## Tuesday Seminar on Topology

Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) KOHNO Toshitake, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya Tea: 16:30 - 17:00 Common Room

Seminar information archive

### 2015/06/30

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Makoto Sakuma (Hiroshima University)
The Cannon-Thurston maps and the canonical decompositions of punctured surface bundles over the circle (JAPANESE)
[ Abstract ]
To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy,
there are associated two tessellations of the complex plane:
one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra,
and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group.
In a joint work with Warren Dicks, I had described the relation between these two tessellations.
This result was recently generalized by Francois Gueritaud to punctured surface bundles
with pseudo-Anosov monodromy where all singuraities of the invariant foliations are at punctures.
In this talk, I will explain Gueritaud's work and related work by Naoki Sakata.

### 2015/06/23

17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Takahiro Matsushita (The University of Tokyo)
Box complexes and model structures on the category of graphs (JAPANESE)
[ Abstract ]
To determine the chromatic numbers of graphs, so-called the graph
coloring problem, is one of the most classical problems in graph theory.
Box complex is a Z_2-space associated to a graph, and it is known that
its equivariant homotopy invariant is related to the chromatic number.

Csorba showed that for each finite Z_2-CW-complex X, there is a graph
whose box complex is Z_2-homotopy equivalent to X. From this result, I
expect that the usual model category of Z_2-topological spaces is
Quillen equivalent to a certain model structure on the category of
graphs, whose weak equivalences are graph homomorphisms inducing Z_2-
homotopy equivalences between their box complexes.

In this talk, we introduce model structures on the category of graphs
whose weak equivalences are described as above. We also compare our
model categories of graphs with the category of Z_2-topological spaces.

### 2015/06/16

17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Masaharu Ishikawa (Tohoku University)
Stable maps and branched shadows of 3-manifolds (JAPANESE)
[ Abstract ]
We study what kind of stable map to the real plane a 3-manifold has. It
is known by O. Saeki that there exists a stable map without certain
singular fibers if and only if the 3-manifold is a graph manifold. According to
F. Costantino and D. Thurston, we identify the Stein factorization of a
stable map with a shadow of the 3-manifold under some modification,
where the above singular fibers correspond to the vertices of the shadow. We
define the notion of stable map complexity by counting the number of
such singular fibers and prove that this equals the branched shadow
complexity. With this equality, we give an estimation of the Gromov norm of the
3-manifold by the stable map complexity. This is a joint work with Yuya Koda.

### 2015/06/09

17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Manabu Akaho (Tokyo Metropolitan University)
Symplectic displacement energy for exact Lagrangian immersions (JAPANESE)
[ Abstract ]
We give an inequality of the displacement energy for exact Lagrangian
immersions and the symplectic area of punctured holomorphic discs. Our
approach is based on Floer homology for Lagrangian immersions and
Chekanov's homotopy technique of continuations. Moreover, we discuss our
inequality and the Hofer--Zehnder capacity.

### 2015/05/26

17:00-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Ken'ichi Kuga (Chiba University)
Introduction to formalization of topology using a proof assistant. (JAPANESE)
[ Abstract ]
Although the program of formalization goes back to David
Hilbert, it is only recently that we can actually formalize
substantial theorems in modern mathematics. It is made possible by the
development of certain type theory and a computer software called a
proof assistant. We begin this talk by showing our formalization of
some basic geometric topology using a proof assistant COQ. Then we
introduce homotopy type theory (HoTT) of Voevodsky et al., which
interprets type theory from abstract homotopy theoretic perspective.
HoTT proposes "univalent" foundation of mathematics which is
particularly suited for computer formalization.

### 2015/05/19

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

### 2015/05/12

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

### 2015/05/07

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

### 2015/04/28

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

### 2015/04/21

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

### 2015/04/14

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

### 2015/04/07

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

### 2015/03/24

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

### 2015/03/10

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

### 2015/01/20

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

### 2015/01/13

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

### 2014/12/16

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

### 2014/12/09

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

### 2014/12/02

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

### 2014/11/25

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

### 2014/11/18

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

### 2014/11/11

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

### 2014/11/04

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

### 2014/10/21

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

### 2014/10/07

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