Alladi Ramakrishnan Hall

Henon maps, Short $\mathbb{C}^2$ and beyond

Ratna Pal

IISER Mohali

The broad research area of my talk lies in Several Complex Variables (SCV). In this talk, I shall sketch out some of my recent results, a large part of which are obtained in several joint works with Sayani Bera, John Erik Fornaess, Kaushal Verma and Erlend Wold. 

Results we shall see in the first part of the talk comes under the umbrella of Holomorphic Dynamics.  We shall see a couple of rigidity properties of Henon maps, which happen to be the most important class of polynomial automorphisms of $\mathbb{C}^2$ from the perspective of
holomorphic dynamics. Loosely speaking, by rigidity properties, we mean those properties of Henon maps which determine the underlying Henon maps almost uniquely. 

Describing the structure of the final union in terms of its exhausting domains is referred to in literature as union problem. The genesis of this problem goes back to the early days of classical SCV and a complete answer to this problem seems to be a tangled one.  Domains which are an increasing union of unit balls hold a special stature in literature. In the next part of the talk, we shall survey a few recent results obtained for one such class of domains in $\mathbb{C}^k$, namely the Short $\mathbb{C}^k$'s. If time permits then   we shall also address the union problem for more general exhausting domains.