## Seminar information archive

Seminar information archive ～05/20｜Today's seminar 05/21 | Future seminars 05/22～

### 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.

#### Lectures

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

Enumeration of fully commutative elements in classical Coxeter groups (English)

http://math.univ-lyon1.fr/homes-www/jouhet/

**Frédéric Jouhet**(Université Claude Bernard Lyon 1 / Institut Camille Jordan)Enumeration of fully commutative elements in classical Coxeter groups (English)

[ Abstract ]

An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. They index naturally a basis of the (generalized) Temperley-Lieb algebra associated with W. In this talk, focusing on the (affine) type A, I will describe how to

enumerate these elements according to their Coxeter length, in all classical finite and affine Coxeter groups. The methods, which generalize previous work of Stembridge,

involve many combinatorial objects, such as heaps, walks, or parallelogram

polyominoes. This talk is based on joint works with R. Biagioli, M. Bousquet-Mélou and

P. Nadeau.

[ Reference URL ]An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. They index naturally a basis of the (generalized) Temperley-Lieb algebra associated with W. In this talk, focusing on the (affine) type A, I will describe how to

enumerate these elements according to their Coxeter length, in all classical finite and affine Coxeter groups. The methods, which generalize previous work of Stembridge,

involve many combinatorial objects, such as heaps, walks, or parallelogram

polyominoes. This talk is based on joint works with R. Biagioli, M. Bousquet-Mélou and

P. Nadeau.

http://math.univ-lyon1.fr/homes-www/jouhet/

### 2017/05/22

#### Seminar on Geometric Complex Analysis

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

Complex K3 surfaces containing Levi-flat hypersurfaces

**Takayuki Koike**(Kyoto University)Complex K3 surfaces containing Levi-flat hypersurfaces

[ Abstract ]

We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.

We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.

#### Operator Algebra Seminars

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

(English)

**Toshihiko Masuda**(Kyushu Univ.)(English)

#### Tokyo Probability Seminar

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

Compactness of Markov and Shcroedinger semigroups (JAPANESE)

**Yoshihiro Tawara**(National Institute of Technology, Nagaoka College)Compactness of Markov and Shcroedinger semigroups (JAPANESE)

### 2017/05/18

#### Seminar on Probability and Statistics

15:00-16:10 Room #117 (Graduate School of Math. Sci. Bldg.)

On a representation of fractional Brownian motion and the limit distributions of statistics arising in cusp statistical models

**Alexander A. Novikov**(University of Technology Sydney)On a representation of fractional Brownian motion and the limit distributions of statistics arising in cusp statistical models

[ Abstract ]

We discuss some extensions of results from the recent paper by Chernoyarov et al. (Ann. Inst. Stat. Math. October 2016) concerning limit distributions of Bayesian and maximum likelihood estimators in the model "signal plus white noise" with irregular cusp-type signals. Using a new representation of fractional Brownian motion (fBm) in terms of cusp functions we show that as the noise intensity tends to zero, the limit distributions are expressed in terms of fBm for the full range of asymmetric cusp-type signals correspondingly with the Hurst parameter H, 0＜H＜1. Simulation results for the densities and variances of the limit distributions of Bayesian and maximum likelihood estimators are also provided.

We discuss some extensions of results from the recent paper by Chernoyarov et al. (Ann. Inst. Stat. Math. October 2016) concerning limit distributions of Bayesian and maximum likelihood estimators in the model "signal plus white noise" with irregular cusp-type signals. Using a new representation of fractional Brownian motion (fBm) in terms of cusp functions we show that as the noise intensity tends to zero, the limit distributions are expressed in terms of fBm for the full range of asymmetric cusp-type signals correspondingly with the Hurst parameter H, 0＜H＜1. Simulation results for the densities and variances of the limit distributions of Bayesian and maximum likelihood estimators are also provided.

### 2017/05/17

#### Number Theory Seminar

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

The Equivariant Tamagawa Number Conjecture for modular motives with coefficients in Hecke algebras (ENGLISH)

**Olivier Fouquet**(Université Paris-Sud)The Equivariant Tamagawa Number Conjecture for modular motives with coefficients in Hecke algebras (ENGLISH)

[ Abstract ]

The Equivariant Tamagawa Number Conjecture (ETNC) of Kato is an awe-inspiring web of conjectures predicting the special values of L-functions of motives as well as their behaviors under the action of algebras acting on motives. In this talk, I will explain the statement of the ETNC with coefficients in Hecke algebras for motives attached to modular forms, show some consequences in Iwasawa theory and outline a proof (under mild hypotheses on the residual representation) using a combination of the methods of Euler and Taylor-Wiles systems.

