Tuesday Seminar on Topology

Seminar information archive ~01/29Next seminarFuture seminars 01/30~

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


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.


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.


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.


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.


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.


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).


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.


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.


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.


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).


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Masamichi Takase (Seikei University)
Desingularizing special generic maps (JAPANESE)
[ Abstract ]
This is a joint work with Osamu Saeki (IMI, Kyushu University).
A special generic map is a generic map which has only definite
fold as its singularities.
We study the condition for a special generic map from a closed
n-manifold to the p-space (n+1>p), to factor through a codimension
one immersion (or an embedding). In particular, for the cases
where p = 1 and 2 we obtain complete results.
Our techniques are related to Smale-Hirsch theory,
topology of the space of immersions, relation between the space
of topological immersions and that of smooth immersions,
sphere eversions, differentiable structures of homotopy spheres,
diffeomorphism group of spheres, free group actions on the sphere, etc.


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Tatsuro Shimizu (The Univesity of Tokyo)
An invariant of rational homology 3-spheres via vector fields. (JAPANESE)
[ Abstract ]
In this talk, we define an invariant of rational homology 3-spheres with
values in a space $\\mathcal A(\\emptyset)$ of Jacobi diagrams by using
vector fields.
The construction of our invariant is a generalization of both that of
the Kontsevich-Kuperberg-Thurston invariant $z^{KKT}$
and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$.
As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for
integral homology 3-spheres.


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Naoyuki Monden (Tokyo University of Science)
The geography problem of Lefschetz fibrations (JAPANESE)
[ Abstract ]
To consider holomorphic fibrations complex surfaces over complex curves
and Lefschetz fibrations over surfaces is one method for the study of
complex surfaces of general type and symplectic 4-manifods, respectively.
In this talk, by comparing the geography problem of relatively minimal
holomorphic fibrations with that of relatively minimal Lefschetz
fibrations (i.e., the characterization of pairs $(x,y)$ of certain
invariants $x$ and $y$ corresponding to relatively minimal holomorphic
fibrations and relatively minimal Lefschetz fibrations), we observe the
difference between complex surfaces of general type and symplectic
4-manifolds. In particular, we construct Lefschetz fibrations violating
the ``slope inequality" which holds for any relatively minimal holomorphic


17:10-18:10   Room #056 (Graduate School of Math. Sci. Bldg.)
Sumio Yamada (Gakushuin University)
On new models of real hyperbolic spaces (JAPANESE)
[ Abstract ]
In this talk, I will introduce several new realization of the real hyperbolic spaces, using classical tools. The constructions will involve aspects of convex geometry as well as projective geometry, and they are interesting from the view point of the history of mathematics. This work belongs to a joint project with Athanase Papadopoulos.


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ryan Budney (University of Victoria)
Smooth 3-manifolds in the 4-sphere (ENGLISH)
[ Abstract ]
Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.


17:10-18:10   Room #056 (Graduate School of Math. Sci. Bldg.)
Tadayuki Watanabe (Shimane University)
Higher-order generalization of Fukaya's Morse homotopy
invariant of 3-manifolds (JAPANESE)
[ Abstract ]
In his article published in 1996, K. Fukaya constructed
a 3-manifold invariant by using Morse homotopy theory. Roughly, his
invariant is defined by considering several Morse functions on a
3-manifold and counting with weights the ways that the theta-graph can
be immersed such that edges follow gradient lines. We generalize his
construction to 3-valent graphs with arbitrary number of loops for
integral homology 3-spheres. I will also discuss extension of our method
to 3-manifolds with positive first Betti numbers.


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Kimihiko Motegi (Nihon University)
Left-orderable, non-L-space surgeries on knots (JAPANESE)
[ Abstract ]
A Dehn surgery is said to be left-orderable
if the resulting manifold of the surgery has the left-orderable fundamental group,
and a Dehn surgery is called an L-space surgery
if the resulting manifold of the surgery is an L-space.
We will focus on left-orderable, non-L-space surgeries on knots in the 3-sphere.
Once we have a knot with left-orderable surgeries,
the ``periodic construction" enables us to provide infinitely many knots with
left-orderable, non-L-space surgeries.
We apply the construction to present infinitely many hyperbolic knots on each
of which every nontrivial surgery is a left-orderable, non-L-space surgery.
This is a joint work with Masakazu Teragaito.


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Takahiro Kitayama (The University of Tokyo)
On an analogue of Culler-Shalen theory for higher-dimensional
[ Abstract ]
Culler and Shalen established a way to construct incompressible surfaces
in a 3-manifold from ideal points of the SL_2-character variety. We
present an analogous theory to construct certain kinds of branched
surfaces from limit points of the SL_n-character variety. Such a
branched surface induces a nontrivial presentation of the fundamental
group as a 2-dimensional complex of groups. This is a joint work with
Takashi Hara (Osaka University).


