Topics in Model Theory

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

The subject of this project is model theory, a branch of mathematical logic which is about the ways in which mathematical objects or classes of objects are defined linguistically. Model theory develops various ways of measuring the complexity of a class of functions or sets, one of which is called Vapnik-Chervonenkis (or VC) dimension and was independently discovered by researchers in statistical learning theory in the 1970's. One part of the project is studying the fine structure of systems with finite VC dimension. A related measure of the complexity of a class of sets is called stability and another aspect of the project is using the model-theoretic understanding of stable systems to compute the equations satisfied by special functions such as the exponential function. Describing the symmetries of spaces and systems is a pervasive aspect of the research which ties together its various components. The project has four interrelated aspects, (i) stability theory and theories without the independence property, (ii) topological dynamics, model theory, and pseudo-finite groups, (iii) tame theories of differential fields, (iv) Ax-Lindemann for semiabelian families and differential Galois theory. At the abstract level model theory is about the classification of first order theories, and part (i) is firmly of this nature. The methods and results obtained in "pure" model theory are often mathematically meaningful in concrete contexts, leading to new interrelations with other areas of mathematics. In particular in (iv) the PI will study transcendence properties of "exponential" functions relative to certain families of commutative algebraic groups, making use of a certain Galois theory of differential equations which is given by constructions in stability-theory (part of model theory). The project also connects to and impacts on topological dynamics, and also potentially and indirectly combinatorics (via the study of certain measures on definable sets in theories without independence property).