Geometry and Arithmetic mini-courses

Speaker: Omid Amini (Ecole Polytechnique) will give a 4 hours mini-course.

Title: Hybrid geometry of curves and their moduli spaces.

Abstract: The aim of these lectures is to introduce geometric objects called hybrid curves and their moduli spaces which mix features from higher rank non-Archimedean, tropical, and complex geometries. Applications to questions with root in arithmetic geometry and mathematical physics, around the asymptotic geometry of Riemann surfaces close to the boundary of their moduli spaces will be discussed. Based on joint works with Noema Nicolussi.

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Speaker: Farbod Shokrieh (U. Washington) will give a 4 hours mini-course.

Title: Tropical geometry, heights, and arithmetic.

Abstract: The theory of "heights" is essential in studying finiteness questions in diophantine geometry. For example, one of the most spectacular results in number theory is Faltings's theorem (Mordell's conjecture): a curve of genus greater or equal to 2 over the field of rational numbers has only finitely many rational points. A crucial tool in Faltings's proof was a notion of "height of an abelian variety".
In these lectures, we study the interplay between non-archimedean/tropical geometry, arithmetic/Arakelov geometry, and combinatorics/convex geometry arising from the theory of heights of abelian varieties. In particular, we discuss how Faltings's height can be related to invariants arising from tropical geometry and Berkovich analytic spaces. In the case of Jacobians, these invariants relate to combinatorics, convex geometry, and potential theory (electrical network) of metric graphs. If time permits, we will also discuss some applications in the study of analytic invariants on degenerating families of curves and abelian varieties.

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Sponsors: IRN MaDeF (CNRS), GeoTop Center (KU)