The talk is a report on the recent completion of a project dating from 2010. Our goal is an explicit description of the additive and multiplicative structure of the cohomology of a polyhedral product. The result, for field coefficients and general CW pairs, comes with a transparency sufficient to allow for computation. The problem has an extensive history which I shall try to outline briefly. Some of the earliest results in this direction, for the case of moment-angle complexes, appeared in the work of M. Franz, I. Baskakov, V. Buchstaber and T. Panov and also S. Lopez de Medrano. The focus in this lecture will be on the additive results. This is joint work with M. Bendersky, F.R. Cohen and S. Gitler.

O-minimality is a branch of model theory with roots in real algebraic geometry which provides a family of settings for "tame topology": flexible enough to include most functions used in homotopy theory but without pathologies such as space-filling curves.

I will introduce a model category of spaces based on the definable sets of any o-minimal structure. These model categories resemble the Serre--Quillen model structure on topological spaces but inherit technical advantages from their construction. At the same time, they provide a context in which to better understand the weak polytopes of Knebusch (generalized to the o-minimal setting by Piekosz). This talk is based on joint work with Johan Commelin.

Given a global function field of characteristic p > 0 and an elliptic curve over it, one can study its L-function. The L-function of an elliptic curve is an interesting object at the interface of complex analysis and algebra and lead to the famous Birch and Swinnerton-Dyer conjecture.

Moreover, explicitly computing an L-function is a non-trivial task. Naive methods involve point-counting and are inefficient. Therefore, alternatives should be welcome.

In this talk, we will first introduce the necessary background on elliptic curves and L-functions. Then, we will discuss a possible approach to effectively tackle the problem of computing L-functions of elliptic curves over global function fields.