Abstract: 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.

Abstract: I'll talk about generic 1-dimensional foliations of Stein manifolds that are locally given by vector fields. (The foliations of $\mathbb{C}^n$ serve as the main example.) The leaves of such foliations are Riemann surfaces. I'll describe the topological type of leaves for a generic foliation. The main results can be summarized in the following theorems:

1) For a generic foliation all leaves except for a countable number are homeomorphic to disks, the rest are homeomorphic to cylinders. 2) Generic foliation is Kupka-Smale.

Multidimensional complex analysis, namely approximation theory on Stein manifolds, is the main tool used. All the results used will be referenced and explained.

Abstract: Natural numbers encompasses at least two way of counting. The first one tells how many objects a collection has: there are 84 students in the class, 4 apples in my lunch box or 223,647,852 inhabitants in Indonesia. In the second way of counting, we care for the position of an event in a sequence of events: the final exam will be the 106th day of the academic year, "trois" is the name of the numeral that comes after "deux" in French and the 8,000,000,000th human birth has already happened. As far as we only consider finite collections, these two notions of counting lead to the same arithmetic. But when we try to generalize them to infinite collections, surprising phenomena appear.