OSU Math Logic Seminar

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

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

SP25

The seminar is nominally on Tuesdays (excluding March 11), 13:50--15:00, in University Hall 43.

March 25

Yutong Duan (UIC). Algebraic independence of solutions of Lotka-Volterra equations

Abstract: Lotka-Volterra equations describe the dynamics of biological systems in which two species interact. Formally, the equations are a parameterized family of planar polynomial vector fields. They are also of interest as model theory by regarding solution sets as being definable in a monster model of the theory of differentially closed field of characteristic zero. I will show that under certain conditions on the parameters, the solutions sets are strongly minimal, geometrically trivial and strictly disintegrated (that is, generic solutions are algebraic independent).

March 4

Nicolas Chavarria Gomez (Waterloo). Internality and binding groups in continuous logic

Abstract: In joint work-in-progress with Moosa, we explore the notion of internality in continuous logic and show that, whenever this phenomenon arises in a metric structure, a definable binding group arises together with it. This requires navigating some of the difficulties of what definability means in the continuous sense. We also explore some related concepts.

Feb 25

(cancelled)

AU24

Nov 19

Leo Jimenez (OSU). Decomposing types in stable theories

Abstract: In stable theories of finite rank, there are two ways to decompose a complete type. The first is the domination equivalence decomposition, which states that the type is domination equivalent to a Morley product of minimal types. The second is the semi-minimal analysis, which builds up the type as a tower of fibrations with fibers internal to a minimal type. The connection between these is not investigated in the literature. In this talk, I will define these concepts and give examples coming from differential fields. I will then explain the relationship between the two, and give applications. This is joint work in progress with Christine Eagles.

Oct 22

Chris Miller (OSU). A tameness result for expansions of (R,<,+,Z)

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.)

Oct 8

Leo Jimenez (OSU). Internality of autonomous differential equations

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.

Oct 1

Francis Wagner (OSU). Malnormal subgroups of finitely presented groups

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.

Sept 24

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.

Sept 17

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.