I will describe some categories of "smooth spaces" which generalize the notion of manifold. The generalizations allow us to form smooth spaces consisting of subsets and quotients of manifolds, as well as loop spaces and other function spaces. In more technical language, these categories of smooth spaces are complete, cocomplete and cartesian closed. I will give examples, discuss possible applications and explain what we have learned about the homotopy theory of these categories. This is work in progress with Enxin Wu.

We develop the basic elements of complex function theory within certain subalgebras of holomorphic functions on coverings of complex manifolds (including holomorphic extension from complex submanifolds, properties of divisors, corona type theorem, holomorphic analogue of Peter-Weyl approximation theorem, Hartogs type theorem, characterization of the uniqueness sets, etc). Our model examples are: (1) subalgebra of Bohr's holomorphic almost periodic functions on tube domains (i.e. the uniform limits of exponential polynomials) (2) subalgebra of all fibrewise bounded holomorphic functions (arising in corona problem for $H^\infty$) (3) subalgebra of holomorphic functions having fibrewise limits.

Our proofs are based on the analogues of Cartan theorems A and B for coherent type sheaves on the maximal ideal spaces of these subalgebras.

This is joint work with Alexander Brudnyi.

Given a polynomial with integer coefficients, one can define invariants of singularities by either considering the polynomial over the complex numbers, or by taking reduction modulo prime integers, and using the Frobenius morphism. I will describe these invariants, and I will discuss some known and conjectural relations between them.

An Einstein manifold is a smooth manifold whose Ricci tensor is proportional to the metric. Many homogeneous spaces can be realized as Einstein manifolds, and have been widely studied for general existence and nonexistence of Einstein metrics. In this talk we will give examples of homogeneous and Einstein manifolds and discuss some of the general underlying theory related to these spaces. We will also briefly discuss how this can be extended to the noncommutative case. Finally, we will show that if we are given a closed, connected, one-dimensional subgroup H of SU(3) that has no nonzero fixed points, then SU(3)/H admits an SU(3)-invariant Riemannian structure of strictly positive curvature. This result was first proven in 1975 by Aloff and Wallach, and it was here that the famous Aloff-Wallach spaces were introduced.

A connected graded algebra is called Koszul if the ground field has a linear resolution, i.e. differentials are defined by matrices that only have linear entries. This condition has less than a million equivalent descriptions. In this survey talk, I will mention a few of these characterizations and examine the resulting homological behavior. As a motivation, I start off by showing the LCS formula for the pure braid group. This is an instance of a more general result about the cohomology ring of a nice class of hyperplane arrangements. I am also planning to describe more examples with origins in quantum groups and show a quick proof for the classical PBW theorem. If there is time left, I will say a few words about the interaction of the Koszul property with the Bloch-Kato conjecture. At last but not least, I will mention the biggest open problem of this area which asks for the correct pronunciation of the word Koszul.