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Mustafa Korkmaz (Middle East Technical University)
Low-dimensional linear representations of mapping class groups. (ENGLISH)
[ Abstract ]
For a compact connected orientable surface, the mapping class group
of it is defined as the group of isotopy classes of orientation-preserving
self-diffeomorphisms of S which are identity on the boundary. The action
of the mapping class group on the first homology of the surface
gives rise to the classical 2g-dimensional symplectic representation.
The existence of a faithful linear representation of the mapping class
group is still unknown. In my talk, I will show the following three results;
there is no lower dimensional (complex) linear representation,
up to conjugation the symplectic representation is the unique nontrivial representation in dimension 2g, and there is no faithful linear representation
of the mapping class group in dimensions up to 3g-3. I will also discuss a few applications of these theorems, including some algebraic consequences.


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yuanyuan Bao (The University of Tokyo)
A Heegaard Floer homology for bipartite spatial graphs and its
properties (ENGLISH)
[ Abstract ]
A spatial graph is a smooth embedding of a graph into a given
3-manifold. We can regard a link as a particular spatial graph.
So it is natural to ask whether it is possible to extend the idea
of link Floer homology to define a Heegaard Floer homology for
spatial graphs. In this talk, we discuss some ideas towards this
question. In particular, we define a Heegaard Floer homology for
bipartite spatial graphs and discuss some further observations
about this construction. We remark that Harvey and O’Donnol
have announced a combinatorial Floer homology for spatial graphs by
considering grid diagrams.


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Kenta Hayano (Osaka University)
Vanishing cycles and homotopies of wrinkled fibrations (JAPANESE)
[ Abstract ]
Wrinkled fibrations on closed 4-manifolds are stable
maps to closed surfaces with only indefinite singularities. Such
fibrations and variants of them have been studied for the past few years
to obtain new descriptions of 4-manifolds using mapping class groups.
Vanishing cycles of wrinkled fibrations play a key role in these studies.
In this talk, we will explain how homotopies of wrinkled fibrtions affect
their vanishing cycles. Part of the results in this talk is a joint work
with Stefan Behrens (Max Planck Institute for Mathematics).


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Tetsuya Ito (RIMS, Kyoto University)
Homological intersection in braid group representation and dual
Garside structure (JAPANESE)
[ Abstract ]
One method to construct linear representations of braid groups is to use
an action of braid groups on certain homology of local system coefficient.
Many famous representations, such as Burau or Lawrence-Krammer-Bigelow
representations are constructed in such a way. We show that homological
intersections on such homology groups are closely related to the dual
Garside structure, a remarkable combinatorial structure of braid, and
prove that some representations detects the length of braids in a
surprisingly simple way.
This work is partially joint with Bert Wiest (Univ. Rennes1).


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Francis Sergeraert (L'Institut Fourier, Univ. de Grenoble)
Discrete vector fields and fundamental algebraic topology.
[ Abstract ]
Robin Forman invented the notion of Discrete Vector Field in 1997.
A recent common work with Ana Romero allowed us to discover the notion
of Eilenberg-Zilber discrete vector field. Giving the topologist a
totally new understanding of the fundamental tools of combinatorial
algebraic topology: Eilenberg-Zilber theorem, twisted Eilenberg-Zilber
theorem, Serre and Eilenberg-Moore spectral sequences,
Eilenberg-MacLane correspondence between topological and algebraic
classifying spaces. Gives also new efficient algorithms for Algebraic
Topology, considerably improving our computer program Kenzo, devoted
to Constructive Algebraic Topology. The talk is devoted to an
introduction to discrete vector fields, the very simple definition of
the Eilenberg-Zilber vector field, and how it can be used in various


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Andrei Pajitnov (Univ. de Nantes)
Twisted Novikov homology and jump loci in formal and hyperformal spaces (ENGLISH)
[ Abstract ]
Let X be a CW-complex, G its fundamental group, and R a repesentation of G.
Any element of the first cohomology group of X gives rise to an exponential
deformation of R, which can be considered as a curve in the space of
representations. We show that the cohomology of X with local coefficients
corresponding to the generic point of this curve is computable from a spectral
sequence starting from the cohomology of X with R-twisted coefficients. We
compute the differentials of the spectral sequence in terms of Massey products,
and discuss some particular cases arising in Kaehler geometry when the spectral
sequence degenerates. We explain the relation of these invariants and the
twisted Novikov homology. This is a joint work with Toshitake Kohno.


16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Hiroyuki Fuji (The University of Tokyo)
Colored HOMFLY Homology and Super-A-polynomial (JAPANESE)
[ Abstract ]
This talk is based on works in collaboration with S. Gukov, M. Stosic,
and P. Sulkowski. We study the colored HOMFLY homology for knots and
its asymptotic behavior. In recent years, the categorification of the
colored HOMFLY polynomial is proposed in term of homological
discussions via spectral sequence and physical discussions via refined
topological string, and these proposals give the same answer
miraculously. In this talk, we consider the asymptotic behavior of the
colored HOMFLY homology \\`a la the generalized volume conjecture, and
discuss the quantum structure of the colored HOMFLY homology for the
complete symmetric representations via the generalized A-polynomial
which we call “super-A-polynomial”.

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