Tuesday Seminar on Topology

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

Last updated March 11, 2015
Information :@
Toshitake Kohno
Nariya Kawazumi
Takuya Sakasai

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

Kei Irie (RIMS, Kyoto University)

Transversality problems in string topology and de Rham chains

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.

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

Toshiyuki Akita (Hokkaido University)

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

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.

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

Brian Bowditch (University of Warwick)

The coarse geometry of Teichmüller space.

Abstract: We describe some results regarding the coarse geometry of the Teichmüller space of a compact surface. In particular, we describe when the Teichmüller space admits quasi-isometric embeddings of euclidean spaces and half-spaces. We also give some partial results regarding the quasi-isometric rigidity of Teichmüller space. These results are based on the fact that Teichmüller space admits a ternary operation, natural up to bounded distance which endows it with the structure of a coarse median space.

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

Kenneth Baker (University of Miami)

Unifying unexpected exceptional Dehn surgeries

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.

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

Charles Siegel (Kavli IPMU)

A Modular Operad of Embedded Curves

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.

November 25 -- Room 056, 16:30 -- 18:00

Masahico Saito (University of South Florida)

Quandle knot invariants and applications

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.

December 2 -- Room 056, 16:30 -- 18:00

Yosuke Kubota (The University of Tokyo)

The Atiyah-Segal completion theorem in noncommutative topology

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.

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

Koji Fujiwara (Kyoto University)

Stable commutator length on mapping class groups

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.

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

Norio Iwase (Kyushu University)

Differential forms in diffeological spaces

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.

January 13 -- Room 056, 16:30 -- 18:00

Ken'ichi Yoshida (The University of Tokyo)

Stable presentation length of 3-manifold groups

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.

January 20 -- Room 056, 16:30 -- 17:30

Toru Yoshiyasu (The University of Tokyo)

On Lagrangian caps and their applications

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.

March 10 -- Room 056, 16:30 -- 18:00

Andrei Pajitnov (Université de Nantes)

Arnold conjecture, Floer homology, and augmentation ideals of finite groups.

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.

March 24 -- Room 056, 17:00 -- 18:30

Mina Aganagic (University of California, Berkeley)

Knots and Mirror Symmetry

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.

April 7 -- Room 056, 17:00 -- 18:30

Kazushi Ueda (Osaka University)

Potential functions for Grassmannians

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.