The contact person for the seminar and mailing list is Chris Miller.

The seminar receives some financial support from the Mathematics Research Institute.

The seminar is nominally on Tuesdays, 13:50--14:45, in Dulles Hall 024 (behind the Math Tower, in the basement).

Abstract. I will present a tameness result for the expansion of (R,<,+,Z) by all bounded sets of reals (of any arity) whose closures are countable and have finite Cantor-Bendixson rank. Loosely, the definable sets are as well behaved as one could reasonably expect. (This is a special case of some ongoing work with Masato Fujita.)

Abstract. When solving a differential equation, one sometimes finds that solutions can be expressed using a finite number of fixed, particular solutions, and some complex numbers. As an example, the set of solutions of a linear differential equation is a finite-dimensional complex vector space. A model-theoretic incarnation of this phenomenon is internality to the constants in a differentially closed field of characteristic zero. In this talk, I will define what this means, and discuss some recent progress, joint with Christine Eagles, on finding concrete methods to determine whether or not the solution set of a differential equation is internal. A corollary of our method also gives a criteria for solutions to be Liouvillian: I will show a concrete application to Lotka-Volterra systems.

Introduced by Sapir in the late 1990s, the `S-machine' is a computational model which resembles a multi-tape, non-deterministic Turing machine. This model was carefully conceived in order to be both computationally robust and interpretable as a multiple HNN extension of a free group. As such, S-machines have proved to be a remarkable tool in the study of groups. I will discuss a generalization of the S-machine which yields the following refinement of Higman's embedding theorem: Every finitely generated recursively presented group may be quasi-isometrically embedded as a malnormal subgroup of a finitely presented group; moreover, the decidability of the Word problem is preserved by this embedding.

Michael Bersudsky (OSU). Equidistribution of polynomially
bounded o-minimal curves in homogenous spaces

Abstract. Given a real algebraic
group G and an o-minimal structure on the real field,
one can naturally define subsets of G that are
definable in this structure. Peterzil and Starchenko recently
showed that when G is the group of upper-triangular
matrices and L is a lattice in G, the closure of
the image in G/L of a definable set in G is the closed
image of a potentially larger definable set in G. Moreover, they
showed that if a curve in G is definable in a
polynomially bounded o-minimal structure and its image
in G/L is dense, then the curve is uniformly
distributed in G/L. In this talk, I will present recent joint
work with Nimish Shah and Hao Xing, where we extend these
results for curves definable in a polynomially bounded o-minimal
structure in a general real algebraic group G and a general
lattice L in G, under a 'non-contraction' condition on
the curves. This work builds upon Shah's earlier technique for
polynomial curves in homogeneous spaces, ‘tangency at infinity’
property of o-minimal curves shown by Peterzil and Steinhorn,
and Ratner's groundbreaking theorems. A key innovation in our
analysis is a proof of a certain growth property for families of
polynomially bounded o-minimal functions.

Chris Miller (OSU). A brief introduction to polynomially bounded
o-minimality.

Abstract. I will present some basic results about polynomially
bounded o-minimal expansions of the real field that might be
needed for some subsequent seminars.