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 and discuss possible applications.

We obtain a new obstruction to symplectic embeddings derived using the Symplectic Field Theory.

Although we might not be aware of it, Number Theory has entered our daily lives: encrypted Internet connections (e.g., for online banking) are based on algorithms involving giant prime numbers with several hundred or more digits.

In this talk, I will explain how RSA, the most prominent of these public key encryption schemes, works. Then I will address the important question of how one can actually find the large primes needed for these applications. The primality tests used in practice turn out to be probabilistic, which means that they may sometimes give a wrong answer. Hopefully, I will convince you that this is no reason to be worried about the money in your account.

We consider von Neumann algebras generated by regular representations of the infinite-dimensional group B_0^\mathbb Z of infinite, finite-order upper triangular matrices. These regular representations were defined and studied by Alexander Kosyak. They depend on a Gaussian measure on the group of infinite (arbitrary order) upper triangular matrices. A certain condition on the measure implies that the right regular representation is reducible and that the von Neumann algebra generated by the right regular representation is the commutant of the left one. In this case we prove that these von Neumann algebras are type III_1 factors, according to the classification of Alain Connes.

Hamiltonian (or symplectic) diffeomorphisms are well-known to preserve volume. Gromov's work in the eighties implied that the smallest factor in a polydisk also corresponds to an invariant of symplectic embeddings. In dimensions at least 6 we will ask to what extent the areas of other factors place restrictions on symplectic embeddings. Specific constructions will show that any such restrictions must be fairly weak, but they nevertheless exist and remarkably our constructions turn out to be in some sense sharp. We will derive some implications for Hofer's metric on groups of Hamiltonian diffeomorphisms.

In this talk we give a geometric interpretation of characters of certain representations of p-adic GL(n), which are usually called depth zero supercuspidal representations, in terms of Euler characteristics of certain character sheaves. Our work relies on Lusztig's theory of character sheaves, and in some sense suggests a general framework for relating character sheaves on groups over a nonarchimedean local field to smooth representations. (Joint with Clifton Cunningham.)