Tuesday Seminar on Topology

Seminar information archive ~09/17Next seminarFuture seminars 09/18~

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

Seminar information archive

2014/07/22

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

2014/07/08

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

2014/07/01

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

2014/06/24

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

2014/06/17

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

2014/06/10

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yuka Kotorii (The University of Tokyo)
On relation between the Milnor's $¥mu$-invariant and HOMFLYPT
polynomial (JAPANESE)
[ Abstract ]
Milnor introduced a family of invariants for ordered oriented link,
called $¥bar{¥mu}$-invariants. Polyak showed a relation between the $¥
bar{¥mu}$-invariant of length 3 sequence and Conway polynomial.
Moreover, Habegger-Lin showed that Milnor's invariants are invariants of
string link, called $¥mu$-invariants. We show that any $¥mu$-invariant
of length $¥leq k$ can be represented as a combination of HOMFLYPT
polynomials if all $¥mu$-invariant of length $¥leq k-2$ vanish.
This result is an extension of Polyak's result.

2014/06/03

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Tatsuru Takakura (Chuo University)
Vector partition functions and the topology of multiple weight varieties
(JAPANESE)
[ Abstract ]
A multiple weight variety is a symplectic quotient of a direct product
of several coadjoint orbits of a compact Lie group $G$, with respect to
the diagonal action of the maximal torus. Its geometry and topology are
closely related to the combinatorics concerned with the weight space
decomposition of a tensor product of irreducible representations of $G$.
For example, when considering the Riemann-Roch index, we are naturally
lead to the study of vector partition functions with multiplicities.
In this talk, we discuss some formulas for vector partition functions,
especially a generalization of the formula of Brion-Vergne. Then, by
using
them, we investigate the structure of the cohomology of certain multiple
weight varieties of type $A$ in detail.

2014/05/27

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ege Fujikawa (Chiba University)
The Teichmuller space and the stable quasiconformal mapping class group for a Riemann surface of infinite type (JAPANESE)
[ Abstract ]
We explain recent developments of the theory of infinite dimensional Teichmuller space. In particular, we observe the dynamics of the orbits by the action of the stable quasiconformal mapping class group on the Teichmuller space and consider the relationship with the asymptotic Teichmuller space. We also introduce the generalized fixed point theorem and the Nielsen realization theorem. Furthermore, we investigate the moduli space of Riemann surface of infinite type.

2014/05/20

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Shintaro Kuroki (The Univeristy of Tokyo)
An application of torus graphs to characterize torus manifolds
with extended actions (JAPANESE)
[ Abstract ]
A torus manifold is a compact, oriented 2n-dimensional T^n-
manifolds with fixed points. This notion is introduced by Hattori and
Masuda as a topological generalization of toric manifolds. For a given
torus manifold, we can define a labelled graph called a torus graph (
this may be regarded as a generalization of some class of GKM graphs).
It is known that the equivariant cohomology ring of some nice class of
torus manifolds can be computed by using a combinatorial data of torus
graphs. In this talk, we study which torus action of torus manifolds can
be extended to a non-abelian compact connected Lie group. To do this, we
introduce root systems of (abstract) torus graphs and characterize
extended actions of torus manifolds. This is a joint work with Mikiya
Masuda.

2014/05/13

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Taro Asuke (The University of Tokyo)
Transverse projective structures of foliations and deformations of the Godbillon-Vey class (JAPANESE)
[ Abstract ]
Given a smooth family of foliations, we can define the derivative of the Godbillon-Vey class
with respect to the family. The derivative is known to be represented in terms of the projective
Schwarzians of holonomy maps. In this talk, we will study transverse projective structures
and connections, and show that the derivative is in fact determined by the projective structure
and the family.

2014/04/15

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Takahito Naito (The University of Tokyo)
On the rational string operations of classifying spaces and the
Hochschild cohomology (JAPANESE)
[ Abstract ]
Chataur and Menichi initiated the theory of string topology of
classifying spaces.
In particular, the cohomology of the free loop space of a classifying
space is endowed with a product
called the dual loop coproduct. In this talk, I will discuss the
algebraic structure and relate the rational dual loop coproduct to the
cup product on the Hochschild cohomology via the Van den Bergh isomorphism.

