Tuesday Seminar on Topology (April -- July, 2013)

[Japanese] [Past Programs]

16:30 -- 18:00 Graduate School of Mathematical Sciences, The University of Tokyo
Tea: 16:00 -- 16:30 Common Room

Last updated June 21, 2013
Information :　
Toshitake Kohno
Nariya Kawazumi

April 9 -- Room 056, 16:30 -- 18:00

Hiroyuki Fuji (The University of Tokyo)

Colored HOMFLY Homology and Super-A-polynomial

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 à 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”.

April 23 -- Room 056, 16:30 -- 18:00

Andrei Pajitnov (Université de Nantes)

Twisted Novikov homology and jump loci in formal and hyperformal spaces

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.

April 30 -- Room 056, 16:30 -- 18:00

Francis Sergeraert (L'Institut Fourier, Université 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 contexts.

May 7 -- Room 056, 16:30 -- 18:00

Tetsuya Ito (RIMS, Kyoto University)

Homological intersection in braid group representation and dual Garside structure

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).

May 14 -- Room 056, 16:30 -- 18:00

Kenta Hayano (Osaka University)

Vanishing cycles and homotopies of wrinkled fibrations

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).

May 21 -- Room 056, 16:30 -- 18:00

Yuanyuan Bao (The University of Tokyo)

A Heegaard Floer homology for bipartite spatial graphs and its properties

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.

June 4 -- Room 056, 16:30 -- 18:00

Mustafa Korkmaz (Middle East Technical University)

Low-dimensional linear representations of mapping class groups.

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.

June 11 -- Room 056, 16:30 -- 18:00

Takahiro Kitayama (The University of Tokyo)

On an analogue of Culler-Shalen theory for higher-dimensional representations

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).

June 18 -- Room 056, 16:30 -- 18:00

Kimihiko Motegi (Nihon University)

Left-orderable, non-L-space surgeries on knots

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.

June 25 -- Room 056, 17:10 -- 18:10

Tadayuki Watanabe (Shimane University)

Higher-order generalization of Fukaya's Morse homotopy invariant of 3-manifolds

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.

July 9 -- Room 056, 16:30 -- 18:00

Ryan Budney (University of Victoria)

Smooth 3-manifolds in the 4-sphere

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.

July 16 -- Room 056, 17:10 -- 18:10

Sumio Yamada (Gakushuin University)

On new models of real hyperbolic spaces

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.