Metric noncommutative geometry: In this series of talks, I am going to review some metric aspects of noncommutative geometry due to Alain Connes. To be more specific, I will discuss four formulas in Riemannian geometry and formulate them in algebraic forms, so that they can be considered in “noncommutative spaces” as well. These four formulas are concerned about geodesic distance, volume form, space of gauge potentials and Yang-Mills functional action. In the first talk last week, we looked at the spectral triple of a Riemannian manifold which in a sense captures our algebraic data. We also saw a formula for geodesic distance using just this piece of information. For the next talk, I'm planning to discuss volume forms and space of gauge potentials.

I will try to cover the following topics: 1. Equivariant cohomology: Definition, motivation, 2. Two constructions of E, 3. Cartan Model and some examples.

Beginning with Fermat's characterization of primes which are the sum of two squares, I would like to show how this naturally leads to Waring's Problem for Integers and then to Waring's Problem for Homogeneous Polynomials.

One half of this latter problem has been solved in recent years and I will explain the nature of the approach to that solution through the study of the higher secant varieties of Veronese varieties and the study of non-reduced subschemes of projective n-space.

Despite the technical sounding terms above, the talk is aimed at a general audience and I will keep the technicalities to a minimum.

The notion of rank for matrices (or $2$-tensors) is well understood but the generalizations to higher order tensors is much less well understood and is an active area of research with applications in Biology, Statistics, Computing and Signal Processing.

I will explain the nature of the problem, the notions of "tensor rank" and "border rank" and how problems in this area have been studied through the lens of Higher Secant Varieties of Segre Varieties and the cohomology of non-reduced varieties whose support is a union of linear spaces. I will mention some recent progress on the problems that have been done by me and my collaborators M. V. Catalisano and A. Gimigliano and also some conjectures and open problems. If time permits I will also explain some recent work of Landsberg, Weyman and ourselves on the defining ideals of these higher order secant varieties.