2024 Tarski Lecture

In a series of papers we examined first the following question:  Given X \subseteq R^n definable in an o minimal structure, and given a lattice L in R^n, with pi: R^n> R^n / L, what can be said about the closure of pi(X) in the torus?

We then generalize the question in two directions: We replace R^n with a unipotent group G, replace the set X with a definable family {X_s} of subsets of G, and ask about the possible Hausdorff limits of {pi(X_s)} in the nilmanifold G /L.

As we shall describe in our talks, the answers combine the model theoretic machinery of types in o-minimal groups, with some cases of Ratner’s theorem in unipotent groups. The “flatness” of such types at infinity, together with the co-compactness of the lattices, gives rise to linear subspaces of R^n, or in the unipotent case, to algebraic subgroups of G associated to the set X, and the closure  and Hausdorff limits could be understood using those subgroups.

In the first talk we  describe the closure problem in tori,  using many examples.

In the second talk we explain some of the model theoretic ingredients which come up in the solution.

In the third talk we discuss generalizations to closures  in nil-manifolds and 
to Hausdorff limits.

Sergei Starchenko received his Ph.D. from Novosibirsk University in 1987. His doctoral dissertation Number of models of Horn theories was written under the supervision of Evgenii Andreevich Palyutin. Starchenko became a Fellow of the American Mathematical Societyin the class of 2017. He is now a full professor at the University of Notre Dame.