## Tuesday Seminar on Topology

Seminar information archive ～09/26｜Next seminar｜Future seminars 09/27～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

**Seminar information archive**

### 2015/05/19

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

Quiver mutation loops and partition q-series (JAPANESE)

**Akishi Kato**(The University of Tokyo)Quiver mutation loops and partition q-series (JAPANESE)

[ Abstract ]

Quivers and their mutations are ubiquitous in mathematics and

mathematical physics; they play a key role in cluster algebras,

wall-crossing phenomena, gluing of ideal tetrahedra, etc.

Recently, we introduced a partition q-series for a quiver mutation loop

(a loop in a quiver exchange graph) using the idea of state sum of statistical

mechanics. The partition q-series enjoy some nice properties such

as pentagon move invariance. We also discuss their relation with combinatorial

Donaldson-Thomas invariants, as well as fermionic character formulas of

certain conformal field theories.

This is a joint work with Yuji Terashima.

Quivers and their mutations are ubiquitous in mathematics and

mathematical physics; they play a key role in cluster algebras,

wall-crossing phenomena, gluing of ideal tetrahedra, etc.

Recently, we introduced a partition q-series for a quiver mutation loop

(a loop in a quiver exchange graph) using the idea of state sum of statistical

mechanics. The partition q-series enjoy some nice properties such

as pentagon move invariance. We also discuss their relation with combinatorial

Donaldson-Thomas invariants, as well as fermionic character formulas of

certain conformal field theories.

This is a joint work with Yuji Terashima.

### 2015/05/12

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

Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)

**Masayuki Asaoka**(Kyoto University)Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)

[ Abstract ]

For any hyperbolic dynamical system, the number of periodic

points grows at most exponentially and the growth rate

reflects statistic property of the system. For dynamics far

from hyperbolicity, the situation is different. In 1999,

Kaloshin proved genericity of super-exponential growth in the

region where dense set of dynamical systems exhibits homoclinic

tangency (so called the Newhouse region).

How does the number of periodic points grow for generic

partially hyperbolic dynamical systems? Such systems are known

to be far from homoclinic tangency. Is the growth at most

exponential like hyperbolic system, or super-exponential by

a mechanism different from homoclinic tangency?

The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved

super-exponential growth of the number of periodic points for

generic one-dimensional iterated function systems under some

reasonable conditions. Such systems are models of dynamics

of partially hyperbolic systems in neutral direction. So, we

expect genericity of super-exponential growth in a region of

partially hyperbolic systems.

In this talk, we start with a brief history of the problem on

growth rate of the number of periodic point and discuss two

mechanisms which lead to genericity of super-exponential growth,

Kaloshin's one and ours.

For any hyperbolic dynamical system, the number of periodic

points grows at most exponentially and the growth rate

reflects statistic property of the system. For dynamics far

from hyperbolicity, the situation is different. In 1999,

Kaloshin proved genericity of super-exponential growth in the

region where dense set of dynamical systems exhibits homoclinic

tangency (so called the Newhouse region).

How does the number of periodic points grow for generic

partially hyperbolic dynamical systems? Such systems are known

to be far from homoclinic tangency. Is the growth at most

exponential like hyperbolic system, or super-exponential by

a mechanism different from homoclinic tangency?

The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved

super-exponential growth of the number of periodic points for

generic one-dimensional iterated function systems under some

reasonable conditions. Such systems are models of dynamics

of partially hyperbolic systems in neutral direction. So, we

expect genericity of super-exponential growth in a region of

partially hyperbolic systems.

In this talk, we start with a brief history of the problem on

growth rate of the number of periodic point and discuss two

mechanisms which lead to genericity of super-exponential growth,

Kaloshin's one and ours.

### 2015/05/07

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

The group of parenthesized braids (ENGLISH)

**Patrick Dehornoy**(Univ. de Caen)The group of parenthesized braids (ENGLISH)

[ Abstract ]

We describe a group B obtained by gluing in a natural way two well-known

groups, namely Artin's braid group B_infty and Thompson's group F. The

elements of B correspond to braid diagrams in which the distances

between the strands are non uniform and some rescaling operators may

change these distances. The group B shares many properties with B_infty:

as the latter, it can be realized as a subgroup of a mapping class

group, namely that of a sphere with a Cantor set removed, and as a group

of automorphisms of a free group. Technically, the key point is the

existence of a self-distributive operation on B.

