פעילויות השבוע

Model theory of Abelian groups is extensively studied in the literature also in recent years. An identical inclusion is a formula that can be expressed as a (possibly infinitary) disjunctive identity u = v1 ∨ u = v2 ∨ u = v3 ∨ . . . , or, equivalently, as a universally closed identical equality of subsets of words (terms). For groups and rings, the classes defined by identical inclusions and by infinitary disjunctive identities are coincide, for semigroups they do not coincide. A class of algebras defined by a set of identical inclusions is called an inclusive variety. An inclusive variety that can not be defined by first order formulas is called a nonelementary inclusive variety. An inclusive variety defined by a system of identical inclusions - each depending on a finite set of variables - is called a quasielementary inclusive variety.

We describe elementary, nonelementary and quasielementary inclusive varieties of Abelian groups. There exist continuum many inclusive varieties of each of these kinds. We also determine Abelian groups defined by identical inclusions up to isomorphism and classify Abelian groups up to inclusive equivalence.

The discovery of the Fargues–Fontaine curve has led to major advances in the geometrization of $p$-adic Hodge theory. In this talk, we explain how several $p$-adic cohomology theories can be realized as vector bundles on the Fargues–Fontaine curve. We then present a motivic approach to show the uniqueness of such vector bundles, which, in particular, yields comparison theorems for them. Moreover, we show that one can choose a canonical comparison isomorphism between these vector bundles.

Nuclear dimension, introduced by Winter and Zacharias, is an invariant for C-algebras which generalizes covering dimension for compact Hausdorff spaces, and plays an important role in structure theory for amenable C-algebras. It is usually mainly interesting to show that it is finite, as opposed to computing its actual value. Given an action of a group G on a locally compact Hausdorff space X, one forms the crossed product C*-algebra C_0(X) \rtimes G; this construction has been heavily studied in the field.

I will discuss joint work with Jianchao Wu, in which we find bounds on the nuclear dimension of nuclear dimension of the crossed product for a large class of group actions, including arbitrary actions of finitely generated virtually nilpotent groups on finite dimensional spaces and certain boundary actions of hyperbolic groups. This involves introducing a notion of “long and thin covers” which serves as the appropriate generalization of Rokhlin-type towers for non-free actions. As another application of the result, we generalize a result of Joseph and construct a family of profinite actions of wreath products of finite abelian groups by Z^d which are allosteric (that is, are minimal and topologically free, but not essentially free, meaning that fixed points sets are meager but have non-zero measure with respect to the unique invariant measure), and show that the resulting crossed product are well behaved from the perspective of structure and classification of C*-algebras.

As the paper is rather long, in the talk I will just give an overview of some of the definitions and techniques, intended for people from dynamical systems who are not experts in C*-algebras.