Tuesday Seminar on Topology (April -- July, 2023)
[Japanese] [Past Programs]
17:00 -- 18:30 Graduate School of Mathematical Sciences,
The University of Tokyo (with live streaming on Zoom)
or
17:00 -- 18:00 Online seminar on Zoom.
Last updated September 26, 2023
Information :
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai
April 11, 17:00-18:30 -- Room 056 with live streaming on Zoom
Kazuo Habiro (The University of Tokyo)
On the stable cohomology of the (IA-)automorphism groups of free groups
Abstract: By combining Borel's stability and vanishing theorem for the stable cohomology of GL(n,Z) with coefficients in algebraic
GL(n,Z)-representations and the Hochschild-Serre spectral sequence,
we compute the twisted first cohomology of the automorphism group Aut(F_n) of the free group F_n of rank n.
This method is used also in the study of the stable rational cohomology of the IA-automorphism group IA_n of F_n.
We propose a conjectural algebraic structure of the stable rational cohomology of IA_n,
and consider some relations to known results and conjectures.
We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces.
This is a joint work with Mai Katada.
April 18, 17:00-18:00 -- Online on Zoom
Shuhei Maruyama (Chuo University)
A crossed homomorphism on a big mapping class group
Abstract: Big mapping class groups are mapping class groups of surfaces of infinite type.
Calegari and Chen determined the second (co)homology group of the mapping class group of the sphere minus a Cantor set.
They also raised related questions:
one of the questions asks an explicit form of certain crossed homomorphisms on the big mapping class group.
In this talk, we provide a construction of crossed homomorphisms via group actions on the circle,
which answers the question of Calegari and Chen.
April 25, 17:00-18:00 -- Online on Zoom
Hiraku Nozawa (Ritsumeikan University)
Harmonic measures and rigidity of surface group actions on the circle
Abstract: We study rigidity properties of surface group actions on the circle via harmonic measures on the suspension bundles,
which are measures invariant under the heat diffusion along leaves.
We will explain a curvature estimate and a Gauss-Bonnet formula for an
S^1-connection obtained by taking the average of the flat connection on the suspension bundle with respect to a harmonic measure.
As consequences,
we give a precise description of the harmonic measure on suspension foliations with maximal Euler number
and an alternative proof of semiconjugacy rigidity theorems of Matsumoto and Burger-Iozzi-Wienhard
for actions with maximal Euler number.
This is joint work with Masanori Adachi and Yoshifumi Matsuda.
May 9, 17:00-18:00 -- Online on Zoom
Michihisa Wakui (Kansai University)
Knots and frieze patterns
Abstract: (joint work with Prof. Takeyoshi Kogiso (Josai University))
In the early 1970s, Conway and Coxeter introduced frieze patterns of positive integers arranged under the unimodular rule ad-bc=1,
and showed that they are classified by triangulations of convex polygons.
Currently, the frieze patterns by Conway and Coxeter are spotlighted in connection with cluster algebras which are introduced by Fomin and Zelevinsky in the early 2000s.
Working with Takeyoshi Kogiso in Josai University
the speaker study on relationship between rational links and Conway-Coxeter friezes through ancestor triangles of rational numbers introduced by Shuji Yamada in Kyoto Sangyo University,
and show that rational links are characterized by Conway-Coxeter friezes of zigzag type.
At nearly the same time Morier-Genoud and Ovsienko also introduce the concept of q-deformation of rational numbers based on continued fraction expansions,
and derive closely related results to our research.
In this seminar we will talk about an outline of these results.
May 16, 17:00-18:30 -- Room 056 with live streaming on Zoom
Mayuko Yamashita (Kyoto University)
Anderson self-duality of topological modular forms and heretoric string theory
Abstract: Topological Modular Forms (TMF) is an E-infinity ring spectrum
which is conjectured by Stolz-Teichner to classify two-dimensional supersymmetric quantum field theories in physics.
In the previous work with Y. Tachikawa, we proved the vanishing of anomalies in heterotic string theory mathematically by using TMF.