We describe a group B obtained by gluing in a natural way two well-known

groups, namely Artin's braid group B_infty and Thompson's group F. The

elements of B correspond to braid diagrams in which the distances

between the strands are non uniform and some rescaling operators may

change these distances. The group B shares many properties with B_infty:

as the latter, it can be realized as a subgroup of a mapping class

group, namely that of a sphere with a Cantor set removed, and as a group

of automorphisms of a free group. Technically, the key point is the

existence of a self-distributive operation on B.

### 2015/04/28

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

Verify hyperbolicity of 3-manifolds by computer and its applications. (JAPANESE)

**Hidetoshi Masai**(The University of Tokyo, JSPS)Verify hyperbolicity of 3-manifolds by computer and its applications. (JAPANESE)

[ Abstract ]

In this talk I will talk about the program called HIKMOT which

rigorously proves hyperbolicity of a given triangulated 3-manifold. To

prove hyperbolicity of a given triangulated 3-manifold, it suffices to

get a solution of Thurston's gluing equation. We use the notion called

interval arithmetic to overcome two types errors; round-off errors,

and truncated errors. I will also talk about its application to

exceptional surgeries along alternating knots. This talk is based on

joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and

A. Takayasu.

In this talk I will talk about the program called HIKMOT which

rigorously proves hyperbolicity of a given triangulated 3-manifold. To

prove hyperbolicity of a given triangulated 3-manifold, it suffices to

get a solution of Thurston's gluing equation. We use the notion called

interval arithmetic to overcome two types errors; round-off errors,

and truncated errors. I will also talk about its application to

exceptional surgeries along alternating knots. This talk is based on

joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and

A. Takayasu.

### 2015/04/21

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

Orbit equivalence relations arising from Baumslag-Solitar groups (JAPANESE)

**Yoshikata Kida**(The University of Tokyo)Orbit equivalence relations arising from Baumslag-Solitar groups (JAPANESE)

[ Abstract ]

This talk is about measure-preserving actions of countable groups on probability

measure spaces and their orbit structure. Two such actions are called orbit equivalent

if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus

on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation

ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial

and geometric group theory. Whether Baumslag-Solitar groups with different p and q can

have orbit-equivalent actions is still a big open problem. I will discuss invariants under

orbit equivalence, motivating background and some results toward this problem.

This talk is about measure-preserving actions of countable groups on probability

measure spaces and their orbit structure. Two such actions are called orbit equivalent

if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus

on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation

ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial

and geometric group theory. Whether Baumslag-Solitar groups with different p and q can

have orbit-equivalent actions is still a big open problem. I will discuss invariants under

orbit equivalence, motivating background and some results toward this problem.

### 2015/04/14

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

Pin(2)-monopole invariants for 4-manifolds (JAPANESE)

**Nobuhiro Nakamura**(Gakushuin University)Pin(2)-monopole invariants for 4-manifolds (JAPANESE)

[ Abstract ]

The Pin(2)-monopole equations are a variant of the Seiberg-Witten equations

which can be considered as a real version of the SW equations. A Pin(2)-mono

pole version of the Seiberg-Witten invariants is defined, and a special feature of

this is that the Pin(2)-monopole invariant can be nontrivial even when all of

the Donaldson and Seiberg-Witten invariants vanish. As an application, we

construct a new series of exotic 4-manifolds.

The Pin(2)-monopole equations are a variant of the Seiberg-Witten equations

which can be considered as a real version of the SW equations. A Pin(2)-mono

pole version of the Seiberg-Witten invariants is defined, and a special feature of

this is that the Pin(2)-monopole invariant can be nontrivial even when all of

the Donaldson and Seiberg-Witten invariants vanish. As an application, we

construct a new series of exotic 4-manifolds.

