Abstract: Let $T$ be a compact torus. Goresky, Kottwitz and Macpherson showed that for a large and interesting class of $T$-equivariant projective varieties $X$, the equivariant cohomology ring $H_T^*(X)$ may be encoded in a graph, now called the GKM-graph, with vertices corresponding to the fixed points of $X$ and edges labeled by the weights, $Hom(T, U(1))$.

In this lecture, we explain how the GKM construction can be generalized to any finite $T$-CW complex. This generalization gives rise to new mathematical objects: GKM-hypergraphs and GKM-sheaves. If time permits, we will show how these methods were used to resolve a conjecture concerning the moduli space of flat connections over a non-orientable surface.

Abstract: This series of lectures provides an introduction to the basic calculus of pseudodifferential operators defined on Euclidean spaces. We will start by reviewing the space of Schwartz functions, the convolution, the Fourier transform, and their basic properties. Then we prove two important results for studying pseudodifferential operators: the Fourier inversion formula and the Plancherel theorem. We will proceed by finding an asymptotic expansion for the symbol of formal adjoint and composition of pseudodifferential operators. We will end the lectures by introducing a notion of ellipticity and constructing parametrices for elliptic pseudodifferential operators.

Abstract: When resolving singularities of an algebraic variety one produces a smooth model and a birational map to the original variety. The desingularization is said to be strict when this map only changes singular points, i.e. it is an isomorphism over the smooth points. Sometimes it is needed to preserve other singularities besides the smooth points. One may want to get an isomorphism over the simple normal crossings points, or over the normal crossings points, or any other family of singularity types. These desingularizations may or may not exist. We will talk about a way to approach the construction of these desingularizations in the case of semi simple normal crossings singularities (the analogue of simple normal crossings on a non normal space).

Abstract: In this talk, we will discuss some of the history behind the Riemann hypothesis, including its relation to the distribution of primes, attempts at a proof over the years and its appearance and importance in many areas of mathematics. This will lead to surprising real life examples where the Riemann hypothesis applies, such as quantum physics.