We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.
The moduli spaces of surfaces assemble into a modular $\infty$-operad, closely related to the $2$-dimensional bordism category. I will establish an obstruction theory for algebras over this surface modular operad where the obstruction for extending from genus $g-1$ to $g$ is controlled by the curve complex $C(\Sigma_g)$.
A Yang-Baxter element in a monoidal category gives a weak form of braiding. We explain how such an element allows to define a semi-simplicial set whose connectivity rules homological stability for certain automorphism groups in the category, and how this can be used in the category of bimarked surfaces to give a quite direct proof of slope 2/3 stability of the homology of the mapping class groups of surfaces. This is joint work with Oscar Harr and Max Vistrup.
In this talk I will give an overview of two related projects. The first project concerns the high-degree rational cohomology of the special linear group of a number ring $R$. Church--Farb--Putman conjectured that, when $R$ is the integers, these cohomology groups vanish in a range close to the virtual cohomological dimension. The groups $SL_n(R)$ satisfy a twisted analogue of Poincaré duality called virtual Bieri--Eckmann duality, and their rational cohomology groups are governed by $SL_n(R)$-representations called the Steinberg modules.
Hirzebruch's signature theorem relates the signature of the intersection form of a manifold with the integral over the manifold of a certain characteristic class, namely the L-class. This was extended to families of smooth manifolds (i.e. smooth fibre bundles) by Atiyah, using the family index theorem for the fibrewise signature operator. In this setting it relates the Chern character of a certain vector bundle constructed from the local system of intersection forms of the fibres, with the fibre integral of the L-class.
I want to give a report on the recent progress on the high (=near the virtual cohomological dimension) cohomology of arithmetic groups such as special linear groups of number rings and their congruence subgroups. These cohomology groups have important connections to algebraic number theory, especially when considering their Hecke actions.
In this talk, I am going to explain the main results of my recent preprint (arXiv:2105.12591). The primary goal will be to prove that for every affinoid analytic adic space $X$, pseudocoherent complexes, perfect complexes, and finite projective modules over $\mathcal{O}_X(X)$ form a stack with respect to the analytic topology on $X$.
Classical Morita equivalence defines an Euler class for modules that generalizes the Euler characteristic for spaces. Using this understanding of the Euler class, fundamental structure of the class becomes accessible. Among many choices, in this talk I'll focus on compatibility of the Euler class with a familiar pairing on Hochschild homology since it also allows the exploration for different forms of duality.
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