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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. In joint work with Alexander Gorodnik and Amos Nevo, we develop a systematic metric theory for dense lattice orbits, including analogues

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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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## Dynamics on abelian varieties in positive characteristic

## Dynamics on abelian varieties in positive characteristic

We show that the dynamics of endomorphisms of abelian varieties (e.g., elliptic curves) over field of characteristic $p>0$ is sharply divided according to two possible scenario’s, reflected in properties of the Artin-Mazur zeta function (transcendental/rational) and orbit growth, somewhat similar to the mixing/non-mixing dichotomy in measurable dynamics. The two scenario’s are distinguished by the action on the $p$-torsion subgroup scheme.

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## Teichmuller curves mod $p$

A Teichmuller curve is a totally geodesic curve in the moduli space of Riemann surfaces. These curves are defined by polynomials with integer coefficients which are irreducible over $\mathbb C$. We will show that these polynomials have surprising factorizations mod $p$. This is joint work with Keerthi Madapusi Pera.

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## Audience recollections of Sergiy Kolyada

## Life and Mathematics of Sergiy Kolyada

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We will discuss an equidistribution problem concerning rational planes in four-space. Due to an accidental isomorphism this problem relates in a natural manner to the simultaneous study of four CM-points on the modular surface and two points on the two-dimensional sphere. Using Duke's theorem and a joining classification we obtain joint equidistribution under suitable congruence conditions on the covolume of the planes.

This is joint work with Menny Aka and Andreas Wieser, and relies on a joint theorem with Elon Lindenstrauss.

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