### The Tarski Lecture in 2024 will be given by Kobi Peterzil and Sergei Starchenko

**Abstract:**

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.

1. Closures and flows in real tori: a model theoretic approach.

Monday April 22, 2024, Physics 3 - reception to follow in 1015 Evans

2. The interplay of o-minimality and discrete subgroups.

Wednesday April 24, 2024 Physics 3

3. From closures to Hausdorff limits, in tori and nilmanifolds.

Thursday April 25, Evans 60