Abstracts of upcoming talks at the MPIM. For an overview see also the calendar.

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## New Mirzakhani-type identities for the simple hyperbolic length spectrum

McShane obtained in 1991 an identity expressing the function 1 on the Teichmueller space of the once-punctured torus as a sum over simple closed curves. It was generalized to bordered surfaces of all topologies by Mirzakhani in 2005, from which she deduced a topological recursion for the Weil-Petersson volumes.

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## The metric theory of dense lattice orbits

The classical theory of metric Diophantine approximation is very well developed and has, in recent years, seen significant advances, partly due to connections with homogeneous dynamics. Several problems in this subject can be viewed as particular examples of a very general setup, that of lattice actions on homogeneous varieties of semisimple groups. The latter setup presents significant challenges, including but not limited to, the non-abelian nature of the objects under study.

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## Rigidity for planes in hyperbolic $3$-manifolds and slices of Sierpinski carpets

## Sphere Packings and Arithmetic

We define a general class of "crystallographic" sphere packings, and study the subclass of "superintegral" such. We prove an "arithmeticity" theorem, connecting these to the theory of hyperbolic arithmetic reflection lattices. As a consequence, superintegral packings only exist in finitely many dimensions, and in fact in finitely many commensurability classes, in principle allowing for a complete classification.

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