Tropical Geometry, Toric Geometry and Beyond

A (complex) algebraic variety X is the solution set of a number of polynomial equations in several variables (over complex numbers) and is the main object of study in algebraic geometry. In this talk we give an introduction to “tropical geometry” which aims to study "exponential behavior of X at infinity”. This behavior at infinity can be encoded in a finite union of convex polyhedral cones known as the “tropical fan” of X. With origins in optimization theory, tropical geometry is usually introduced as the study of the “tropical semi-field”, i.e. real numbers equipped with operations of taking minimum (in place of addition) and addition (in place of multiplication). Tropical geometry has close connections with convex geometry and is an active area of research in recent years with applications in several different areas such as algebraic geometry, symplectic geometry, representation theory, optimization, phylogenetics etc.

We will touch on recent far extensions of the notion of “tropical variety” in two different directions: (1) tropical varieties for subvarieties in algebraic groups G and their homogeneous spaces G/H, (2) study of tropical varieties in semi-field of “piecewise linear functions” and semi-ring of convex polytopes, and utilizing them to classify equivariant vector bundles, principal bundles and more generally torus equivariant flat families over toric varieties. This is a far generalizations of the famous work of Klyachko which classifies equivariant vector bundles on toric varieties.

While the abstract may contain some fancy-scary words, the talk for the most part (at least 51% :) should be understandable for audience outside of algebra and geometry. You will need to know what a polynomial with complex coefficients is :)