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 many geometric situations we may encounter the action of a group G on a manifold M, e.g., in the context of foliations. If the action is free and proper, then the quotient M/G is a smooth manifold. However, in general the quotient M/G need not even be Hausdorff. Under these conditions how can we do diffeomorphism-invariant geometry?

Noncommutative geometry provides us with a solution by trading the badly behaved space M/G for a non-commutative algebra, which essentially plays the role of the algebra of smooth functions on that space. The local index formula of Atiyah and Singer ultimately holds in the setting of noncommutative geometry. This enabled Connes and Moscovici to reformulation of the local index formula in the setting of diffeomorphism-invariant geometry.

The first part of the lectures will be a review of noncommutative geometry and Connes-Moscovici's index theorem in diffeomorphism-invariant geometry.

In the 2nd part, I will hint to on-going projects on the reformulation of the local index formula in two new geometric settings: biholomorphism-invariant geometry of strictly pseudo-convex domains and contactomorphism-invariant geometry of contact manifolds.

A Gerstenhaber algebra is a special kind of graded Poisson algebra. A homotopy Gerstenhaber algebra is a specific "up to homotopy" version of the former. Important examples of homotopy Gerstenhaber algebras are the Hochschild cochains of an associative algebra and the cochain complex of a simplicial set.

In this talk I will address the following problem: What structure does the tensor product of two homotopy Gerstenhaber algebras have? If time permits, I will also talk about formality results for homotopy Gerstenhaber algebras.

In many geometric situations we may encounter the action of a group G on a manifold M, e.g., in the context of foliations. If the action is free and proper, then the quotient M/G is a smooth manifold. However, in general the quotient M/G need not even be Hausdorff. Under these conditions how can we do diffeomorphism-invariant geometry?

Noncommutative geometry provides us with a solution by trading the badly behaved space M/G for a non-commutative algebra, which essentially plays the role of the algebra of smooth functions on that space. The local index formula of Atiyah and Singer ultimately holds in the setting of noncommutative geometry. This enabled Connes and Moscovici to reformulation of the local index formula in the setting of diffeomorphism-invariant geometry.

The first part of the lectures will be a review of noncommutative geometry and Connes-Moscovici's index theorem in diffeomorphism-invariant geometry.

In the 2nd part, I will hint to on-going projects on the reformulation of the local index formula in two new geometric settings: biholomorphism-invariant geometry of strictly pseudo-convex domains and contactomorphism-invariant geometry of contact manifolds.

We define the Lebesgue integral directly by extending the Riemann integral. We will see that the passage of limit under the integral sign can be used effectively to define the Lebesgue integral, and avoids all the usual definitions of measures and measurable sets. This talk is also available to all undergrad students who have good backgroud in calculus.

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.

We 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 will present results in measure theory that are needed in a generalization of the Shurn-Horn theorem to symplectomorphism groups of toric manifolds.

I will review the classification of representations of the symmetric and unitary groups, and how they are related to each other. In particular, I will describe the Young projection operators whose images give the irreducible representations. Then I will give new formulas which use the Young projection operators to construct a family of orthogonal projections which are convenient for computations. Finally, I will describe how computations of traces of maps of symmetric group representations can be used to compute traces of maps of $U(n)$ representations for all $n$ at once. If I have time, I hope to package this up in the language of traced monoidal categories.