The Equivariant Tamagawa Number Conjecture (ETNC) of Kato is an awe-inspiring web of conjectures predicting the special values of L-functions of motives as well as their behaviors under the action of algebras acting on motives. In this talk, I will explain the statement of the ETNC with coefficients in Hecke algebras for motives attached to modular forms, show some consequences in Iwasawa theory and outline a proof (under mild hypotheses on the residual representation) using a combination of the methods of Euler and Taylor-Wiles systems.

### 2017/05/16

#### Algebraic Geometry Seminar

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

On separable higher Gauss maps (English)

**Katsuhisa Furukawa**(The University of Tokyo)On separable higher Gauss maps (English)

[ Abstract ]

We study the $m$-th Gauss map in the sense of F. L. Zak of a projective variety $X ¥subset P^N$ over an algebraically closed field in any characteristic, where $m$ is an integer with $n:= ¥dim(X) ¥leq m < N$. It is known that the contact locus on $X$ of a general tangent $m$-plane can be non-linear in positive characteristic, if the $m$-th Gauss map is inseparable.

In this talk, I will explain that for any $m$, the locus is a linear variety if the $m$-th Gauss map is separable. I will also explain that for smooth $X$ with $n < N-2$, the $(n+1)$-th Gauss

map is birational if it is separable, unless $X$ is the Segre embedding $P^1 ¥times P^n ¥subset P^{2n-1}$. This is related to L. Ein's classification of varieties with small dual varieties in characteristic zero.

This talk is based on a joint work with Atsushi Ito.

We study the $m$-th Gauss map in the sense of F. L. Zak of a projective variety $X ¥subset P^N$ over an algebraically closed field in any characteristic, where $m$ is an integer with $n:= ¥dim(X) ¥leq m < N$. It is known that the contact locus on $X$ of a general tangent $m$-plane can be non-linear in positive characteristic, if the $m$-th Gauss map is inseparable.

In this talk, I will explain that for any $m$, the locus is a linear variety if the $m$-th Gauss map is separable. I will also explain that for smooth $X$ with $n < N-2$, the $(n+1)$-th Gauss

map is birational if it is separable, unless $X$ is the Segre embedding $P^1 ¥times P^n ¥subset P^{2n-1}$. This is related to L. Ein's classification of varieties with small dual varieties in characteristic zero.

This talk is based on a joint work with Atsushi Ito.

#### Tuesday Seminar on Topology

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

Twisted Alexander invariants and Hyperbolic volume of knots (JAPANESE)

**Hiroshi Goda**(Tokyo University of Agriculture and Technology)Twisted Alexander invariants and Hyperbolic volume of knots (JAPANESE)

[ Abstract ]

In [1], Müller investigated the asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, and then Menal-Ferrer and Porti [2] have obtained a formula on the volume of a hyperbolic 3-manifold using the Higher-dimensional Reidemeister torsion.

On the other hand, Yoshikazu Yamaguchi has shown that a relationship between the twisted Alexander polynomial and the Reidemeister torsion associated with the adjoint representation of the holonomy representation of a hyperbolic 3-manifold in his thesis [3].

In this talk, we observe that Yamaguchi's idea is applicable to the Higher-dimensional Reidemeister torsion, then we give a volume formula of a hyperbolic knot using the twisted Alexander polynomial.

References

[1] Müller, W., The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric and differential geometry, 317--352, Progr. Math., 297, Birkhäuser/Springer, Basel, 2012.

[2] Menal-Ferrer, P. and Porti, J., Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds. J. Topol., 7 (2014), no. 1, 69--119.

[3] Yamaguchi, Y., On the non-acyclic Reidemeister torsion for knots, Dissertation at the University of Tokyo, 2007.

In [1], Müller investigated the asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, and then Menal-Ferrer and Porti [2] have obtained a formula on the volume of a hyperbolic 3-manifold using the Higher-dimensional Reidemeister torsion.

On the other hand, Yoshikazu Yamaguchi has shown that a relationship between the twisted Alexander polynomial and the Reidemeister torsion associated with the adjoint representation of the holonomy representation of a hyperbolic 3-manifold in his thesis [3].

In this talk, we observe that Yamaguchi's idea is applicable to the Higher-dimensional Reidemeister torsion, then we give a volume formula of a hyperbolic knot using the twisted Alexander polynomial.

References

[1] Müller, W., The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric and differential geometry, 317--352, Progr. Math., 297, Birkhäuser/Springer, Basel, 2012.

[2] Menal-Ferrer, P. and Porti, J., Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds. J. Topol., 7 (2014), no. 1, 69--119.

[3] Yamaguchi, Y., On the non-acyclic Reidemeister torsion for knots, Dissertation at the University of Tokyo, 2007.

