Abstract: The concepts of train track was introduced by
W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface.
In this talk, we translate some concepts of train tracks into the
language of cluster algebras using the tropicalization of
Goncharov--Shen's potential function.
Using this, we translate a combinatorial property of a train track associated with a pseudo-Anosov mapping class into the combinatorial property in cluster algebras,
called the sign stability which was introduced by Tsukasa Ishibashi and the speaker.
10月24日 -- オンライン開催, 17:00 -- 18:00
林 晋 (青山学院大学)
Index theory for quarter-plane Toeplitz operators via extended symbols
Abstract: We consider index theory for some Toeplitz operators on a discrete quarter-plane.
Index theory for such operators has been investigated by Simonenko, Douglas-Howe,
Park and index formulas are obtained by Coburn-Douglas-Singer, Duducava.
In this talk, we revisit Duducava’s idea and discuss an index formula for quarter-plane Toeplitz operators of two-variable rational matrix function symbols from a geometric viewpoint.
By using Gohberg-Krein theory for matrix factorizations and analytic continuation,
we see that the symbols of Fredholm quarter-plane Toeplitz operators defined originally on a two-dimensional torus can canonically be extended to some three-sphere,
and show that their Fredholm indices coincides with the three-dimensional winding number of extended symbols.
If time permits, we briefly mention a contact with a topic in condensed matter physics,
called (higher-order) topological insulators.
Hodge numbers of moduli spaces of principal bundles on curves
Abstract: The Poincaré series of moduli stacks of semistable G-bundles on curves has been computed by Laumon and Rapoport.
In this joint work with Melissa Liu,
we show that the Hodge-Poincaré series of these moduli stacks can be computed in a similar way.
As an application, we obtain a new proof of a joint result of the speaker with Erwan Brugallé,
on the maximality on moduli spaces of vector bundles over real algebraic curves.
Abstract: In 2008, Costantino and D.Thurston revealed that
the combinatorial structure of the Stein factorizations of stable maps from 3-manifolds into the real plane can be used
to describe the hyperbolic structures of the complement of the set of definite fold points, which is a link.
The key was that the Stein factorizations can naturally be embedded into 4-manifolds,
and nice ideal polyhedral decompositions become visible on their boundaries.
In this talk, we consider divides, which are the images of a proper and generic immersions of compact 1-manifolds into the 2-disk.
Due to A'Campo's theory, each divide is associated with a link in the 3-sphere.
By embedding a polyhedron induced from a given divide into the 4-ball as was done to Stein factorization,
we can read off the ideal polyhedral decompositions on the boundary.
We then show that the complement of the link of the divide can be obtained by Dehn filling a hyperbolic 3-manifold that admits a decomposition into several ideal regular hyperbolic polyhedra,
where the number of each polyhedron is determined by types of the double points of the divide.
This immediately gives an upper bound of the hyperbolic volume of the links of divides,
which is shown to be asymptotically sharp.
As in the case of Stein factorizations, an idea from the theory of Turaev's shadows plays an important role here.
This talk is based on joint work with Ryoga Furutani (Hiroshima University).
11月28日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
Gwénaël Massuyeau (Université de Bourgogne)
An analogue of the Johnson-Morita theory for the handlebody group
Abstract: The Johnson-Morita theory provides an approach for the mapping class group of a surface by considering its actions on the successive nilpotent quotients of the fundamental group of the surface.
In this talk, after an outline of the original theory, we will present an analogue of the Johnson-Morita theory for the handlebody group,
i.e. the mapping class group of a handlebody.
This is joint work with Kazuo Habiro;
as we shall explain if time allows, our motivation is to recover the "tree reduction"
of a certain functor on the category of bottom tangles in handlebodies that we introduced (a few years ago) using the Kontsevich integral.
12月5日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
北野 晃朗 (創価大学)
On the Euler class for flat S1-bundles, C∞ vs Cω
Abstract: We describe low dimensional homology groups of the real analytic,
orientation preserving diffeomorphism group of S1 in terms of
BΓ1 by applying a theorem of Thurston.
It is an open problem whether some power of the rational Euler class vanishes for real analytic flat S1 bundles.
In this talk we discuss that if it does, then the homology group should contain many torsion classes that vanish in the smooth case.
Along this line we can give a new proof for the non-triviality of any power of the rational Euler class in the smooth case.
If time permits, we will mention some attempts to study a Mather-Thurston map in the analytic case.
This talk is based on a joint work with Shigeyuki Morita and Yoshihiko Mitsumatsu.
12月12日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
Stavros Garoufalidis (南方科技大学)
Multivariable knot polynomials from braided Hopf algebras with automorphisms
Abstract: We will discuss a unified approach to define
multivariable polynomial invariants of knots that include the colored Jones polynomials,
the ADO polynomials and the invariants defined using the theory of quantum groups.
Our construction uses braided Hopf algebras with automorphisms.
We will give examples of 2-variable invariants,
and discuss their structural properties. Joint work with Rinat Kashaev.
Abstract: The study of torus orbit closures in the flag variety
was initiated by Gelfand-Serganova and Klyachko in 1980’s
but has not been studied much since then.
Recently, I have studied its geometry and topology jointly with Eunjeong Lee,
Seonjeong Park, Jongbaek Song in connection with combinatorics of polytopes,
Coxeter matroids, and polygonal triangulations.
In this talk I will report on the development of this subject.
12月19日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
河東 泰之 (東京大学大学院数理科学研究科)
Topological quantum computing, tensor networks and operator algebras
Abstract: Modular tensor categories have caught much attention in connection to topological quantum computing based on anyons recently.
Condensed matter physicists recently try to understand structures of modular tensor categories appearing in two-dimensional topological order using tensor networks.
We present understanding of their tools in terms of operator algebras.
For example, 4-tensors they use are exactly bi-unitary connections in the Jones theory of subfactors and their sequence of finite dimensional Hilbert spaces on which their gapped Hamiltonians act is given by the so-called higher relative commutants of a subfactor.
No knowledge on operator algebras are assumed.
