Tuesday Seminar on Topology

17:00 -- 18:30 Graduate School of Mathematical Sciences, The University of Tokyo
Tea: 16:30 -- 17:00 Common Room

Last updated June 22, 2015
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Toshitake Kohno
Nariya Kawazumi
Takuya Sakasai

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

Kazushi Ueda (The University of Tokyo)

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.

April 14 -- Room 056, 17:00 -- 18:30

Nobuhiro Nakamura (Gakushuin University)

Pin(2)-monopole invariants for 4-manifolds

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.

April 21 -- Room 056, 17:00 -- 18:30

Yoshikata Kida (The University of Tokyo)

Orbit equivalence relations arising from Baumslag-Solitar groups

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.

April 28 -- Room 056, 17:00 -- 18:30

Hidetoshi Masai (The University of Tokyo, JSPS)

Verify hyperbolicity of 3-manifolds by computer and its applications.

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.

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

Patrick Dehornoy (Université de Caen)

The group of parenthesized braids

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.

This seminar will be held on Thursday.

May 12 -- Room 056, 17:30 -- 18:30

Masayuki Asaoka (Kyoto University)

Growth rate of the number of periodic points for generic dynamical systems

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.

May 19 -- Room 056, 17:00 -- 18:30

Akishi Kato (The University of Tokyo)

Quiver mutation loops and partition q-series

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.

May 26 -- Room 056, 17:00 -- 18:30

Ken'ichi Kuga (Chiba University)

Introduction to formalization of topology using a proof assistant.

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.

June 9 -- Room 056, 17:00 -- 18:30

Manabu Akaho (Tokyo Metropolitan University)

Symplectic displacement energy for exact Lagrangian immersions

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.

June 16 -- Room 056, 17:00 -- 18:30

Masaharu Ishikawa (Tohoku University)

Stable maps and branched shadows of 3-manifolds

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.

June 23 -- Room 056, 17:00 -- 18:30

Takahiro Matsushita (The University of Tokyo)

Box complexes and model structures on the category of graphs

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.

June 30 -- Room 056, 17:30 -- 18:30

Makoto Sakuma (Hiroshima University)

The Cannon-Thurston maps and the canonical decompositions of punctured surface bundles over the circle

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.

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

Takahiro Kitayama (Tokyo Institute of Technology)

Representation varieties detect essential surfaces

Abstract: Extending Culler-Shalen theory, Hara and I presented a way to construct certain kinds of branched surfaces (possibly without any branch) in a 3- manifold from an ideal point of a curve in the SL_n-character variety. There exists an essential surface in some 3-manifold known to be not detected in the classical SL_2-theory. We show that every essential surface in a 3-manifold is given by the ideal point of a line in the SL_ n-character variety for some n. The talk is partially based on joint works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.

July 14 -- Room 056, 17:00 -- 18:30

Carlos Moraga Ferrándiz (The University of Tokyo, JSPS)

How homoclinic orbits explain some algebraic relations holding in Novikov rings.

Abstract: Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space Fu of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each in Fu with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (,X) with in Fu . Here, X is a descending pseudo-gradient, which is said to be adapted to . The morphism 1(M) R induced by u (given by the integral of any in Fu over a loop of M) determines a set of u-negative loops. We show that for every u-negative g in 1(M), there exists a co-dimension 1 C-stratum Sg of Fu which is naturally co-oriented. The stratum Sg is made of elements (, X) such that X has exactly one homoclinic orbit L whose homotopy class is g. The goal of this talk is to show that there exists a co-dimension 1 C-stratum Sg (0) of Sg which lies in the closure of Sg2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u. We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.

July 21 -- Room 056, 17:00 -- 18:30

Keiji Tagami (Tokyo Institute of Technology)

Ribbon concordance and 0-surgeries along knots

Abstract: Akbulut and Kirby conjectured that two knots with the same 0-surgery are concordant. Recently, Yasui gave a counterexample of this conjecture. In this talk, we introduce a technique to construct non-ribbon concordant knots with the same 0-surgery. Moreover, we give a potential counterexample of the slice-ribbon conjecture. This is a joint work with Tetsuya Abe (Osaka City University, OCAMI).