Model theory studies the ways in which classes of mathematical objects can be defined in some restricted formal language, and what structural properties are implied by these definability assumptions. This method of study originated in questions around the philosophy and foundations of mathematics, but in recent years it has found striking applications in the study of some central objects of classical mathematics and computer science. This project investigates further these connections, in particular in the context of graphs (mathematical ways of describing networks and related systems) and group actions (mathematical ways of describing collections of symmetries of a space). This study has the potential to open up a route for applications of the powerful infinitary model-theoretic machinery to open questions in finite graph combinatorics, and conversely for applications of deep results in combinatorics to open questions in model theory.

Motivated by Morley's conjecture on the possible number of uncountable models of first-order theories, Shelah isolated several important classes of "tame" theories and developed a rich machinery for analyzing models and definable sets for some of those classes, particularly for stable theories. Later work by many researchers demonstrated that notions and methods of generalized stability reflect important phenomena in other areas of mathematics. This project will investigate two such connections. 1) Improved Ramsey-type bounds and strong regularity lemmas were obtained for semi-algebraic graphs by Fox et al. and for algebraic graphs in large finite fields by Tao, with numerous applications in the corresponding areas. These results can naturally be viewed as results about graphs definable in certain structures fitting into the classification picture. 2) Study of definable group actions turns out to be closely related to certain questions in topological dynamics, especially around weakly almost periodic dynamical systems and tame systems studied by Glasner and others. The investigator will work on developing further methods of generalized stability and applying them to questions in combinatorics of graphs definable in various "tame" structures (stable, o-minimal, distal, dependent) and to study dynamical properties of definable group actions in those structures.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Application #
1600796
Program Officer
Tomek Bartoszynski
Project Start
Project End
Budget Start
2016-07-01
Budget End
2019-06-30
Support Year
Fiscal Year
2016
Total Cost
$179,998
Indirect Cost
Name
University of California Los Angeles
Department
Type
DUNS #
City
Los Angeles
State
CA
Country
United States
Zip Code
90095