### 2015/04/07

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

Potential functions for Grassmannians (JAPANESE)

**Kazushi Ueda**(The University of Tokyo)Potential functions for Grassmannians (JAPANESE)

[ Abstract ]

Potential functions are Floer-theoretic invariants

obtained by counting Maslov index 2 disks

with Lagrangian boundary conditions.

In the talk, we will discuss our joint work

with Yanki Lekili and Yuichi Nohara

on Lagrangian torus fibrations on the Grassmannian

of 2-planes in an n-space,

the potential functions of their Lagrangian torus fibers,

and their relation with mirror symmetry for Grassmannians.

Potential functions are Floer-theoretic invariants

obtained by counting Maslov index 2 disks

with Lagrangian boundary conditions.

In the talk, we will discuss our joint work

with Yanki Lekili and Yuichi Nohara

on Lagrangian torus fibrations on the Grassmannian

of 2-planes in an n-space,

the potential functions of their Lagrangian torus fibers,

and their relation with mirror symmetry for Grassmannians.

### 2015/03/24

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

Knots and Mirror Symmetry (ENGLISH)

**Mina Aganagic**(University of California, Berkeley)Knots and Mirror Symmetry (ENGLISH)

[ Abstract ]

I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.

I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.

### 2015/03/10

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

Arnold conjecture, Floer homology,

and augmentation ideals of finite groups.

(ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes)Arnold conjecture, Floer homology,

and augmentation ideals of finite groups.

(ENGLISH)

[ Abstract ]

Let H be a generic time-dependent 1-periodic

Hamiltonian on a closed weakly monotone

symplectic manifold M. We construct a refined version

of the Floer chain complex associated to (M,H),

and use it to obtain new lower bounds for the number P(H)

of the 1-periodic orbits of the corresponding hamiltonian

vector field. We prove in particular that

if the fundamental group of M is finite

and solvable or simple, then P(H)

is not less than the minimal number

of generators of the fundamental group.

This is joint work with Kaoru Ono.

Let H be a generic time-dependent 1-periodic

Hamiltonian on a closed weakly monotone

symplectic manifold M. We construct a refined version

of the Floer chain complex associated to (M,H),

and use it to obtain new lower bounds for the number P(H)

of the 1-periodic orbits of the corresponding hamiltonian

vector field. We prove in particular that

if the fundamental group of M is finite

and solvable or simple, then P(H)

is not less than the minimal number

of generators of the fundamental group.

This is joint work with Kaoru Ono.

### 2015/01/20

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

On Lagrangian caps and their applications (JAPANESE)

**Toru Yoshiyasu**(The University of Tokyo)On Lagrangian caps and their applications (JAPANESE)

[ Abstract ]

In 2013, Y. Eliashberg and E. Murphy established the $h$-principle for

exact Lagrangian embeddings with a concave Legendrian boundary. In this

talk, I will explain a modification of their $h$-principle and show

applications to Lagrangian submanifolds in the complex projective spaces.

In 2013, Y. Eliashberg and E. Murphy established the $h$-principle for

exact Lagrangian embeddings with a concave Legendrian boundary. In this

talk, I will explain a modification of their $h$-principle and show

applications to Lagrangian submanifolds in the complex projective spaces.

### 2015/01/13

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

Stable presentation length of 3-manifold groups (JAPANESE)

**Ken'ichi Yoshida**(The University of Tokyo)Stable presentation length of 3-manifold groups (JAPANESE)

[ Abstract ]

We will introduce the stable presentation length

of a finitely presented group, which is defined

by stabilizing the presentation length for the

finite index subgroups. The stable presentation

length of the fundamental group of a 3-manifold

is an analogue of the simplicial volume and the

stable complexity introduced by Francaviglia,

Frigerio and Martelli. We will explain some

similarities of stable presentation length with

simplicial volume and stable complexity.

We will introduce the stable presentation length

of a finitely presented group, which is defined

by stabilizing the presentation length for the

finite index subgroups. The stable presentation

length of the fundamental group of a 3-manifold

is an analogue of the simplicial volume and the

stable complexity introduced by Francaviglia,

Frigerio and Martelli. We will explain some

similarities of stable presentation length with

simplicial volume and stable complexity.

