a point. They

showed that the moduli spaces are connected by a sequence

of flip-like diagrams. The key ingredient of the proof is

to use perverse coherent sheaves in the sense of T.Bridgeland

and M.Van den Bergh.

In this talk, I will explain how to generalize their theorem

to the case when Y is a smooth projective variety of arbitrary

dimension and C is its codimension two subvariety.

I would like to talk about my recent work in progress.

Let us consider the blow-up X of Y along a subvariety C.

Then the following natural question arises:

What is the relation between moduli space of sheaves on Y

and that of X?

H.Nakajima and K.Yoshioka answered the above question

in the case when Y is a surface and C is a point. They

showed that the moduli spaces are connected by a sequence

of flip-like diagrams. The key ingredient of the proof is

to use perverse coherent sheaves in the sense of T.Bridgeland

and M.Van den Bergh.

In this talk, I will explain how to generalize their theorem

to the case when Y is a smooth projective variety of arbitrary

dimension and C is its codimension two subvariety.

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