Tuesday Seminar on Topology (September, 2020 -- January, 2021)
[Japanese] [Past Programs]
This is an online seminar on Zoom.
Last updated March 23, 2021
Information :
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai
September 29, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Kohei Iwaki (The University of Tokyo)
Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds
Abstract: In 1998, Lawrence-Zagier introduced a certain q-series and proved that its limit value at root of unity
q=exp(2π i / K) coincides with the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant of the Poincare homology sphere Σ(2,3,5) at the level K.
Employing the idea of Gukov-Marino-Putrov based on resurgent analysis,
we generalize the result of Lawrence-Zagier for the Seifert loops (Seifert manifolds with a single loop inside).
That is, for each Seifert loop, we introduce an explicit q-series (WRT function)
and show that its limit value at the root of unity coincides with the WRT invariant of the Seifert loop.
We will also discuss a q-difference equation satisfied by the WRT function.
This talk is based on a joint work with H. Fuji, H. Murakami and Y. Terashima which is available on arXiv:2007.15872.
Slides
October 6, 17:30-18:30 -- Online on Zoom.
Pre-registration required.
Shinichiroh Matsuo (Nagoya University)
The Atiyah-Patodi-Singer index of manifolds with boundary and domain-wall fermions
Abstract: We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index.
In a previous work, motivated by the study of lattice gauge theory,
we derived a formula expressing the Atiyah-Patodi-Singer index in terms of the eta invariant of
“domain-wall fermion Dirac operators” when the base manifold is a flat 4-dimensional torus.
Now we generalise this formula to any even dimensional closed Riemannian manifolds, and prove it mathematically rigorously.
Our proof uses a Witten localisation argument combined with a devised embedding into a cylinder of one dimension higher.
Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.
This talk is based on a joint paper arXiv:1910.01987, to appear in CMP,
with H. Fukaya, M. Furuta, T. Onogi, S. Yamaguchi, and M. Yamashita.
Slides
October 20, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Alexandru Oancea (Sorbonne Université)
Poincaré duality for free loop spaces
Abstract: A certain number of dualities between homological and cohomological invariants of free loop spaces have been observed over the years,
having the flavour of Poincaré duality but nevertheless holding in an infinite dimensional setting.
The goal of the talk will be to explain these through a new duality theorem, whose proof uses symplectic methods.
The talk will report on joint work with Kai Cieliebak and Nancy Hingston.
October 27, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Jun Yoshida (The University of Tokyo)
Vassiliev derivatives of Khovanov homology and its applications
Abstract: Khovanov homology is a categorification of the Jones polynomial.
It is known that Khovanov homology also arises from a categorical representation of braid groups,
so we can regard it as a kind of quantum knot invariant.
However, in contrast to the case of classical quantum invariants, its relation to Vassiliev invariants remains unclear.
In this talk, aiming at the problem,
we discuss a categorified version of Vassiliev skein relation on Khovanov homology.
Namely, we extend Khovanov homology to singular links so that extended ones can be seen as
"derivatives" in view of Vassiliev theory.
As an application, we compute first derivatives to determine Khovanov homologies of twist knots.
This talk is based on papers arXiv:2005.12664 (joint work with N.Ito) and arXiv:2007.15867.
Slides
November 17, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Yoshihiko Mitsumatsu (Chuo University)
Lefschetz fibration on the Milnor fibers of simple elliptic and cusp singularities
Abstract: In this talk a joint work with Naohiko Kasuya(Kyoto Sangyo U.),
Hiroki Kodama(Tohoku U.), and Atsuhide Mori(Osaka Dental U.) is reported.
The main result is the following.
There exist a Lefschetz fibration of the Milnor fiber of T_{pqr}-singularity
(1/p + 1/q + 1/r ≦ 1) to the unit disk with regular fiber diffeomorphic to T^2.
An outline of the construction will be explained, through which,
the space of 2-jets of (R^4, 0) to (R^2, 0) is analysed.
This is motivated by F. Presas' suggestion that the speaker's construction of regular Poisson structures
(=leafwise symplectic foliations) on S^5 might be interpreted by ``leafwise Lefschetz fibration''.
