UCLA

A central goal of modern model theory is to classify and understand first-order structures through studying the combinatorial properties of their definable sets, Conversely, model-theoretic properties can shed light on the behavior of interesting families of sets, such as semialgebraic families, which happen to be definable in well-behaved structures. This dissertation studies one such model-theoretic property, distality, as it manifests in the combinatorics of definable sets, and in continuous logic, where definable sets are generalized to real-valued functions.In Chapter 2, we calculate explicit bounds on the sizes of distal cell decompositions for definable sets in a variety of distal structures. These lead to incidence combinatorics bounds in exponential-polynomial and p-adic settings. In Chapter 3 and Chapter 4, we develop a theory of distality in continuous logic. This begins with a study of the combinatorics of NIP metric structures, building on earlier work by Ben Yaacov and results from statistical learning theory. We also develop a theory of generically stable Keisler measures in continuous logic, allowing us to generalize combinatorial statements from just pertaining to finite counting measures. We then generalize many definitions of distal structures to the continuous logic context, showing that under an NIP assumption, they are all equivalent. These allow us to study distal metric structures through the perspectives of indiscernible sequences, strong honest definitions, distal cell decompositions, Keisler measures, and an analytic regularity lemma. Finally in Chapter 5, joint work with Itaï Ben Yaacov, we present examples of distal metric structures that are unique to continuous logic, including real closed metric valued fields and dual linear continua.