Let $k$ be an algebraicallly closed field of characteristic $p\geq 0$ and let $G$ be a linear algebraic group of rank $\ell\geq 1$ over $k$. Let $V$ be a rational finite-dimensional $kG$-module and let $V_g(\mu)$ denote the eigenspace corresponding to the eigenvalue $\mu\in k^*$ of $g \in G$ on $V$. We set $\nu_G(V)=\min\{\dim(V)-\dim(V_g(\mu))| g \in G \setminus Z(G), \mu \in k^*\}$.

Let $G$ be a subgroup of $S_n$. What can be said on the number of conjugacy classes of $G$, in terms of $n$?

I will review many results from the literature and give examples. I will then present an upper bound for the case where $G$ is primitive with nonabelian socle. This states that either $G$ belongs to explicit families of examples, or the number of conjugacy classes is smaller than $n/2$, and in fact, it is $o(n)$. I will finish with a few questions. Joint work with Nick Gill. 

A celebrated theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the derived group $G'$  is finite.  In this talk we will discuss a stronger version of  Neumann's result and some consequences for finite and profinite groups.  Based on a joint work with Pavel Shumyatsky.

This talk concerns numerical semigroups, i.e. cofinite submonoids $S$ of $\mathbb{N}$. Wilf's conjecture (1987) on numerical semigroups $S$ states that $e \cdot \ell \ge c$, where $e$ is the embedding dimension of $S$ and $\ell$ is the number of elements of $S$ which are smaller than its conductor $c$. We shall present the main ideas of a recently published proof of Wilf's conjecture in the particular case $e \ge m/3$, where $m$ is the smallest nonzero element of $S$. Asymptotically, most numerical semigroups seem to satisfy $e \ge m/3$ as the genus goes to infinity.

Using graphs as a tool to encode properties of groups is a well established approach to many problems nowadays.

Let G be a permutation group acting on a finite set Ω. A subset B of Ω is called a base for G if the pointwise stabilizer of B in G is trivial.

In the 19th century, bounding the order of a finite primitive permutation group G was a problem that attracted a lot of attention. Early investigations of bases then arose because such a problem reduces to that of bounding the minimal size of a base of G. 

A Steiner triple system STS(v) is a set of triples of {1, 2, . . . , v} such that every pair of points belongs to exactly one of these triples. A Kirkman triple system KTS(v) is a STS(v) whose triples can be partitioned into parallel classes, each of which is a partition of the point set. A KTS(v) is called 3-pyramidal if it admits a group of automorphisms that fixes 3 points and acts regularly on the other points. I will present recent results we obtained about 3-pyramidal Kirkman triple systems. This is joint work with  S. Bonvicini, M. Buratti, G. Rinaldi and T. Traetta.

If G is a finite group and k a field of characteristic p, the group algebra kG can be written uniquely as a direct product of indecomposable algebras, known as the "blocks'' of G.  The representation theory of kG can now be treated one block at a time, and some blocks may be easier than others.  To each block B one may associate a p-subgroup of G, called its "defect group'', which measures the difficulty of B.  Very little is known in general, but blocks whose defect group is cyclic are completely understood.  Working with Ricardo Franquiz Flores, we have begun to extend block theory to pro

Building on the classification of modules for algebraic groups with finitely many orbits on subspaces, we determine all irreducible modules for simple algebraic groups that are self-dual and have finitely many orbits on totally singular k-spaces (k=1 or k=2). This question is naturally connected with the problem of finding for which pairs of subgroups H,J of an algebraic group G there are finitely many (H,J)-double cosets. We provide a solution to the question when J is a maximal parabolic subgroup P_k of a classical group.

A central polynomial of an algebra A is a polynomial f in non-commutative variables that takes central values when evaluated in A. In case it vanishes in A, f is called a polynomial identities of A, otherwise f is a proper central polynomial of A. 

The purpose of this talk is to survey some old and new results on the growth of the spaces of central polynomials, proper central polynomials and polynomial identities of  an algebra  over a field of characteristic zero.