Kourovka Notebook

On Thursday the 17-th of February, 2022, Dan Segal (All Souls College, Oxford) will give an online talk “Groups, rings, logic” at 20:00 Novosibirsk time (13:00 London time).

Abstract: In group theory, interesting statements about a group usually can’t be expressed in the language of first-order logic. It turns out, however, that some groups can actually be determined by their first-order properties, or, even more strongly, by a single first-order sentence. In the latter case the group is said to be finitely axiomatizable. I will describe some examples of this phenomenon (joint work with A. Nies and K. Tent). One family of results concerns axiomatizability of p-adic analytic pro-p groups, within the class of all profinite groups. Another main result is that for an adjoint simple Chevalley group of rank at least 2 and an integral domain R; the group G(R) is bi-interpretable with the ring R. This means in particular that first-order properties of the group G(R) correspond to first-order properties of the ring R. As many rings are known to be finitely axiomatizable we obtain the corresponding result for many groups; this holds in particular for every finitely generated group of the form G(R).

Chair of the session: Evgeny Khukhro

The link to the Zoom meeting: https://us02web.zoom.us/j/3167567840?pwd=aExXZmUxVmE5blYxYTVRMHhrakxuQT09

(ID 316 756 7840, Password gD11t2).

See the program updates at http://mca.nsu.ru/kourovkaforum/.

If you wish to be added to the seminar mailing list and receive regular announcements, please contact the organizers at ilygor8@gmail.ru .

On Thursday the 3rd of February, 2022, Alex Lubotzky (Weizmann Institute and Hebrew University, Israel) will give an online talk “Stability and testability of permutations’ equations” at 16:00 Novosibirsk time (9:00 London time).

Abstract: Let A and B be two permutations in Sym(n) which “almost commute” — are they a small deformation of permutations that truly commute? More generally, if R is a system of word-equations in variables X=(x_1,….,x_d) and A=(A_1,…., A_d) permutations which are nearly solution; are they near true solutions? It turns out that the answer to this question depends only on the group presented by the generators X and relations R. This leads to the notions of “stable groups” and “testable groups”. We will present a few results and methods which were developed in recent years to check whether a group is stable\testable. We will also describe the connection of this subject with property testing in computer science, with the long-standing problem of whether every group is sofic and with IRS’s ( =invariant random subgroups).
A number of open questions will be presented.

Chair of the session: Mikhail Belolipetsky

The link to the Zoom meeting:  https://us02web.zoom.us/j/3167567840?pwd=aExXZmUxVmE5blYxYTVRMHhrakxuQT09 (ID 316 756 7840, Password gD11t2).

See the program updates at http://mca.nsu.ru/kourovkaforum/.