Speaker
Mikhail Shkolnikov, International Center for Mathematical Sciences
Abstract
Hyperbolic amoebas are images of complex subvarieties in PSL(2,C) under the phase-forgetting map, a quotient by the maximal compact subgroup PSU(2). In the joint work with Grigory Mikhalkin, we have shown that tropical limits of hyperbolic amoebas of curves are given by unions of spheres and geodesic segments. In my talk, I will prove that the tropical limit of hyperbolic amoebas of surfaces is a complement to a geometric ball. This result motivates the use of an enhanced version of tropicalization, which is performed via rescaling without forgetting the phase. In collaboration with Peter Petrov, we have realized that such a procedure makes sense in any dimension, and can be seen as a radial degeneration in the projective space or a quadric. The corresponding limits appear to be similar to buildings in symplectic field theory, with spherical coamoebas inside PSU(2) playing a special role. I will conclude by discussing the topology of generic radial degenerations following the work in progress with Ilia Zharkov.