学术报告（Mikhail Shkolnikov 2025.1.10）

Abstract: A standard approach to tropical geometry is via amoebas of subvarieties in a complex algebraic torus, where an amoeba is the image of a subvariety under the real part of the coordinate wise logarithm, which may be thought of as taking the quotient by the maximal compact subgroup, i.e. forgetting the phase. In the joint work with Grigory Mikhalkin, we investigated the possibility of replacing the algebraic torus by a non-commutative complex algebraic group, with the principal example being PSL2(C). The quotient of this group by its maximal compact subgroup PSU(2) is naturally the three-dimensional hyperbolic space. The corresponding PSL2 tropical limits of curves were described in terms of certain versions of floor diagrams, however the case of surfaces remained until recently unresolved. In my talk, based on a joint work with Peter Petrov, I will explain that such a tropical limit of hyperbolic amoebas of surfaces is always a complement to an open geometric ball. This result motivates a refinement of this procedure which consists of restoring the phase, i.e. PSL2 phase tropicalization, which will be described in detail and some examples will be considered.