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 February 11, 2017
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Toshitake Kohno
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai

September 27 -- Room 056, 17:00 -- 18:30

Shouta Tounai (The University of Tokyo)

CAT(0) properties for orthoscheme complexes

Abstract: Gromov showed that a cubical complex is locally CAT(0) if and only if the link of every vertex is a flag complex. Brady and MacCammond introduced an orthoscheme complex as a generalization of cubical complexes. It is, however, difficult to tell whether an orthoscheme complex is (locally) CAT(0) or not. In this talk, I will discuss a translation of Gromov's characterization for orthoscheme complexes. As a generalization of Gromov's characterization, I will show that the orthoscheme complex of locally distributive semilattice is CAT(0) if and only if it is a flag semilattice.

October 11 -- Room 056, 17:00 -- 18:30

Nariya Kawazumi (The University of Tokyo)

The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebra in genus zero

Abstract: In view of results of Goldman and Turaev, the free vector space over the free loops on an oriented surface has a natural Lie bialgebra structure. The Goldman bracket has a formal description by using a special (or symplectic) expansion of the fundamental group of the surface. It is natural to ask for a formal description of the Turaev cobracket. We will show how to obtain a formal description of the Goldman-Turaev Lie bialgebra for genus 0 using a solution of the Kashiwara-Vergne problem. A similar description was recently obtained by Massuyeau using the Kontsevich integral. Moreover we propose a generalization of the Kashiwara-Vergne problem in the context of the Goldman-Turaev Lie bialgebra. This talk is based on a joint work with A. Alekseev, Y. Kuno and F. Naef.

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

Yoshitake Hashimoto (Tokyo City University)

Conformal field theory for extended W-algebras

Abstract: This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of "infinitesimal deformation of parameter" and Jack symmetric polynomials.

In this talk I will discuss the following subjects:
1. "factorization" in conformal field theory,
2. tensor structure of the category of M-modules and "module-bimodule correspondence".

November 1 -- Room 056, 17:00 -- 18:30

Takahiro Oba (Tokyo Institute of Technology)

Higher-dimensional contact manifolds with infinitely many Stein fillings

Abstract: A Stein fillings of a given contact manifold is a Stein domain whose boundary is contactomorphic to the given contact manifold. Open books, Lefschetz fibrations, and mapping class groups of their fibers in particular help us to produce various contact manifolds and their Stein fillings. However, little is known about mapping class groups of higher-dimensional manifolds. This is one of the reasons that it was unknown whether there is a contact manifold of dimension > 3 with infinitely many Stein fillings. In this talk, I will choose a certain symplectic manifold as fibers of open books and Lefschetz fibrations and by using them, construct an infinite family of higher-dimensional contact manifolds with infinitely many Stein fillings.

November 8 -- Room 056, 17:00 -- 18:30

Toshiyuki Akita (Hokkaido University)

Second mod 2 homology of Artin groups

Abstract: After a brief survey on the K(π,1) conjecture and homology of Artin groups, I will introduce our recent result: we determined second mod 2 homology of arbitrary Artin groups without assuming the K(π,1)-conjecture. The key ingredients are Hopf's formula and a result of Howlett on Schur multipliers of Coxeter groups. This is a joint work with Ye Liu.

November 15 -- Room 056, 17:00 -- 18:30

Takuya Sakasai (The University of Tokyo)

Cohomology of the moduli space of graphs and groups of homology cobordisms of surfaces

Abstract: We construct an abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra generated by the fundamental representation of the symplectic group. It gives an alternative proof of the fact first shown by Bartholdi that the top rational homology group of the moduli space of metric graphs of rank 7 is one dimensional. As an application, we construct a non-trivial abelian quotient of the homology cobordism group of a surface of positive genus. This talk is based on joint works with Shigeyuki Morita, Masaaki Suzuki and Gwénaël Massuyeau.

November 22 -- Room 056, 17:00 -- 18:30

Takahito Naito (The University of Tokyo)

Sullivan's coproduct on the reduced loop homology

Abstract: In string topology, Sullivan introduced a coproduct on the reduced loop homology and showed that the homology has an infinitesimal bialgebra structure with respect to the coproduct and Chas-Sullivan loop product. In this talk, I will give a homotopy theoretic description of Sullivan's coproduct. By using the description, we obtain some computational examples of the structure over the rational number field. Moreover, I will also discuss a based loop space version of the coproduct.