2014/04/08

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Hidetoshi Masai (The University of Tokyo)
On the number of commensurable fibrations on a hyperbolic 3-manifold. (JAPANESE)
[ Abstract ]
By work of Thurston, it is known that if a hyperbolic fibred
$3$-manifold $M$ has Betti number greater than 1, then
$M$ admits infinitely many distinct fibrations.
For any fibration $\\omega$ on a hyperbolic $3$-manifold $M$,
the number of fibrations on $M$ that are commensurable in the sense of
Calegari-Sun-Wang to $\\omega$ is known to be finite.
In this talk, we prove that the number can be arbitrarily large.

2014/01/21

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Naohiko Kasuya (The University of Tokyo)
On contact submanifolds of the odd dimensional Euclidean space (JAPANESE)
[ Abstract ]
We prove that the Chern class of a closed contact manifold is an
obstruction for codimension two contact embeddings in the odd
dimensional Euclidean space.
By Gromov's h-principle,
for any closed contact $3$-manifold with trivial first Chern class,
there is a contact structure on $\\mathbb{R}^5$ which admits a contact
embedding.

2014/01/21

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Xiaolong Li (The University of Tokyo)
Weak eigenvalues in homoclinic classes: perturbations from saddles
with small angles (ENGLISH)
[ Abstract ]
For 3-dimensional homoclinic classes of saddles with index 2, a
new sufficient condition for creating weak contracting eigenvalues is
provided. Our perturbation makes use of small angles between stable and
unstable subspaces of saddles. In particular, by recovering the unstable
eigenvector, we can designate that the newly created weak eigenvalue is
contracting. As applications, we obtain C^1-generic non-trivial index-
intervals of homoclinic classes and the C^1-approximation of robust
heterodimensional cycles. In particular, this sufficient condition is
satisfied by a substantial class of saddles with homoclinic tangencies.

2014/01/14

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Rinat Kashaev (University of Geneva)
State-integral partition functions on shaped triangulations (ENGLISH)
[ Abstract ]
Quantum Teichm\\"uller theory can be promoted to a
generalized TQFT within the combinatorial framework of shaped
triangulations with the tetrahedral weight functions given in
terms of the Weil-Gelfand-Zak transformation of Faddeev.FN"s
quantum dilogarithm. By using simple examples, I will
illustrate the connection of this theory with the hyperbolic
geometry in three dimensions.

2013/12/24

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Tirasan Khandhawit (Kavli IPMU)
Stable homotopy type for monopole Floer homology (ENGLISH)
[ Abstract ]
In this talk, I will try to give an overview of the
construction of stable homotopy type for monopole Floer homology. The
construction associates a stable homotopy object to 3-manifolds, which
will recover the Floer groups by appropriate homology theory. The main
ingredients are finite dimensional approximation technique and Conley
index theory. In addition, I will demonstrate construction for certain
3-manifolds such as the 3-torus.

2013/12/17

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Inasa Nakamura (The University of Tokyo)
Satellites of an oriented surface link and their local moves (JAPANESE)
[ Abstract ]
For an oriented surface link $F$ in $\\mathbb{R}^4$,
we consider a satellite construction of a surface link, called a
2-dimensional braid over $F$, which is in the form of a covering over
$F$. We introduce the notion of an $m$-chart on a surface diagram
$p(F)\\subset \\mathbb{R}^3$ of $F$, which is a finite graph on $p(F)$
satisfying certain conditions and is an extended notion of an
$m$-chart on a 2-disk presenting a surface braid.
A 2-dimensional braid over $F$ is presented by an $m$-chart on $p(F)$.
It is known that two surface links are equivalent if and only if their
surface diagrams are related by a finite sequence of ambient isotopies
of $\\mathbb{R}^3$ and local moves called Roseman moves.
We show that Roseman moves for surface diagrams with $m$-charts can be
well-defined. Further, we give some applications.

2013/12/10

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Motoo Tange (University of Tsukuba)
Corks, plugs, and local moves of 4-manifolds. (JAPANESE)
[ Abstract ]
Akbulut and Yasui defined cork, and plug
to produce many exotic pairs.
In this talk, we introduce a plug
with respect to Fintushel-Stern's knot surgery
or more 4-dimensional local moves and
and argue by using Heegaard Fleor theory.

