Tuesday Seminar on Topology (April -- July, 2014)

[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 July 4, 2014
Information :　
Toshitake Kohno
Nariya Kawazumi
Takuya Sakasai

April 8 -- Room 056, 16:30 -- 18:00

Hidetoshi Masai (The Univesity of Tokyo)

On the number of commensurable fibrations on a hyperbolic 3-manifold.

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 ω on a hyperbolic 3-manifold M, the number of fibrations on M that are commensurable in the sense of Calegari-Sun-Wang to ω is known to be finite. In this talk, we prove that the number can be arbitrarily large.

April 15 -- Room 056, 16:30 -- 18:00

Takahito Naito (The Univesity of Tokyo)

On the rational string operations of classifying spaces and the Hochschild cohomology

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.

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

Taro Asuke (The University of Tokyo)

Transverse projective structures of foliations and deformations of the Godbillon-Vey class

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.

May 20 -- Room 056, 16:30 -- 18:00

Shintaro Kuroki (The Univeristy of Tokyo)

An application of torus graphs to characterize torus manifolds with extended actions

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.

May 27 -- Room 056, 16:30 -- 18:00

Ege Fujikawa (Chiba University)

The Teichmüller space and the stable quasiconformal mapping class group for a Riemann surface of infinite type

Abstract: We explain recent developments of the theory of infinite dimensional Teichmüller space. In particular, we observe the dynamics of the orbits by the action of the stable quasiconformal mapping class group on the Teichmüller space and consider the relationship with the asymptotic Teichmüller 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.

June 3 -- Room 056, 16:30 -- 18:00

Tatsuru Takakura (Chuo University)

Vector partition functions and the topology of multiple weight varieties

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.

June 10 -- Room 056, 14:40 -- 16:10

Special Lecture

Sergei Duzhin (Steklov Institute of Mathematics)

Bipartite knots

Abstract: We give a solution to a part of Problem 1.60 in Kirby's list of open problems in topology thus proving a conjecture raised in 1987 by J.Przytycki. A knot is said to be bipartite if it has a "matched" diagram, that is, a plane diagram that has an even number of crossings which can be split into pairs that look like a simple braid on two strands with two crossings. The conjecture was that there exist knots that do not have such diagrams. I will prove this fact using higher Alexander ideals. This talk is based on a joint work with my student M.Shkolnikov.

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

Yuka Kotorii (The University of Tokyo)

On relation between the Milnor's μ-invariant and HOMFLYPT polynomial

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.

June 17 -- Room 002, 16:30 -- 18:00

Yoshifumi Matsuda (Aoyama Gakuin University)

Bounded Euler number of actions of 2-orbifold groups on the circle

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.

June 24 -- Room 056, 17:10 -- 18:10

Takefumi Nosaka (Faculty of Mathematics, Kyushu University)

On third homologies of quandles and of groups via Inoue-Kabaya map

Abstract: In this talk, we demonstrate certain quandles, which are defined from a group G and an isomorphism ρ : G --> G, and introduce the following results: First, "Inoue-Kabaya chain map" is formulated as a map from quandle homology to group homology. For example, with respect to every Alexander quandle over F_q, the all of Mochizuki 3-cocycle is derived from some group 3-cocycle, and mostly interpreted by a Massey products. In addition, for universal centrally extended quandles, the chain map induces an isomorphism between the 3-rd homologies (up to certain torsion parts).

July 1 -- Room 056, 16:30 -- 18:00

Yohsuke Imagi (Kavli IPMU)

Singularities of special Lagrangian submanifolds

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.

July 8 -- Room 056, 16:30 -- 18:00

Ingrid Irmer (JSPS, the University of Tokyo)

The Johnson homomorphism and a family of curve graphs

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 π₁(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."

July 22 -- Room 056, 16:30 -- 18:00

Jesse Wolfson (Northwestern University)

The Index Map and Reciprocity Laws for Contou-Carrere Symbols

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.