### 2014/12/16

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Differential forms in diffeological spaces (JAPANESE)

**Norio Iwase**(Kyushu University)Differential forms in diffeological spaces (JAPANESE)

[ Abstract ]

The idea of a space with smooth structure is first introduced by K. T. Chen in his study of a loop space to employ the idea of iterated path integrals.

Following the pattern established by Chen, J. M. Souriau introduced his version of a space with smooth structure which is now called diffeology and become one of the most exciting topics in Algebraic Topology. Following Souriau, P. I.-Zenmour presented de Rham theory associated to a diffeology of a space. However, if one tries to show a version of de Rham theorem for a general diffeological space, he must encounter a difficulty to show the existence of a partition of unity and thus the exactness of the Mayer-Vietoris sequence. To resolve such difficulties, we introduce a new definition of differential forms.

The idea of a space with smooth structure is first introduced by K. T. Chen in his study of a loop space to employ the idea of iterated path integrals.

Following the pattern established by Chen, J. M. Souriau introduced his version of a space with smooth structure which is now called diffeology and become one of the most exciting topics in Algebraic Topology. Following Souriau, P. I.-Zenmour presented de Rham theory associated to a diffeology of a space. However, if one tries to show a version of de Rham theorem for a general diffeological space, he must encounter a difficulty to show the existence of a partition of unity and thus the exactness of the Mayer-Vietoris sequence. To resolve such difficulties, we introduce a new definition of differential forms.

### 2014/12/09

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

Stable commutator length on mapping class groups (JAPANESE)

**Koji Fujiwara**(Kyoto University)Stable commutator length on mapping class groups (JAPANESE)

[ Abstract ]

Let MCG(S) be the mapping class group of a closed orientable surface S.

We give a precise condition (in terms of the Nielsen-Thurston

decomposition) when an element

in MCG(S) has positive stable commutator length.

Stable commutator length tends to be positive if there is "negative

curvature".

The proofs use our earlier construction in the paper "Constructing group

actions on quasi-trees and applications to mapping class groups" of

group actions on quasi-trees.

This is a joint work with Bestvina and Bromberg.

Let MCG(S) be the mapping class group of a closed orientable surface S.

We give a precise condition (in terms of the Nielsen-Thurston

decomposition) when an element

in MCG(S) has positive stable commutator length.

Stable commutator length tends to be positive if there is "negative

curvature".

The proofs use our earlier construction in the paper "Constructing group

actions on quasi-trees and applications to mapping class groups" of

group actions on quasi-trees.

This is a joint work with Bestvina and Bromberg.

### 2014/12/02

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

The Atiyah-Segal completion theorem in noncommutative topology (JAPANESE)

**Yosuke Kubota**(The University of Tokyo)The Atiyah-Segal completion theorem in noncommutative topology (JAPANESE)

[ Abstract ]

We introduce a new perspevtive on the Atiyah-Segal completion

theorem applying the "noncommutative" topology, which deals with

topological properties of C*-algebras. The homological algebra of the

Kasparov category as a triangulated category, which is developed by R.

Meyer and R. Nest, plays a central role. It contains the Atiyah-Segal

type completion theorems for equivariant K-homology and twisted K-theory.

This is a joint work with Yuki Arano.

We introduce a new perspevtive on the Atiyah-Segal completion

theorem applying the "noncommutative" topology, which deals with

topological properties of C*-algebras. The homological algebra of the

Kasparov category as a triangulated category, which is developed by R.

Meyer and R. Nest, plays a central role. It contains the Atiyah-Segal

type completion theorems for equivariant K-homology and twisted K-theory.

This is a joint work with Yuki Arano.

### 2014/11/25

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

Quandle knot invariants and applications (JAPANESE)

**Masahico Saito**(University of South Florida)Quandle knot invariants and applications (JAPANESE)

[ Abstract ]

A quandles is an algebraic structure closely related to knots. Homology theories of

quandles have been defined, and their cocycles are used to construct invariants for

classical knots, spatial graphs and knotted surfaces. In this talk, an overview is given

for quandle cocycle invariants and their applications to geometric properties of knots.

The current status of computations, recent developments and open problems will also

be discussed.

