Boris Kunyavskiĭ

The Bogomolov Multiplier of Finite Simple Groups

The subgroup of the Schur multiplier of a finite group G consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero is called the Bogomolov multiplier of G. We prove that if G is quasisimple or almost simple, its Bogomolov multiplier is trivial except for the case of certain covers of PSL(3, 4).

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

Bicyclic algebras of prime exponent over function fields

We examine some properties of bicyclic algebras, i.e. the tensor product of two cyclic algebras, defined over a purely transcendental function field in one variable. We focus on the following problem: When does the set of local invariants of such an algebra coincide with the set of local invariants of some cyclic algebra? Although we show this is not always the case, we determine when it happens for the case where all degeneration points are defined over the ground field. Our main tool is Faddeevs theory. We also study a geometric counterpart of this problem (pencils of Severi-Brauer varieties with prescribed degeneration data).

Division algebras that ramify only on a plane quartic curve

Let k be an algebraically closed field of characteristic 0. We prove that any division algebra over k(x, y) whose ramification locus lies on a quartic curve is cyclic.

Deterministic primality tests based on tori and elliptic curves

Abstract We develop a general framework for producing deterministic primality tests based on commutative group schemes over rings of integers. Our focus is on the cases of algebraic tori and elliptic curves. The proposed general machinery provides several series of tests which include, as special cases, tests discovered by Gross and by Denomme and Savin for Mersenne and Fermat primes, primes of the form 2 2 l + 1 − 2 l + 1 , as well as some new ones.