Tropical geometry is a powerful tool that allows us to study algebraic varieties through the lens of combinatorics. For algebraic subvarieties X of the complex algebraic torus (C^*)^n, these “combinatorial shadows” can be thought of as limit sets of amoebas Log_t(X) as t approaches 0. An amoeba is the image of the variety under the logarithm map Log_t: (C^*)^n → R^n.
In this talk, I’ll report on joint work in progress with Victor Batyrev, Megumi Harada, and Kiumars Kaveh, where we take initial steps towards generalising the above to the non-abelian setting. We choose to view the complex torus (C^∗)^n as a special case of a spherical homogeneous space G/H in which the group G is abelian. Spherical tropical geometry was introduced and developed by Tevelev-Vogiannou and Kaveh-Manon, and we propose that there should be a theory of spherical amoebas for G/H which correspond to the classical amoebae in the abelian case. Our approach follows Akhiezer’s definition of spherical functions to introduce a spherical logarithm map that parametrises K-orbits in G/H for K a maximal compact subgroup of G. From this point of view the spherical logarithm map can be viewed as a generalisation of the classical Cartan decomposition. By using this spherical logarithm map, we define spherical amoebas of subvarieties of G/H and ask for conditions under which these converge to their spherical tropicalizations. This leads to several conjectural topological descriptions of spherical homogeneous spaces.