Several results in convex geometry establish asymptotic estimates for the gap between a convex domain and approximating polyhedra of increasing complexity. Is it possible to do the same for approximations of pseudoconvex domains by analytic polyhedra? In this talk, I will discuss a class of polyhedral objects in strongly pseudoconvex domains that allow for such estimates. Connections with both the Fefferman hypersurface measure and a tiling problem on the Heisenberg group will be discussed. I will also indicate how our formulation suggests a way to discuss volume approximations of more general pseudoconvex domains.

In this series of talks we will review Quillen's celebrated determinant line bundle construction on the space of Fredholm operators and study the geometry of this line bundle over the space of Cauchy-Riemann operators on a Riemann surface. Quillen defines a Hermitian metric using zeta regularized determinants on this line bundle and computes its curvature. This computation is then used to define a holomorphic determinant for Cauchy-Riemann operators. It is fairly easy to see that one cannot define a determinant function which is both holomorphic and gauge invariant (conformal anomaly).

Then we will move to a noncommutative setting and review our recent work, with A. Fathi and A. Ghorbanpour, in which we studied the curvature of the determinant line bundle over a space of Dirac operators on the noncommutative two torus. We developed the tools that are needed in our computation of the curvature, including an algebra of logarithmic pseudodifferential symbols and a Konstsevich-Vishik type trace on this algebra. These talks will move slowly and the idea is to develop the necessary tools for further study of the determinant line bundle in noncommutative geometry.

The theory of Lie groups originated from Sophus Lie's work on symmetry groups of differential equations. Those are transformation groups acting on the space of independent and dependent variables of a differential equation, which transform a solution to another solution.

Our focus is on 1-parameter connected Lie groups and we use the generating vector field as a powerful tool for the calculations. Once we obtain a symmetry group for a differential equation, we are able to construct new solutions from already known ones.

We will use (primary and) secondary cohomology operations to describe the structure of the stable stems in low dimensions and compute a few Toda brackets in the stable stems.

I will give a general introduction to the subject of Massey products and Toda brackets, suitable for a general mathematical audience. I will describe some of their many uses in algebra and topology, and I will present some new results about the general theory of fourfold Massey products.

We will introduce the Clifford algebra associated to a vector space equipped a quadratic form. As an important case, then we give a description of the Clifford algebra for $\mathbb{R}^n$ and $\mathbb{C}^n$, equipped with the standard quadratic form, as a subalgebra of matrix algebras and prove the periodicity theorem for these algebras. Moreover, representation theory of the Clifford algebras will be discussed.

Nakajima constructed highest weight representations of A-type affine Kac-Moody algebras by using the (equivariant) cohomology of cyclic quiver varieties. In the present talk I will describe a geometric realization of level one highest weight representations of A-type affine Kac-Moody algebras by using moduli spaces of sheaves on a 2-dimensional root toric stack over the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}_k$. If time permits, I will explain the conjectural relation between these two geometric constructions.