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.
4月18日 -- オンライン開催, 17:00 -- 18:00
丸山 修平 (中央大学)
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.
4月25日 -- オンライン開催, 17:00 -- 18:00
野澤 啓 (立命館大学)
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.
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.
5月30日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
児玉 悠弥 (東京都立大学)
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.
