In this talk we will discuss the relation between homological properties of Stanley--Reisner rings of simplicial complexes and topology of polyhedral products. A key result in this direction is the characterization of Golod rings over rationals in terms of their Poincaré series and loop homology of the corresponding moment-angle-complexes. Much more can be said if only flag simplicial complexes are considered. We will see how the methods and objects of toric topology allow us to interpret the results on Poincaré series and Koszul homology of face rings as well as to get new results.

An exciting theme in combinatorics is degenerating a continuous theory into a discrete one and asking which features of the original are preserved. This talk will focus on our effort to replicate classical theorems of topology in the setting of discrete homotopy and singular homology theories for graphs. These combinatorial theories have a distinct cubical flavor, with the roles of spheres and simplices played by grids and hypercube graphs. A major goal has been to connect the two by way of a discrete Hurewicz theorem. Our first result marks progress toward this goal: We will describe a natural map from discrete homotopy to discrete homology, and show that it is surjective in a large number of cases. As a corollary, we prove the existence of nontrivial higher discrete homotopy groups. Our second result is a discrete version of a theorem of P. A. Smith, which says that the fundamental group of a nontrivial symmetric product of $X$ is isomorphic to the first homology group of $X$.

Chow groups of classifying spaces of tori arise in the theory of cohomological invariants. In this talk we will give an overview of a computer-assisted proof that \({\rm CH}^2\) groups of the classifying space \(BT\) are torsion free when \({\rm dim}(T) <= 5\)