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.

Profinite groups in which the centralizer of any non-identity element  is abelian (i.e., profinite CA-groups) are also known as profinite  commutativity-transitivity groups.

In this talk I shall present a dichotomy theorem obtained with  P. Shumyatsky and P. Zalesskii (2019, Israel J. Math, v. 230):  Any profinite CA-group has a finite index closed subgroup that  is either abelian or pro-p.

Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm locally at every completion of K is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions. 

In this series of three lectures, we will discuss two important and relatively new methods in combinatorics. Firstly, the probabilistic method developed by Erdős and which has now seen numerous applications in various areas of mathematics such as number theory, linear algebra, additive combinatorics, real analysis, as well as in computer science. Secondly, we will give an introduction to the polynomial method and present few surprising applications of linear algebra type-arguments to problems in combinatorics and geometry.