November 29 -- Room 056, 17:30 -- 18:30

Hayato Chiba (Kyushu University)

Generalized spectral theory and its application to coupled oscillators on networks

Abstract: For a system of large coupled oscillators on networks, we show that the transition from the de-synchronous state to the synchronization occurs as the coupling strength increases. For the proof, the generalized spectral theory of linear operators is employed.

December 6 -- Room 056, 17:00 -- 18:30

Ken'ichi Yoshida (The University of Tokyo)

Union of 3-punctured spheres in a hyperbolic 3-manifold

Abstract: An essential 3-punctured sphere in a hyperbolic 3-manifold is isotopic to a totally geodesic one. We will classify the topological types for components of union of the totally geodesic 3-punctured spheres in an orientable hyperbolic 3-manifold. There are special types each of which appears in precisely one manifold.

December 13 -- Room 056, 17:00 -- 18:30

Yoshihiko Mitsumatsu (Chuo University)

Plane fields on 3-manifolds and asymptotic linking of the tangential incompressible flows

Abstract: This is a report on a project in (a very slow) progress which aims to prove the tightness of contact structures associated with algebraic Anosov flows without using Bennequin's nor Gromov's results.
After introducing an interpretation of asymptotic linking pairing in terms of differential forms, we attach a subspaces of exact 2-forms to each plane field. We analyze this space in the case where the plane field is an algebraic Anosov foliation and explain what can be done using results from foliated cohomology and frameworks for secondary characteristic classes. We also show some explicit computations.
To close the talk, a quantization phenomenon which happens when a foliation is deformed into a contact structure is explained and we state some perspectives on applying the results on foliations to the tightness.

December 20 -- Room 056, 17:00 -- 18:30

Irene Pasquinelli (Durham University)

Deligne-Mostow lattices and cone metrics on the sphere

Abstract: Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.
One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.
In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with three folding symmetry.

January 10 -- Room 056

17:00 -- 18:00

Shunsuke Saito (The University of Tokyo)

Stability of anti-canonically balanced metrics

Abstract: Donaldson introduced "anti-canonically balanced metrics" on Fano manifolds, which is a finite dimensional analogue of Kähler-Einstein metrics. It is proved that anti-canonically balanced metrics are critical points of the quantized Ding functional.
We first study the slope at infinity of the quantized Ding functional along Bergman geodesic rays. Then, we introduce a new algebro-geometric stability of Fano manifolds based on the slope formula, and show that the existence of anti-canonically balanced metrics implies our stability. The relationship between the stability and others is also discussed.
This talk is based on a joint work with R. Takahashi (Tohoku Univ).

18:00 -- 19:00

Shin Hayashi (The University of Tokyo)

Topological Invariants and Corner States for Hamiltonians on a Three Dimensional Lattice

Abstract: In condensed matter physics, a correspondence between two topological invariants defined for a gapped Hamiltonian is well-known. One is defined for such a Hamiltonian on a lattice (bulk invariant), and the other is defined for its restriction onto a subsemigroup (edge invariant). The edge invariant is related to the wave functions localized near the edge. This correspondence is known as the bulk-edge correspondence. In this talk, we consider a variant of this correspondence. We consider a periodic Hamiltonian on a three dimensional lattice (bulk) and its restrictions onto two subsemigroups (edges) and their intersection (corner). We will show that, if our Hamiltonian is "gapped" in some sense, we can define a topological invariant for the bulk and edges. We will also define another topological invariant related to the wave functions localized near the corner. We will explain that there is a correspondence between these two topological invariants by using the six-term exact sequence associated to the quarter-plane Toeplitz extension obtained by E. Park.

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

Satoshi Sugiyama (The University of Tokyo)

On an application of the Fukaya categories to the Koszul duality

Abstract: In this talk, we compute an A-Koszul dual of path algebras with relations over the directed An-type quivers via the Fukaya categories of exact Riemann surfaces.
The Koszul duality is originally a duality between certain quadratic algebras called Koszul algebras. In this talk, we are interested in the case when A is not a quadratic algebra, i.e. the case when A is defined as a quotient algebra of tensor algebra devided by higher degree relations.
The definition of Koszul duals for such algebras, A-Koszul duals, are given by some people, for example, D. M. Lu, J. H. Palmieri, Q. S. Wu, J. J. Zhang. However, the computation for a concrete examples is hard. In this talk, we use the Fukaya categories of exact Riemann surfaces to compute A-Koszul duals. Then, we understand the Koszul duality as a duality between higher products and relations.