A quandles is an algebraic structure closely related to knots. Homology theories of

quandles have been defined, and their cocycles are used to construct invariants for

classical knots, spatial graphs and knotted surfaces. In this talk, an overview is given

for quandle cocycle invariants and their applications to geometric properties of knots.

The current status of computations, recent developments and open problems will also

be discussed.

### 2014/11/18

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

A Modular Operad of Embedded Curves (ENGLISH)

**Charles Siegel**(Kavli IPMU)A Modular Operad of Embedded Curves (ENGLISH)

[ Abstract ]

Modular operads were introduced by Getzler and Kapranov to formalize the structure of gluing maps between moduli of stable marked curves. We present a construction of analogous gluing maps between moduli of pluri-log-canonically embedded marked curves, which fit together to give a modular operad of embedded curves. This is joint work with Satoshi Kondo and Jesse Wolfson.

Modular operads were introduced by Getzler and Kapranov to formalize the structure of gluing maps between moduli of stable marked curves. We present a construction of analogous gluing maps between moduli of pluri-log-canonically embedded marked curves, which fit together to give a modular operad of embedded curves. This is joint work with Satoshi Kondo and Jesse Wolfson.

### 2014/11/11

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

Unifying unexpected exceptional Dehn surgeries (ENGLISH)

**Kenneth Baker**(University of Miami)Unifying unexpected exceptional Dehn surgeries (ENGLISH)

[ Abstract ]

This past summer Dunfield-Hoffman-Licata produced examples of asymmetric, hyperbolic, 1-cusped 3-manifolds with pairs of lens space Dehn fillings through a search of the extended SnapPea census.

Examinations of these examples with Hoffman and Licata lead us to coincidences with other work in progress that gives a simple holistic topological approach towards producing and extending many of these families. In this talk we'll explicitly describe our construction and discuss related applications of the technique.

This past summer Dunfield-Hoffman-Licata produced examples of asymmetric, hyperbolic, 1-cusped 3-manifolds with pairs of lens space Dehn fillings through a search of the extended SnapPea census.

Examinations of these examples with Hoffman and Licata lead us to coincidences with other work in progress that gives a simple holistic topological approach towards producing and extending many of these families. In this talk we'll explicitly describe our construction and discuss related applications of the technique.

### 2014/11/04

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

The coarse geometry of Teichmuller space. (ENGLISH)

**Brian Bowditch**(University of Warwick)The coarse geometry of Teichmuller space. (ENGLISH)

[ Abstract ]

We describe some results regarding the coarse geometry of the

Teichmuller space

of a compact surface. In particular, we describe when the Teichmuller

space admits quasi-isometric embeddings of euclidean spaces and

half-spaces.

We also give some partial results regarding the quasi-isometric rigidity

of Teichmuller space. These results are based on the fact that Teichmuller

space admits a ternary operation, natural up to bounded distance

which endows it with the structure of a coarse median space.

We describe some results regarding the coarse geometry of the

Teichmuller space

of a compact surface. In particular, we describe when the Teichmuller

space admits quasi-isometric embeddings of euclidean spaces and

half-spaces.

We also give some partial results regarding the quasi-isometric rigidity

of Teichmuller space. These results are based on the fact that Teichmuller

space admits a ternary operation, natural up to bounded distance

which endows it with the structure of a coarse median space.

### 2014/10/21

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

Vanishing theorems for p-local homology of Coxeter groups and their alternating subgroups (JAPANESE)

**Toshiyuki Akita**(Hokkaido University)Vanishing theorems for p-local homology of Coxeter groups and their alternating subgroups (JAPANESE)

[ Abstract ]

Given a prime number $p$, we estimate vanishing ranges of $p$-local homology groups of Coxeter groups (of possibly infinite order) and alternating subgroups of finite reflection groups. Our results generalize those by Nakaoka for symmetric groups and Kleshchev-Nakano and Burichenko for alternating groups. The key ingredient is the equivariant homology of Coxeter complexes.

Given a prime number $p$, we estimate vanishing ranges of $p$-local homology groups of Coxeter groups (of possibly infinite order) and alternating subgroups of finite reflection groups. Our results generalize those by Nakaoka for symmetric groups and Kleshchev-Nakano and Burichenko for alternating groups. The key ingredient is the equivariant homology of Coxeter complexes.