2013/12/03

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Bruno Martelli (Univ. di Pisa)
Hyperbolic four-manifolds with one cusp (cancelled) (JAPANESE)
[ Abstract ]
(joint work with A. Kolpakov)

We introduce a simple algorithm which transforms every
four-dimensional cubulation into a cusped finite-volume hyperbolic
four-manifold. Combinatorially distinct cubulations give rise to
topologically distinct manifolds. Using this algorithm we construct
the first examples of finite-volume hyperbolic four-manifolds with one
cusp. More generally, we show that the number of k-cusped hyperbolic
four-manifolds with volume smaller than V grows like C^{V log V} for
any fixed k. As a corollary, we deduce that the 3-torus bounds
geometrically a hyperbolic manifold.

2013/11/26

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Hiroo Tokunaga (Tokyo Metropolitan University)
Rational elliptic surfaces and certain line-conic arrangements (JAPANESE)
[ Abstract ]
Let S be a rational elliptic surface. The generic
fiber of S can be considered as an elliptic curve over
the rational function field of one variable. We can make
use of its group structure in order to cook up a curve C_2 on
S from a given section C_1.
In this talk, we consider certain line-conic arrangements of
degree 7 based on this method.

2013/11/19

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Hiroki Kodama (The University of Tokyo)
Minimal $C^1$-diffeomorphisms of the circle which admit
measurable fundamental domains (JAPANESE)
[ Abstract ]
We construct, for each irrational number $\\alpha$, a minimal
$C^1$-diffeomorphism of the circle with rotation number $\\alpha$
which admits a measurable fundamental domain with respect to
the Lebesgue measure.
This is a joint work with Shigenori Matsumoto (Nihon University).

2013/11/12

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Alexander Voronov (University of Minnesota)
The Batalin-Vilkovisky Formalism and Cohomology of Moduli Spaces (ENGLISH)
[ Abstract ]
We use the Batalin-Vilkovisky formalism to give a new proof of Costello's theorem on the existence and uniqueness of solution to the Quantum Master Equation. We also make a physically motivated conjecture on the rational homology of moduli spaces. This is a joint work with Domenico D'Alessandro.

2013/11/05

16:30-18:00   Room #123 (Graduate School of Math. Sci. Bldg.)
Carlos Moraga Ferrandiz (The University of Tokyo, JSPS)
The isotopy problem of non-singular closed 1-forms. (ENGLISH)
[ Abstract ]
Given alpha_0, alpha_1 two cohomologous non-singular closed 1-forms of a compact manifold M, are they always isotopic? We expect a negative answer to this question, at least in high dimensions by the work of Laudenbach, as well as an obstruction living in the algebraic K-theory of the Novikov ring associated to the underlying cohomology class.
A similar problem for functions N x [0,1] --> [0,1] without critical points was treated by Hatcher and Wagoner in the 70s.

The first goal of this talk is to explain how we can carry a part of the strategy of Hatcher and Wagoner into the context of closed 1-forms and to indicate the main difficulties that appear by doing so. The second goal is to show the techniques to treat this difficulties and the progress in defining the expected obstruction.

2013/10/29

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Daniel Matei (IMAR, Bucharest)
Fundamental groups of algebraic varieties (ENGLISH)
[ Abstract ]
We discuss restrictions imposed by the complex
structure on fundamental groups of quasi-projective
algebraic varieties with mild singularities.
We investigate quasi-projectivity of various geometric
classes of finitely presented groups.

2013/10/22

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Rei Inoue (Chiba University)
Cluster algebra and complex volume of knots (JAPANESE)
[ Abstract ]
The cluster algebra was introduced by Fomin and Zelevinsky around
2000. The characteristic operation in the algebra called `mutation' is
related to various notions in mathematics and mathematical physics. In
this talk I review a basics of the cluster algebra, and introduce its
application to study the complex volume of knot complements in S^3.
Here a mutation corresponds to an ideal tetrahedron.
This talk is based on joint work with Kazuhiro Hikami (Kyushu University).

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