First meeting, Weizmann Institute, 25.12.23
Stable invariants and their role in word measures on groups
Let w be a word in a free group. A few years ago, Magee and I, relying on the work of Calegari, discovered that the stable commutator length of w, which is a topological invariant, can also be defined in terms of certain Fourier coefficients of w-random unitary matrices. But there are additional Fourier coefficients of w-random unitary matrices. Is there an analogous result for arbitrary coefficients? Is there an analogous result for orthogonal groups or symmetric groups? I will present a new set of conjectures and results involving a plethora of new topological "stable invariants", some of which defined recently by Wilton, which we show/believe play the role of scl in these more general settings. All notions will be defined.
This is joint work with Yotam Shomroni.
The cubical route to understanding groups
Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.
What can pushforward measures tell us about the geometry and singularities of polynomial maps?
Polynomial equations and polynomial maps are central objects in modern mathematics, and understanding their geometry and singularities is of great importance. In this talk, I will pitch the idea that polynomial maps can be studied by investigating analytic properties of regular measures pushed-forward by them (over local and finite fields). Such pushforward measures are amenable to analytic and model-theoretic tools, and the rule of thumb is that singular maps produce pushforward measures with bad analytic behavior. I will discuss some results in this direction, as well as some applications to group theory and representation theory.
Based on joint projects with R. Cluckers, I. Glazer, J. Gordon and S. Sodin.
Dessins d'enfants and visualization of some finite and profinite groups
The Grothendieck-Belyi theory establishes the equivalences between certain combinatorial-topological and arithmetico-geometric categories. These equivalencies define the faithful action of the absolute Galois group on the visualizable objects. Some finite groups turn out to be the Galois invariants; it will be illustrated by the examples of the Mathieu Groups M_11 and M_23. All these concepts and results will be explained (with some examples) in the first part of the talk, which is going to be fairly elementary. In the second part we are going to touch on some more advanced topics -- the relations of the theory with the moduli space of curves and the Grothendieck-Teichmuller group.
You are cordially invited.