Abstract: K theory is a study of the Abelian group constructed from vector bundles $E\to X$ over some topological space $X$ with respect to the direct sum of the vector bundles then taking the Grothendieck closure. KG group is the K theory of $E\to X$ and a space X equipped both with continuous G actions. Moreover, it forms a commutative ring with multiplication $\otimes$ . Of our par- ticular interest to the index seminar I will define equivariant K theory and allude to its applications to index of elliptic operators. The goal of this talk is to develop sufficient tools to understand the index theory of elliptic operators.

Abstract: Large random spaces can be used to model large systems which arise in applications in computer science, biology and engineering. Large random spaces can be also used in pure mathematics to test probabilistically challenging mathematical problems. In my talk I will focus on large random 2-dimensional simplicial complexes which are generalisations of random graphs of Erdös and Rényi. One wants to understand topological properties of random complexes, in particular those which are satisfied with probability close to 1. In the talk I will also describe properties of the fundamental groups of random 2-complexes (cohomological dimension, torsion and others). Aspherical subcomplexes of random 2-complex satisfy the Whitehead conjecture, i.e. all their subcomplexes are also aspherical. The proofs exploit strong hyperbolicity property of random 2-complexes and use inequalities for Cheeger constants and systoles of simplicial surfaces.

The talk is based on a joint work with A. E. Costa.