### 2014/10/07

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

Transversality problems in string topology and de Rham chains (JAPANESE)

**Kei Irie**(RIMS, Kyoto University)Transversality problems in string topology and de Rham chains (JAPANESE)

[ Abstract ]

The starting point of string topology is the work of Chas-Sullivan, which uncovered the Batalin-Vilkovisky(BV) structure on homology of the free loop space of a manifold.

It is important to define chain level structures beneath the BV structure on homology, however this problem is yet to be settled.

One of difficulties is that, to define intersection products on chain level, we have to address the transversality issue.

In this talk, we introduce a notion of "de Rham chain" to bypass this trouble, and partially realize expected chain level structures.

The starting point of string topology is the work of Chas-Sullivan, which uncovered the Batalin-Vilkovisky(BV) structure on homology of the free loop space of a manifold.

It is important to define chain level structures beneath the BV structure on homology, however this problem is yet to be settled.

One of difficulties is that, to define intersection products on chain level, we have to address the transversality issue.

In this talk, we introduce a notion of "de Rham chain" to bypass this trouble, and partially realize expected chain level structures.

### 2014/07/22

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

The Index Map and Reciprocity Laws for Contou-Carrere Symbols (ENGLISH)

**Jesse Wolfson**(Northwestern University)The Index Map and Reciprocity Laws for Contou-Carrere Symbols (ENGLISH)

[ Abstract ]

In the 1960s, Atiyah and Janich constructed the families index as a natural map from the space of Fredholm operators to the classifying space of topological K-theory, and showed it to be an equivalence. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. Building on recent work of Sho Saito, we show this provides an analogue of Atiyah and Janich's equivalence. More significantly, the index map allows us to relate the Contou-Carrere symbol, a local analytic invariant of schemes, to algebraic K-theory. Using this, we provide new proofs of reciprocity laws for Contou-Carrere symbols in dimension 1 (first established by Anderson--Pablos Romo) and 2 (established recently by Osipov--Zhu). We extend these reciprocity laws to all dimensions.

In the 1960s, Atiyah and Janich constructed the families index as a natural map from the space of Fredholm operators to the classifying space of topological K-theory, and showed it to be an equivalence. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. Building on recent work of Sho Saito, we show this provides an analogue of Atiyah and Janich's equivalence. More significantly, the index map allows us to relate the Contou-Carrere symbol, a local analytic invariant of schemes, to algebraic K-theory. Using this, we provide new proofs of reciprocity laws for Contou-Carrere symbols in dimension 1 (first established by Anderson--Pablos Romo) and 2 (established recently by Osipov--Zhu). We extend these reciprocity laws to all dimensions.

### 2014/07/08

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

The Johnson homomorphism and a family of curve graphs (ENGLISH)

**Ingrid Irmer**(JSPS, the University of Tokyo)The Johnson homomorphism and a family of curve graphs (ENGLISH)

[ Abstract ]

Abstract: A family of curve graphs of an oriented surface $S_{g,1}$ will be defined on which there exists a natural orientation, coming from the orientation of subsurfaces. Distances in these graphs represent commutator lengths in $\\pi_{1}(S_{g,1})$. The displacement of vertices in the graphs under the action of the Torelli group is used to give a combinatorial description of the Johnson homomorphism."

Abstract: A family of curve graphs of an oriented surface $S_{g,1}$ will be defined on which there exists a natural orientation, coming from the orientation of subsurfaces. Distances in these graphs represent commutator lengths in $\\pi_{1}(S_{g,1})$. The displacement of vertices in the graphs under the action of the Torelli group is used to give a combinatorial description of the Johnson homomorphism."

### 2014/07/01

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

Singularities of special Lagrangian submanifolds (JAPANESE)

**Yohsuke Imagi**(Kavli IPMU)Singularities of special Lagrangian submanifolds (JAPANESE)

[ Abstract ]

There are interesting invariants defined by "counting" geometric

objects, such as instantons in dimension 4 and pseudo-holomorphic curves

in symplectic manifolds. To do the counting in a sensible way, however,

we have to care about singularities of the geometric objects. Special

Lagrangian submanifolds seem very difficult to "count" as their

singularities may be very complicated. I'll talk about simple

singularities for which we can make an analogy with instantons in

dimension 4 and pseudo-holomorphic curves in symplectic manifolds. To do

it I'll use some techniques from geometric measure theory and Lagrangian

Floer theory, and the Floer-theoretic part is a joint work with Dominic

Joyce and Oliveira dos Santos.

