Abstract: We will discuss the classification of homogeneous finite-dimensional permutation structures, i.e. structures in a language of finitely many linear orders, recently completed in joint work with Pierre Simon. After constructing the catalog of such structures, we will present some of the key concepts in the classification, primarily coming from Simon's work on linear orders in omega-categorical structures. We will also mention an open question concerning the number of linear orders needed to represent any given structure in the catalog.
Abstract: In model theory, one wants to measure the complexity of types in a given theory. One way of measuring the complexity of types in NIP theories is the notion of dp-rank. In this talk, I present a framework for generalizing dp-rank that I am calling âpositive local combinatorial rank.â We define this rank, show that it generalizes dp-rank in distal theories (and op-dimension in NIP theories), and explore how this rank behaves in general. In particular, we examine two test cases and compute some ranks.
Abstract: We study regular types in Baldwin-Shi hypergraphs with rational valued rank functions. We characterize the regular types up to non-orthogonality and study the associated pregeometries. Combining the above results with the work of Laskowski and Brody, we obtain examples of pseudofinite theories with non-locally modular regular types, answering a question of Pillay's.
Abstract: It is known that a complete stable theory may fail to have higher amalgamation properties. A reasonable question is whether or not it is possible to construct a suitable cover T' of the theory T so that T' does have higher amalgamation over algebraically closed sets.
In this talk, I will present recent results, due to Evans, Kirby, and Zander, that describe the construction of a candidate for such a theory T'.
Abstract: Using a general quantifier elimination for ordered Abelian groups due to Cluckers and Halupczok we characterize those ordered Abelian groups which are strong (a strengthening of being NIP). As a consequence we obtain that any strong ordered Abelian group had finite dp-rank and show how to compute the rank from algebraic data. This is joint work with John Goodrick.
Abstract: Two structures are called siblings if they are bi-embeddable. Given a countable structure M, we wish to count its siblings up to isomorphism. Extending work of Laflamme, Pouzet, Sauer, and Woodrow, we give a complete answer to this in the case M is omega-categorical, making use of the recent "Ryll-Nardzewski theorem" for mutually algebraic theories. Joint work with Chris Laskowski.
Abstract: The Feferman-Vaught Theorem is a classical model theoretic result on definability in product-like structures. In this talk, we explain this theorem and derive an elementary but seemingly unpublished combinatorial consequence. We then show that a converse of sorts holds in the case of a product of integral domains. Finally, we give a quantifier reduction for products of finite fields, which arise naturally in a number-theoretic context, using results of Kiefe and Ax. This is joint work with Alice Medvedev (CCNY).
Abstract: A cohesive power of a countable structure can be viewed as a computability-theoretic analog of an ultrapower, where a structure is computable and, instead of an ultrafilter, we use an infinite set that is indecomposable with respect to computably enumerable sets. Such indecomposable sets are called cohesive. A structure for a finite language is computable if its domain is computable and its operations and relations are computable. The elements of cohesive powers are equivalence classes of partial computable functions. We investigate the isomorphism types and other properties of cohesive powers for certain kinds of computable structures.
Abstract: The notion of bounded VC dimension is a property at the intersection of combinatorics and probability. This family has been discovered repeatedly and studied from various perspectives - for instance, in model theory, theories with bounded VC dimension are known as NIP (the theories which do Not have the Independence Property). One useful property is that graphs with bounded VC dimension are the graphs that can be always be finitely approximated in a random-free way: graphs with bounded VC dimension satisfy a strengthening of Szemeredi's Regularity Lemma in which the densities between the pieces of the partition are either close to 0 or close to 1. The generalization of VC dimension to higher arity, known in model theory as k-NIP for various k, has been less well-studied. We summarize some known facts about this generalization, including a new result (joint with Chernikov) showing k-NIP hypergraphs have a similar kind of approximation with only lower order randomness.
Abstract: Morley's categoricity theorem says that a countable first-order theory with a single model in some uncountable cardinal has a single model in all uncountable cardinals. The proof has had numerous consequences, including the development of stability theory, and in particular the invention by Shelah of forking, a far reaching generalization of algebraic independence in fields. Shelah has conjectured that a generalization of Morley's result should hold in any abstract elementary class (AEC). Roughly, an AEC is a category that behaves like the category of models of a first-order theory with elementary embedding. The framework encompasses for example classes of models of logics with infinite conjunctions and disjunctions.
In this talk, I will survey progress on Shelah's eventual categoricity conjecture, including a proof from large cardinals (joint with Shelah) and a full characterization of the categoricity spectrum in AECs with amalgamation. These proofs have already suggested several new directions, including a theory of forking in accessible categories connecting the field with homotopy theory (joint with Lieberman and RosickÃ½).
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