CAREER: The Geography of Tame Ordered Structures

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

At the heart of logic and model theory lies the observation that within mathematics there are certain objects that have to be considered tame, and others that have to be considered wild. Many celebrated results in logic from the first half of the 20th century concerned the existence of objects considered wild. Gödel's proof of the undecidability of arithmetic established that this structure is complicated (or wild) from a logical viewpoint. Such results are negative in spirit as they point to the limitations of mathematical reasoning. However, in the second half of the last century the focus changed. Model theorists found a vast number of tame mathematical structures that exhibit no such wildness, and for often very different reasons are amenable to model-theoretic methods. This program of identifying and analyzing tame classes of structures whose model theory can be understood came to be known as the geography of tame mathematics, and in its various forms has dominated model theory throughout the last thirty years. Although arising as a program of foundational importance, it has led to striking applications of model theory to other areas of mathematics, most recently to the André-Oort conjecture in number theory. Since this program is to explore areas of tame mathematics, these connections are not coincidental at all. The development of such interactions has proven again and again to have far reaching applications outside of logic that could not have been envisioned beforehand. This project continues this line of research in the setting of expansions of the real line. It aims to settle important open questions within model theory, but also naturally develops new connections between model theory and other areas in logic such as neostability and descriptive set theory, and outside of logic such as geometric measure theory and geometric group theory. The investigator will lead a large scale investigation of dividing lines between tame and wild behavior arising in the study of the geometry of definable sets in ordered structures. Building on early advances, the investigator will determine far reaching consequences of various logical tameness conditions on the topological and metric tameness of definable sets. Furthermore, the project comprises new challenges in the classification of classes of structures considered tame. The educational component of this CAREER grant ties together the investigator's research with his teaching and outreach efforts. This project involves undergraduate and graduate students and young researchers in the investigator's research program, strengthens the ties between model theory and other branches of mathematics, and continues the investigator's outreach efforts in the Urbana-Champaign area.