We answer a problem posed by Panov, which is to describe the relationship between the wedge summands in a homotopy decomposition of the moment-angle complex corresponding to a disjoint union of m points and the connected sum factors in a diffeomorphism decomposition of the moment-angle manifold corresponding to the simple polytope obtained by making m vertex cuts on a standard d-simplex.

I shall describe results on degrees of maps between (n-2)-connected (2n-1)-dimensional Poincare complexes which have torsion free integral homology. In particular, necessary and sufficient algebraic conditions for the existence of map degrees between such Poincare complexes will be established and the calculation of the set of all map degrees between certain Poincare complexes will be illustrated. Time permitting, we shall see how the knowledge of possible degrees of self maps can be used to classify, up to homotopy, torsion free (n-2)-connected (2n-1)-dimensional Poincare complexes for low n.

This talk outlines a close connection between the moduli spaces of aperiodic tilings and attractors of certain types of so-called chaotic dynamical systems. We take in a little homological algebra and geometric group theory on the way.

I will discuss several probabilistic models producing simplicial complexes, manifolds and discrete groups. Random simplicial complexes are high dimensional analogues of random graphs and can be used for studying the behaviour of large systems or networks depending on many random parameters. We are interested in properties of random spaces which are satisfies with probability tending to one. Using probabilistic models one may also test probabilistically the validity of open topological questions such as the Whitehead and the Eilenberg Ð Ganea conjectures.

A polyhedral product is a space constructed from an abstract simplicial complex and a sequence of pairs of spaces, which appear in several important constructions such as (higher) Whitehead products, graph products of groups, coordinate subspace arrangements, and so on. I will talk about a homotopy decomposition of polyhedral products motivated by a property of Stanley-Reisner rings called Golod.

In this talk, we consider homotopy theoretic classification problems of gauge groups. For a given principal bundle P, the gauge group Gau(P) is the topological group of automorphisms on P. Our main theorem is as follows: if we fix the base B and the Lie group G, there are only finite gauge groups of principal G-bundles over B up to A_n-equivalence. As an example, we consider the classification of the A_n-types of the gauge groups of principal SU(2)-bundles.