There are interesting invariants defined by "counting" geometric

objects, such as instantons in dimension 4 and pseudo-holomorphic curves

in symplectic manifolds. To do the counting in a sensible way, however,

we have to care about singularities of the geometric objects. Special

Lagrangian submanifolds seem very difficult to "count" as their

singularities may be very complicated. I'll talk about simple

singularities for which we can make an analogy with instantons in

dimension 4 and pseudo-holomorphic curves in symplectic manifolds. To do

it I'll use some techniques from geometric measure theory and Lagrangian

Floer theory, and the Floer-theoretic part is a joint work with Dominic

Joyce and Oliveira dos Santos.

### 2014/06/24

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)

**Takefumi Nosaka**(Faculty of Mathematics, Kyushu University)On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)

[ Abstract ]

本講演では, 群とその群同型の組から定まるカンドルを扱い, 以下の結果を紹介

する. まず, その際Inoue-Kabaya鎖写像が, カンドルホモロジーから群ホモロ

ジーへの写像とし, 定式化される事を見る. 例えば, 有限体上のAlexander

quandleに対し, 望月3-コサイクル全ては, 当写像を通じ, 或る群コホモロジー

から導出され, 殆どがトリプルマッセイ積で解釈できる事をみる. 加えてカンド

ルの普遍中心拡大に対し, Inoue-Kabaya鎖写像が3次において(或る捩れ部分を

除き)同型となる. なお講演内容は当週にある集中講義の聴講を仮定しない.

本講演では, 群とその群同型の組から定まるカンドルを扱い, 以下の結果を紹介

する. まず, その際Inoue-Kabaya鎖写像が, カンドルホモロジーから群ホモロ

ジーへの写像とし, 定式化される事を見る. 例えば, 有限体上のAlexander

quandleに対し, 望月3-コサイクル全ては, 当写像を通じ, 或る群コホモロジー

から導出され, 殆どがトリプルマッセイ積で解釈できる事をみる. 加えてカンド

ルの普遍中心拡大に対し, Inoue-Kabaya鎖写像が3次において(或る捩れ部分を

除き)同型となる. なお講演内容は当週にある集中講義の聴講を仮定しない.

### 2014/06/17

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

Bounded Euler number of actions of 2-orbifold groups on the circle (JAPANESE)

**Yoshifumi Matsuda**(Aoyama Gakuin University)Bounded Euler number of actions of 2-orbifold groups on the circle (JAPANESE)

[ Abstract ]

Burger, Iozzi and Wienhard defined the bounded Euler number for a

continuous action of the fundamental group of a connected oriented

surface of finite type possibly with punctures on the circle. A Milnor-Wood

type inequality involving the bounded Euler number holds and its maximality

characterizes Fuchsian actions up to semiconjugacy. The definition of the

bounded Euler number can be extended to actions of 2-orbifold groups by

considering coverings. A Milnor-Wood type inequality and the characterization

of Fuchsian actions also hold in this case. In this talk, we describe when lifts

of Fuchsian actions of certain 2-orbifold groups, such as the modular group,

are characterized by its bounded Euler number.

Burger, Iozzi and Wienhard defined the bounded Euler number for a

continuous action of the fundamental group of a connected oriented

surface of finite type possibly with punctures on the circle. A Milnor-Wood

type inequality involving the bounded Euler number holds and its maximality

characterizes Fuchsian actions up to semiconjugacy. The definition of the

bounded Euler number can be extended to actions of 2-orbifold groups by

considering coverings. A Milnor-Wood type inequality and the characterization

of Fuchsian actions also hold in this case. In this talk, we describe when lifts

of Fuchsian actions of certain 2-orbifold groups, such as the modular group,

are characterized by its bounded Euler number.