Let $M$ be a $2n$ dimensional manifold with a symplectic form $\omega$. This symplectic form determines a Lie subgroup, $Symp_{\omega}(M)$, of $Diff(M)$ called the symplectomorphism group. There is yet another subgroup of interest called the Hamiltonian group. $Ham_{\omega}(M)$ is an infinite dimensional Lie group yet it has some properties of compact finite dimensional Lie groups. The presence of finite dimensional tori $T^k$, k≤n, inside $Ham_{\omega}(M)$ determines $M$ completely when $n = 2, 4$. I will sketch the proof of the case $n = 4$, a result of my supervisor and his colleagues.

In the present talk I describe a (conjectural) relation between moduli spaces of (framed) sheaves on some two-dimensional root toric stacks and Nakajima quiver varieties of type the affine Dynkin diagram $\hat{A}_{n}$. If time permits, I will discuss an application of this relation to representation theory of Kac-Moody algebras (and vertex algebras).

In his work on Dulac's problem, Ilyashenko uses a quasianalytic class of functions that is a group under composition, but not closed under addition or multiplication. When trying to extend Ilyashenko's ideas to understand certain cases of Hilbert's 16th problem, it seems desirable to be able to define corresponding quasianalytic classes in several variables that are also closed under various algebraic operations, such as addition, multiplication, blow-ups, etc. One possible way to achieve this requires us to first extend the one-variable class into a quasianalytic algebra whose functions have unique asymptotic expansions based on monomials definable in $R_{an,exp}$. I will explain some of the difficulties that arise in constructing such an algebra and how far (or close) we are to obtaining it. (This is joint work with Tobias Kaiser.)

In this talk, we will show a way to develop category theory in Univalent foundations. As it turns out, the naive reformulation of the standard axioms from set theory leads to a rather ill-behaved notion. We show how it can be refined. We also observe that for these redefined categories, two concepts of equivalence and isomorphism are the same.

Operator theory on spaces of holomorphic functions has undergone a rapid development in the last several decades. It started with spaces of functions holomorphic on the unit disk in the complex plane and it kept developing into higher dimensions. Particularly well studied cases are operators acting on Bergman spaces on the ball and polydisk. We use $\overline\partial$-techniques to study compactness of Hankel and Toeplitz operators on Bergman spaces on pseudoconvex domains in $\mathbb{C}^n$. This is joint work with Sonmez Sahutoglu.