I shall discuss prosoluble subgroups of free profinite products towards a complete characterization of them in terms of the intersections with the free factors.

The ring of differential operators on a cuspidal curve whose coordinate ring is a numerical semigroup algebra is shown to be a cocommutative and cocomplete left Hopf algebroid, which essentially means that the category of $D$-modules is closed monoidal. If the semigroup is symmetric so that the curve is Gorenstein, it is a full Hopf algebroid (admits an antipode), which means that the subcategory of those $D$-modules that are finite rank vector bundles over the curve is rigid. Based on joint work with Myriam Mahaman

I will discuss some recent forays into some counting problems for free objects. I will focus on free inverse semigroups and free regular $*$-semigroups. I will first discuss recent results joint with M. Kambites, N. Szakács, and R. Webb giving a precise rate of exponential growth of the free inverse monoid of arbitrary (finite) rank, which turns out to be given by a surprisingly complicated but algebraic number.

Given a group $G$ acting on a set $X$, an element $g$ of $G$ is called a derangement if it acts without fixed points on $X$. The Boston-Shalev conjecture, proved by Fulman and Guralnick, asserts that in a finite simple group $G$ acting transitively on $X$, the proportion of derangements is at least some absolute constant $c>0$. After giving an introduction to the subject, I will present a version of the conjecture for the proportion of   *conjugacy classes* containing derangements in finite groups of Lie type.

The length of a finite system of generators for a finite-dimensional algebra over a field is the least positive integer $k$ such that the products of length not exceeding $k$ span this algebra as a vector space.The maximum length for the systems of generators of an algebra is called the length of this algebra. Length function is an important invariant widely used to study finite dimensional algebras since 1959. The length evaluation is a difficult problem.  For example, the length of the full matrix algebra is still  unknown.

I will present some results that were obtained in collaboration with Csaba Schneider (Universidade Federal de Minas Gerais) concerning groups that have a solvable subgroup of prime-power index. Under weak conditions such groups are solvable and, when they are not, the index of their solvable radical is asymptotically small. 
 

Keywords: Solvability and Fermat primes

Given a group $G$ and a partial factor set $\sigma $ of $G,$ we introduce the twisted partial group algebra $\kappa_{par}^{\sigma}G,$ which governs the partial projective $\sigma $-representations of $G$ into algebras over a field $\kappa .$ Using the relation between partial projective representations and twisted partial actions we endow $\kappa_{par}^{\sigma}G$   with the structure of a crossed product by a twisted partial action of $G$ on a commutative subalgebra of $\kappa_{par}^{\sigma}G.$   Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral se