Dear mathematicians, we are forced to stop the work of the forum. We will resume the workshop as soon as possible.
SEMINAR #1. February 3, 2022, 16:00 Novosibirsk time (UTC +7)
Title: Stability and testability of permutations' equations Speaker: Alex Lubotzky (Weizmann Institute and Hebrew University, Israel) 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.
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 groupG(R) correspond to first-order properties of the ringR. 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).
SEMINAR #1. October 29, 2020, 16:00 Novosibirsk time (UTC +7)
Title: Some problems about bounding length parameters of finite groups Speaker:Evgeny Khukhro (University of Lincoln & Sobolev Institute of Mathematics)
Abstract: Bounding the p-length or/and nilpotent length (Fitting height) of finite groups became a classical area of research since the seminal works of P. Hall and G. Higman of 1956 on Restricted Burnside Problem and of J. Thompson of 1964 on automorphisms. Various generalizations and improvements followed in a torrent of subsequent papers, mainly in the 1960s and 1970s. This area of research became somewhat less fashionable later, and a certain perception was even formed that nothing interesting remains to be discovered, that all the methods are already known. The purpose of my talk is to dispel this perception, to highlight several important open problems results about bounding the Fitting height and p-length of finite soluble groups, as well as other length parameters applicable to not necessarily soluble groups. Among examples, there are problems about finite groups with fixed-point-free and almost fixed-point-free automorphisms, about generalizations of the Restricted Burnside Problem, about coset identities, which have applications in the study of profinite groups, etc.
SEMINAR #2. November 12, 2020, 19:00 Novosibirsk time (UTC +7)
Title: Axial algebras and groups Speaker: Sergey Shpectorov (University of Birmingham)
Abstract: Axial algebras were introduced as a broad generalisation of Majorana algebras of Ivanov, which in turn generalise the properties of the Griess algebra whose automorphism group is the Monster sporadic simple group. Axial algebras with graded fusion laws have inherent symmetry and hence they always have nontrivial automorphism groups. In the talk we survey a number of recent results concerning axial algebras of Jordan and Monster type and mention several interesting open problems.
SEMINAR #3. November 26, 2020, 19:00 Novosibirsk time (UTC +7)
Title: Asymptotic group theory: trends, results, and open questions Speaker: Alexei Miasnikov (Stevens Institute of Technology)
Abstract: I will talk about asymptotic group theory, which concerns with group properties that hold asymptotically almost surely. These include random groups (finitely presented, nilpotent, solvable, periodic, finite, etc.), random subgroups, generic properties, and 0-1 laws.
For questions about the Kourovka notebook, please contact
Evgeny Khukhro e-mail: email@example.com
Victor Mazurov e-mail: firstname.lastname@example.org
The Kourovka notebook is a collection of open problems in group theory published every two to four years at the Sobolev Institute of Mathematics since 1965. For more than fifty years the Kourovka notebook has served as a unique means of communication for researchers in group theory and related topics. Solutions of many problems from the Kourovka notebook led to significant advances in various areas of group theory and related fields of mathematics.
The series of online seminars titled Kourovka forum is an additional feature of the Kourovka notebook. Via this series of seminars, we would like to highlight the most promising research directions in group theory and, more broadly, algebra, along with other fields of mathematics where groups are applied. We believe that clear formulations of the most intriguing problems will help young researchers to focus their efforts and may give them a higher chance of achieving great mathematical results.
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The seminar is supported by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation.