C1

We will explain a result of Bridson, Howie, Miller and Short on the finiteness properties of subgroups of direct products of surface groups. More precisely, we will show that a subgroup of a direct product of n surface groups is of finiteness type $FP_n$ if and only if there is virtually a direct product of at most n finitely generated surface groups. All relevant notions will be explained in the talk.

 

I will describe a method to find negatively curved structures on some groups, by manipulating metrics on piecewise hyperbolic complexes. As an example, I will prove that hyperbolic limit groups are CAT(-1).

The simplicial boundary is another way to study the boundary of CAT(0) cube complexes. I will define this boundary introducing the relevant terminology from CAT(0) cube complexes along the way. There will be many examples and many pictures, hopefully to help understanding but also to improve my (not so great) drawing skills. 

I will discuss the circumstances in which residual finiteness properties of an amalgamated free product $A\ast_c B$ may be deduced from the properties of $A$ and $B$, with particular regard to the pro-p residual properties.

In this lecture we describe the effective Chebotarev Theorem for global function fields and show how this can be used to describe the statistics of a polynomial map f in terms of its monodromy groups. With this tool in hand, we will provide a strategy to remove the remaining heuristic in the quasi-polynomial time algorithm for discrete
logarithm problems over finite fields of small characteristic.

Quasimorphisms (QM) of groups to the reals are well studied and are linked to stable commutator length (scl) via Bavard Duality- Theorem. The notion of QM can be generalized to yield maps  between groups such that each QM from one group pulls back to a QM in the other.

We will give both a short overview of features of scl and investigate these generalized QMs with large scale properties of the commutator group. 

In his ICM address in 1983, Gromov proposed a program of classifying finitely generated groups up to quasi-isometry. One way of approaching this is by breaking a group down into simpler parts by means of a JSJ decomposition. I will give a survey of various JSJ theories and related quasi-isometric rigidity results, including recent work by Cashen and Martin.

in this talk I will try to introduce some key ideas and concepts about random walks on discrete spaces, with special interest on random walks on Cayley graphs.

Using the "trichotomy principle" by Boris Zilber I will give model theoretic proofs of appropriate versions of Torelli theorem and Borel-Tits theorem. The first one has interesting applications to anabelian geometry, I won't assume any prior knowledge in model theory.