Abstracts for Twinned Conference on Homotopy Theory with Applications to Arithmetic and Geometry

I will discuss the problem of constructing a natural rational structure on periodic cyclic homology of dg-algebras and dg-categories. The promising candidate is A. Blanc’s topological K-theory. I will show that it, indeed, provides a rational structure in a number of cases and I will discuss its structural properties and possible applications.

I will report on joint work with Achim Krause and Thomas Nikolaus on an algorithm to compute the higher algebraic $K$-groups of rings such as $\mathbb{Z}/p^n$.

In this talk, I speak about establishing a close connection between three notions attached to a modular subgroup, namely, the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup, and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

This talk will introduce a notion of a stratification of a (stable presentable) category. A stratification of a scheme determines a stratification of its category of quasi-coherent sheaves. A stratification of a topological space determines a stratification of its category of linear sheaves. Such a stratification determines two reconstruction theorems, each reconstructing the category in terms of its strata and gluing data. An abelian group can be reconstructed in terms of its p-completions and its rationalization, as well as from its p-torsion and its corationalization.