The idea of making spacetime into a noncommutative space goes back to the late 60's, and deformation quantization was first introduced in 1978 by Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer. In my talk, I will define a star product on a spacetime $M$, which is a noncommutative deformation of $C^\infty(M)$ and (essentially uniquely) quantize the star product. My goal is to use this to construct a noncommutative field theory on the principal $G$-bundle $P$, considered as a finitely generated projective module in the quantized star product over the star product algebra $C^\infty(M)$.

Classical Tannaka duality is a duality between groups and their categories of representations. The two basic questions it answers are the reconstruction problem (when can a group be reconstructed from its category of representations?) and the recognition problem (can we characterize categories of representations abstractly?).

I will outline how the notion of a Tannakian category can be weakend in order to solve the recognition problem for categories of coherent sheaves of algebraic stacks (if you are an algebraic geometer), respectively categories of comodules of Hopf algebroids (if you are an algebraic topologist). I will end with an application that illustrates one of the differences between these two perspectives.

We will discuss conditions, necessary or sufficient, that the Fourier transform maps one weighted Lebesgue space into another weighted Lebesgue space.

The quantization commutes with reduction problem for Hamiltonian actions of compact Lie groups was solved by Meinrenken in the mid-1990s, and solved again afterwards by many other people using different methods. In this talk, I will consider a generalization of [Q, R]=0 theorem when the manifold is non-compact. In this case, the main issue is that how to quantize a non-compact manifold. I will adopt some ideas from geometric K-homology introduced by Baum and Douglas in 1980s and examine this problem from a topological perspective. One of the applications is that it provides a geometric model for the Kasparov KK group KK(C*(G, X), C).

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.

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.  

Ostrowski showed that there are intimate connections between the gap structure of a Taylor series and the phenomenon of "overconvergence"; that is convergence of subsequences of the partial sums outside the disk of convergence. In this talk we will discuss the corresponding problem for the homogeneous polynomial expansion of a harmonic function. As we will see, the results for harmonic functions display new features in the case of higher dimensions.