This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the nonelementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of the class of models of a complete first order theory with the elementary substructure relation. We examine the symmetry property of splitting (previously isolat...

We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety X over a complete discretely valued field K with perfect residue field k. If K has characteristic zero, we extend the definition to arbitrary K-varieties using Bittner’s presentation of the Grothendieck ring and a process of Néron smoothening of pairs of varieties. The motivic Serre invariant can be considered...

Theorem. For each 2 ≤ k < ω there is an Lω1,ω-sentence φk such that: (1) φk is categorical in μ if μ ≤ אk−2; (2) φk is not אk−2-Galois stable; (3) φk is not categorical in any μ with μ > אk−2; (4) φk has the disjoint amalgamation property; (5) For k > 2, (a) φk is (א0,אk−3)-tame; indeed, syntactic first-order types determine Galois types over models of cardinality at most אk−3; (b) φk is אm-Gal...

OF THE DISSERTATION Simple Groups of Finite Morley Rank of Odd and Degenerate Type by Jeffrey Burdges Dissertation Director: Professor Gregory Cherlin The present thesis aims to drive Borovik’s program towards its final endgame, and to lay out a plan for bringing it to a conclusion. Borovik’s program is an approach to the longstanding Algebraicity Conjecture of Cherlin and Zilber, which propose...