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DTSTART;TZID=UTC+1:20230427T083000
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DTSTAMP:20231004T092600
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UID:42221-1682584200-1682589600@math.u-bourgogne.fr
SUMMARY:Johannes HOFSCHEIER\, "Spherical Amoebae and a Spherical Logarithm Map"
DESCRIPTION: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. \nhttps://indico.math.cnrs.fr/event/8826/ \n
URL:https://math.u-bourgogne.fr/agenda/johannes-hofscheier-tba
CATEGORIES:Géométrie, Algèbre, Dynamique et Topologie
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