In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

In the chromatic take on stable homotopy theory, the homotopy type of a finite $p$-local spectrum $X$ is reassembled from its localizations with respect to the various Morava $K$-theories. In the early 1990s, Hopkins proprosed a brash conjecture for how the reassembly process works. I'll review the conjecture and the state of the art -- including a verfication of the conjecture at $p=3$ and chromatic level $2$, where the question is not simply algebraic and where there has been a proposed counterexample. This is joint work with Hans-Werner Henn.

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.

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.

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).

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.

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well. Arash will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures. The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

We start by considering the notions of a torsor and a gerbe for a discrete Abelian group A (over a discrete set X). We describe central extensions of a group G by A as multiplicative A-torsors on X=G. A categorification of this construction is given by the notion of a gerbal central extension of a group G by an Abelian group A. We classify gerbal A-central extensions of the group G by A-valued 3-cocycles of G.

Then we recall the construction of determinantal gerbe Det(V) of an infinite dimensional Tate vector space V (e.g. V=k((s)) ) due to Kapranov. We explain that the obstruction of Det(V) to be GL(V)-equivariant provides the well known central extension of GL(V) by k^*. Given a 2-Tate vector space V, e.g. V=k((s))((t)), we consider the determinantal 2-gerbe 2-Det(V). The obstruction of 2-Det(V) to be GL(V)-equivariant provides a gerbal central extension of GL(V) by k^*. This construction leads to gerbal central extensions of double loop groups.

TBA