January 24 -- Room 056

17:00 -- 18:00

Ryuma Orita (The University of Tokyo)

On the existence of infinitely many non-contractible periodic trajectories in Hamiltonian dynamics on closed symplectic manifolds

Abstract: We show that the presence of a non-contractible Hamiltonian one-periodic trajectory in a closed symplectic manifold yields the existence of infinitely many non-contractible periodic trajectories, provided that the symplectic form is aspherical and the fundamental group is virtually abelian. Moreover, we also show that a similar statement holds for closed monotone or negative monotone symplectic manifolds having virtually abelian fundamental groups. These results are certain generalizations of works by Ginzburg and Gurel who proved a similar statement holds for atoroidal or toroidally monotone closed symplectic manifolds. The proof is based on the machinery of filtered Floer-Novikov homology for non-contractible periodic trajectories.

18:00 -- 19:00

Noriaki Kawaguchi (The University of Tokyo)

Quantitative shadowing property, shadowable points, and local properties of topological dynamical systems

Abstract: Shadowing property has been one of the key notions in topological hyperbolic dynamics, which is also common since C^0-generic homeomorphisms on a smooth closed manifold satisfy the property for instance. In this talk, the shadowing property in relation to other chaotic or non-chaotic properties of dynamical systems (entropy, sensitivity, equicontinuity, etc.) is discussed. Also, we introduce an idea of localizing and quantifying the shadowing property following the recent work of Morales, and present some of its consequences. The idea is shown to be effective for the description of local properties of dynamical systems.

February 20 (Monday) -- Room 056, 17:00 -- 18:30

Jørgen Ellegaard Andersen (Aarhus University)

The Verlinde formula for Higgs bundles

Abstract: In this talk we will present a Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. We further present a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. We will explain how all these dimensions fit into a one parameter family of 2D TQFT's, encoded in a one parameter family of Frobenius algebras, which we will construct.

March 8 (Wednesday) -- Room 056, 17:00 -- 18:30

Arthur Soulié (Université de Strasbourg)

Action of the Long-Moody Construction on Polynomial Functors

Abstract: In 2016, Randal-Williams and Wahl proved homological stability with certain twisted coefficients for different families of groups, in particular the one of braid groups. In fact, they obtain the stability for coefficients given by functors satisfying polynomial conditions. We only know few examples of such functors. Among them, we have the functor given by the unreduced Burau representations. In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of Bn with a representation of Bn+1. This construction complexifies in a sense the initial representation: for instance, starting from a dimension one representation, one obtains the unreduced Burau representation. In this talk, I will present this construction from a functorial point of view. I will explain that the construction of Long and Moody defines an endofunctor, called the Long-Moody functor, between a suitable category of functors. Then, after defining strong polynomial functors in this context, I will prove that the Long-Moody functor increases by one the degree of strong polynomiality of a strong polynomial functor. Thus, the Long-Moody construction will provide new examples of twisted coefficients entering in the framework of Randal-Williams and Wahl.

March 10 [Joint with Lie Groups and Representation Theory Seminar] (Friday) -- Room 056, 17:00 -- 18:30

Lizhen Ji (University of Michigan)

Satake compactifications and metric Schottky problems

The quotient of the Poincare upper half plane by the modular group SL(2, Z) is a basic locally symmetric space and also the moduli space of compact Riemann surfaces of genus 1, and it admits two important classes of generalization:
(1) Moduli spaces M_g of compact Riemann surfaces of genus g>1,
(2) Arithmetic locally symmetric spaces Γ\G/K such as the Siegel modular variety A_g, which is also the moduli of principally polarized abelian varieties of dimension g.
There have been a lot of fruitful work to explore the similarity between these two classes of spaces, and there is also a direct interaction between them through the Jacobian (or period) map J: M_g --> A_g. In this talk, I will discuss some results along these lines related to the Stake compactifications and the Schottky problems on understanding the image J(M_g) in A_g from the metric perspective.