### 2017/05/15

#### Seminar on Geometric Complex Analysis

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

On the moduli spaces of the tangent cones at infinity of some hyper-Kähler manifolds

**Kota Hattori**(Keio University)On the moduli spaces of the tangent cones at infinity of some hyper-Kähler manifolds

[ Abstract ]

For a metric space $(X,d)$, the Gromov-Hausdorff limit of $(X, a_n d)$ as $a_n \rightarrow 0$ is called the tangent cone at infinity of $(X,d)$. Although the tangent cone at infinity always exists if $(X,d)$ comes from a complete Riemannian metric with nonnegative Ricci curvature, the uniqueness does not hold in general. Colding and Minicozzi showed the uniqueness under the assumption that $(X,d)$ is a Ricci-flat manifold satisfying some additional conditions.

In this talk, I will explain a example of noncompact complete hyper-Kähler manifold who has several tangent cones at infinity, and determine the moduli space of them.

For a metric space $(X,d)$, the Gromov-Hausdorff limit of $(X, a_n d)$ as $a_n \rightarrow 0$ is called the tangent cone at infinity of $(X,d)$. Although the tangent cone at infinity always exists if $(X,d)$ comes from a complete Riemannian metric with nonnegative Ricci curvature, the uniqueness does not hold in general. Colding and Minicozzi showed the uniqueness under the assumption that $(X,d)$ is a Ricci-flat manifold satisfying some additional conditions.

In this talk, I will explain a example of noncompact complete hyper-Kähler manifold who has several tangent cones at infinity, and determine the moduli space of them.

### 2017/05/11

#### Mathematical Biology Seminar

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

Environmental stochasticity and Heterogeniety in structured

population models ~Optimal life schedule in twofold stochasticity (JAPANESE)

**Ryo Oizumi**Environmental stochasticity and Heterogeniety in structured

population models ~Optimal life schedule in twofold stochasticity (JAPANESE)

### 2017/05/10

#### Number Theory Seminar

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

Wild ramification and restrictions to curves (JAPANESE)

**Hiroki Kato**(University of Tokyo)Wild ramification and restrictions to curves (JAPANESE)

### 2017/05/09

#### Tuesday Seminar on Topology

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

Local and global coincidence homology classes (JAPANESE)

**Tatsuo Suwa**(Hokkaido University)Local and global coincidence homology classes (JAPANESE)

[ Abstract ]

We consider two differentiable maps between two oriented manifolds. In the case the manifolds are compact with the same dimension and the coincidence points are isolated, there is the Lefschetz coincidence point formula, which generalizes his fixed point formula. In this talk we discuss the case where the dimensions of the manifolds may possible be different so that the coincidence points are not isolated in general. In fact it seems that Lefschetz himself already considered this case (cf. [4]).

We introduce the local and global coincidence homology classes and state a general coincidence point theorem.

We then give some explicit expressions for the local class. We also take up the case of several maps as considered in [1] and perform similar tasks. These all together lead to a generalization of the aforementioned Lefschetz formula. The key ingredients are the Alexander duality in combinatorial topology, intersection theory with maps and the Thom class in Čech-de Rham cohomology. The contents of the talk are in [2] and [3].

References

[1] C. Biasi, A.K.M. Libardi and T.F.M. Monis,

[2] C. Bisi, F. Bracci, T. Izawa and T. Suwa,

[3] J.-P. Brasselet and T. Suwa,

[4] N.E. Steenrod,

We consider two differentiable maps between two oriented manifolds. In the case the manifolds are compact with the same dimension and the coincidence points are isolated, there is the Lefschetz coincidence point formula, which generalizes his fixed point formula. In this talk we discuss the case where the dimensions of the manifolds may possible be different so that the coincidence points are not isolated in general. In fact it seems that Lefschetz himself already considered this case (cf. [4]).

We introduce the local and global coincidence homology classes and state a general coincidence point theorem.

We then give some explicit expressions for the local class. We also take up the case of several maps as considered in [1] and perform similar tasks. These all together lead to a generalization of the aforementioned Lefschetz formula. The key ingredients are the Alexander duality in combinatorial topology, intersection theory with maps and the Thom class in Čech-de Rham cohomology. The contents of the talk are in [2] and [3].

References

[1] C. Biasi, A.K.M. Libardi and T.F.M. Monis,

*The Lefschetz coincidence class of p maps*, Forum Math. 27 (2015), 1717-1728.[2] C. Bisi, F. Bracci, T. Izawa and T. Suwa,

*Localized intersection of currents and the Lefschetz coincidence point theorem*, Annali di Mat. Pura ed Applicata 195 (2016), 601-621.[3] J.-P. Brasselet and T. Suwa,

*Local and global coincidence homology classes*, arXiv:1612.02105.[4] N.E. Steenrod,

*The work and influence of Professor Lefschetz in algebraic topology*, Algebraic Geometry and Topology: A Symposium in Honor of Solomon Lefschetz, Princeton Univ. Press 1957, 24-43.< Previous 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135 Next >