Consider two domains $D$ and $G$ in $\mathbb{C}^n$, each of which is the product of smoothly bounded domains, and assume that each factor of $D$ satisfies condition R, i.e, the Bergman projection preserves the class of functions smooth up to the boundary. We show that any proper holomorphic map from $D$ to $G$ extends smoothly to the closures, and splits as a product of equidimensional mappings of the factors. We also consider some possible generalization to a class of piecewise smooth domains. This is joint work with Kaushal Verma.

Tropical geometry is the discrete geometry version of algebraic geometry which provides a framework for reducing classical algebraic geometry questions into combinatorial ones. We start by introducing the tropical semiring and some motivational examples from computer science. The main objective of the talk is to explain the equivalent constructions of the tropical varieties.

Holomorphic homotopy theory studies continuous deformations of holomorphic mappings and the major question is when one holomorphic mapping can be continuously deformed into another holomorphic mapping via holomorphic mappings. We call such mappings h-homotopic.

The serious studies of such questions was initiated by M. Gromov in 1989 who was interested in the homotopical Oka principle: when homotopic holomorphic mappings are h-homotopic? It led to the notions of Oka and elliptic manifolds and many interesting applications. Recently h-homotopical constructions appeared on non-elliptic manifolds which are much more general. For example, in the description of B. Joricke of envelopes of holomorphy and the disk formula for plurisubharmonic subextensions by F. Larusson and the speaker. These results raised an interest to h-homotopies on general complex manifolds.

In the talk we will briefly present Gromov's theory and then discuss the h-homotopy theory for general manifolds including the results of Joricke and Larusson-Poletsky. Finally, we will show how an h-analog for the fundamental group can be introduced.

The talk will be accessible to anybody with the knowledge of the first graduate course in complex variables.