The Borel Complexity of Isomorphism for Some First Order Theories

In this work we consider several instances of the following problem: "how complicated can the isomorphism relation for countable models be?"' Using the Borel reducibility framework, we investigate this question with regard to the space of countable models of particular complete first-order theories. We also investigate to what extent this complexity is mirrored in the number of back-and-forth inequivalent models of the theory. We consider this question for two large and related classes of theories. First, we consider o-minimal theories, showing that if T is o-minimal, then the isomorphism relation is either Borel complete or Borel. Further, if it is Borel, we characterize exactly which values can occur, and when they occur. In all cases Borel completeness implies lambda-Borel completeness for all lambda. Second, we consider colored linear orders, which are (complete theories of) a linear order expanded by countably many unary predicates. We discover the same characterization as with o-minimal theories, taking the same values, with the exception that all finite values are possible except two. We characterize exactly when each possibility occurs, which is similar to the o-minimal case. Additionally, we extend Schirrman's theorem, showing that if the language is finite, then T is countably categorical or Borel complete. As before, in all cases Borel completeness implies lambda-Borel completeness for all lambda.

Classical Invariants for Principal Series and Isomorphisms of Root Data

We develop some new techniques to calculate the Schur indicator for self-dual irreducible Langlands quotients of the principal series representations. Using these techniques we derive some new formulas for the Schur indicator and the real-quaternionic indicator. We make progress towards developing an algorithm to decide whether or not two root data are isomorphic. When the derived group has cyclic center, we solve the isomorphism problem completely. An immediate consequence is a clean and precise classification theorem for connected complex reductive groups whose derived groups have cyclic center.