TBA

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.

Topological spaces - such as classifying spaces of small categories and spacetimes - often admit extra temporal structure. Such "directed spaces" often arise as geometric realizations of simplicial sets and cubical sets; the temporal structure encodes orientations of simplices and 1-cubes. Directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Nevertheless, we present simplicial and cubical approximation theorems for a homotopy theory of directed spaces. In our directed setting, ordinal subdivision plays the role of barycentric subdivision and cubical sets equipped with coherent compositions of higher cubes serve as analogues of Kan complexes. We consequently show that geometric realization induces an equivalence between certain weak homotopy diagram categories of cubical sets and directed spaces. As applications, we show that directed analogues of homotopy groups of spheres are uninteresting, sketch constructions of a (more interesting) cubical singular cohomology theory for directed spaces, and calculate such "directed cohomology" monoids for various directed spaces of interest.

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.

A higher order automorphic form is a generalization of the notion of a classical automorphic form. I will discuss the definition and I will review some recent results.

The idea of (left) regular representation has its origins in group theory and a well known theorem of Arthur Cayley from 19th century which is part of any introduction to group theory. The goal of this talk is to highlight a few of its ramifications, extensions, and applications in different areas of mathematics, including analysis and algebra. After a quick discussion of the general idea of representation, I shall try to point out - by way of examples drawn from different fields - how universal, as well as simple and useful, the idea of regular representation is. I shall make every effort to make this talk as self contained as possible for an undergraduate talk.

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

The Kunneth spectral sequence for abelian sheaf cohomology is displayed and discussed. Computational applications of this spectral sequence for the etale cohomology of classifying spaces of algebraic groups will also be displayed.