In this talk, I explain our recent update on the previous work.
Because of the vanishing result, we can consider a secondary transformation of spectra,
which is shown to coincide with the Anderson self-duality morphism of TMF.
This allows us to detect subtle torsion phenomena in TMF by differential-geometric ways.
May 30, 17:00-18:30 -- Room 056 with live streaming on Zoom
Yuya Kodama (Tokyo Metropolitan University)
p-colorable subgroup of Thompson's group F
Abstract: Thompson's group F is a subgroup of Homeo([0, 1]).
In 2017, Jones found a way to construct knots and links from elements in F.
Moreover, any knot (or link) can be obtained in this way.
So the next question is, which elements in F give the same knot (or link)?
In this talk, I define a subgroup of F and show that every element
(except the identity) gives a p-colorable knot (or link).
When p=3, this gives a negative answer to a question by Aiello.
This is a joint work with Akihiro Takano.
June 6, 17:30-18:30 [Joint with Lie Groups and Representation Theory Seminar]
-- Room 056 with live streaming on Zoom
Atsumu Sasaki (Tokai University)
Visible actions on reductive spherical homogeneous spaces and their invariant measures
Abstract: Toshiyuki Kobayashi has established propagation theorem of multiplicity-freeness property.
This theorem enables us to give an unified explanation of multiplicity-freeness of multiplicity-free representations which have been found so far,
and also to find new examples of multiplicity-free representations systematically.
Kobayashi further has introduced the notion of visible actions on complex manifolds as a basic condition for propagation theorem of multiplicity-freeness property.
This notion plays an important role to this theorem and also brings us to find various decomposition theorems of Lie groups and homogeneous spaces.
In this talk,
we explain visible actions on reductive spherical homogeneous spaces.
In particular,
we see that for a visible action on reductive spherical homogeneous space our construction of a submanifold which meets every orbit (called a slice)
is given by an explicit description of a Cartan decomposition for this space.
As a corollary of this study,
we characterize the invariant measure on a reductive spherical homogeneous space by giving an integral formula for a Cartan decomposition explicitly.
June 13, 17:00-18:00 -- Online on Zoom
Shunsuke Usuki (Kyoto University)
On a lower bound of the number of integers in Littlewood's conjecture
Abstract: Littlewood's conjecture is a famous and long-standing open problem on simultaneous Diophantine approximation.
It is closely related to the action of diagonal matrices on SL(3,R)/SL(3,Z),
and M. Einsiedler, A. Katok and E. Lindenstrauss showed in 2000's
that the exceptional set for Littlewood's conjecture has Hausdorff dimension zero by using some rigidity for invariant measures under the diagonal action.
In this talk, I explain that we can obtain some quantitative result on the result of Einsiedler, Katok and Lindenstrauss by studying the empirical measures with respect to the diagonal action.
June 20, 17:00-18:30 -- Room 056 with live streaming on Zoom
Arnaud Maret (Sorbonne Université)
Moduli spaces of triangle chains
Abstract: In this talk, I will describe a moduli space of triangle chains in the hyperbolic plane with prescribed angles.
We will relate this moduli space to a specific character variety of representations of surface groups into PSL(2,R).
This identification provides action-angle coordinates for the Goldman symplectic form on the character variety.
If time permits, I will explain why the mapping class group action on that particular character variety is ergodic.
July 4, 17:00-18:30 -- Room 056 with live streaming on Zoom
Takefumi Nosaka (Tokyo Institute of Technology)
Reciprocity of the Chern-Simons invariants of 3-manifolds
Abstract: Given an oriented closed 3-manifold M and a representation π1(M) → SL2(C),
we can define the Chern-Simons invariant and adjoint Reidemeister torsion.
Recently, several physicists and topologists pose and study reciprocity conjectures of the torsions.
Analogously, I pose reciprocity conjectures of the Chern-Simons invariants of 3-manifolds,
and argue some supporting evidence on the conjectures.
Especially, I show that the conjectures hold if Galois descent of a certain K3-group is satisfied.
In this talk, I will explain the backgrounds and the results in detail.