These Lefschetz fibrations give a way to look at K3 surfaces through an extended class of Arnol'd's strange duality.
These applications are introduced as well.
November 24, 17:30-18:30 -- Online on Zoom.
Pre-registration required.
Shinpei Baba (Osaka University)
Intersection of Poincare holonomy varieties and Bers' simultaneous uniformization theorem
Abstract: Given a marked compact Riemann surface X,
the vector space of holomorphic quadratic differentials on X is identified with the space of CP1-structures on X.
Then, by the holonomy representations of CP1-structures,
this vector space properly embeds into the PSL(2, C)-character variety,
the space of representations of the fundamental group of X into PSL(2,C).
In this manner, different Riemann surfaces structures yield different half-dimensional smooth analytic subvarieties in the character variety.
In this talk, we discuss some properties of their intersection.
To do so, we utilize a cut-and-paste operation, called grafting, of CP1-structures.
December 1, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Yuya Koda (Hiroshima University)
Goeritz groups of bridge decompositions
Abstract: For a bridge decomposition of a link in the 3-sphere,
we define the Goeritz group to be the group of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere
that preserve each of the bridge sphere and link setwise.
The Birman-Hilden theory tells us that this is a
Z/2Z-quotient of a "hyperelliptic Goeritz group".
In this talk, we discuss properties, mainly of dynamical nature,
of this group using a measure of complexity called the distance of the decomposition.
We then give an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings.
This talk is based on a joint work with Susumu Hirose, Daiki Iguchi and Eiko Kin.
December 8, 17:30-18:30 -- Online on Zoom.
Pre-registration required.
Shin Satoh (Kobe University)
The intersection polynomials of a virtual knot
Abstract: We define two kinds of invariants of a virtual knot called the first and second intersection polynomials.
The definition is based on the intersection number of a pair of curves on a closed surface.
We study several properties of the polynomials.
By introducing invariants of long virtual knots,
we give connected sum formulae of the intersection polynomials,
and prove that there are infinitely many connected sums of any two virtual knots as an application.
Furthermore, by studying the behavior under a crossing change,
we show that the intersection polynomials are finite type invariants of order two,
and find an invariant of a flat virtual knot derived from the the intersection polynomials.
This is a joint work with R. Higa, T. Nakamura, and Y. Nakanishi.
Slides
December 15, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Eiko Kin (Osaka University)
Braids, triangles and Lissajous curves
Abstract: The purpose of this talk is to introduce Lissajous 3-braids.
Suppose we have a closed curve on the plane,
and we consider the periodic motion of n points along the closed curve.
If the motion is collision-free,
then we get a braid obtained from the trajectory of the set of n points in question.
In this talk, we consider 3-braids coming from the periodic motion of 3 points on Lissajous curves.
We classify Lissajous 3-braids and present a parametrization in terms of natural numbers together with slopes.
We also discuss some properties of pseudo-Anosov stretch factors for Lissajous 3-braids.
The main tool is the shape sphere --- the configuration space of the oriented similarity classes of triangles. This is a joint work with Hiroaki Nakamura and Hiroyuki Ogawa.
January 12, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Mitsuaki Kimura (The University of Tokyo)
Bounded cohomology of volume-preserving diffeomorphism groups
Abstract: Let M be a complete Riemannian manifold of finite volume.
Brandenbursky and Marcinkowski proved that the third bounded cohomology of the volume-preserving diffeomorphism group of M
is infinite dimensional when the fundamental group of M is "complicated enough".
For example, if M is two-dimensional,
the above condition is satisfied if the Euler characteristic is negative.
Recently, we have extended this result in the following two directions.
(1) When M is two-dimensional and the Euler characteristic is greater than or equal to zero.
(2) When the volume of M is infinite.
In this talk, we will mainly discuss (1).
The key idea is to use the fundamental group of the configuration space of M
(i.e., the braid group), rather than the fundamental group of M.
If time permits, we will also explain (2).
For this extension, we introduce the notion of "norm controlled cohomology".