Activities This Week

Tits’ independence property (P) and geometric density are two properties a group action on a tree can have; Tits showed that the combination of these properties yields interesting simple groups. However, constructing and detecting these properties remained unclear. Burger and Mozes’s introduction of universal groups gave one rich “local-to-global” way to construct groups with (P) by defining their action locally around each vertex. I will present a paper by Colin Reid and Simon Smith which defines local action diagrams, greatly generalizing the Burger-Mozes construction. Local action diagrams turn out to completely classify groups with property (P), as well as to be able to detect geometric density and other global properties of group actions on trees.

Link to the paper: https://link.springer.com/article/10.1007/s00208-026-03412-w

(This talk will be the second part of the Colloquium-Lecture the same week)

We will survey basic tools of project scheduling including Gannt charts, CPM and PERT. We will then consider a new point of view that takes into account the different resources that different potential contractors may have when scheduling the same project. We will also consider the aspects of policies for many similar projects. We will do so taking into account only operational considerations. Nonetheless, we will show that this purely application driven approach can lead to a lot of interesting and diverse mathematics and physics including enumerative combinatorics, Lorentzian geometry, Kardar-Parisi-Zhang processes (Integrable probability) and wave propagation in hyperbolic media. The talk will be self contained.

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.