Hyperbolic amoebas

Abstract: Classical amoebas serve as a tool to study algebraic varieties and as one of the entry points to tropical geometry. The original definition involves a logarithmic projection of a subvariety of a complex algebraic torus, which can be interpreted as forgetting the phase, i.e. the arguments of complex numbers. In group-theoretic terms, this projection map may be thought of as passing to the quotient by the maximal compact subgroup. Suppose one replaces the algebraic torus with a complex three-dimensional matrix group. In that case, the analogous projection naturally has a three-dimensional hyperbolic space as its target, and it still makes sense to consider images, i.e. hyperbolic amoebas, of complex algebraic varieties under this map. I will review some of the basic properties of hyperbolic amoebas, extending a fascinating interplay between complex and hyperbolic geometries.

Based on joint work with Grigory Mikhalkin.