The University of Maryland Logic Seminar will meet on Tuesdays from 3:00-4:20pm in Math 1311.
Tuesday, September 4, Organizational Meeting.
Tuesday, September 11, John Goodrick, UMCP
Title: Finite subcategories of embeddability skeletons
Abstract:
Given a theory T, define its embeddability skeleton to be the
category whose objects are isomorphism classes of models of T,
and whose arrows are given by elementary embeddability. In this
talk we will consider the question: What can we say about
the finite (full) subcategories of its embeddabilty skeleton?
I will give a few partial answers (based on my thesis and some older
results of others) and present some wild speculations.
Tuesday, September 18, Chris Laskowski, UMCP
Title: `Automatic' quantifier elimination
Abstract:
We will survey and discuss a number of recent results
demonstrating that strong stability theoretic assumptions
automatically yield a bound on the quantifier complexity of
the definable sets.
Tuesday, September 25, Hunter Johnson, UMCP
Title: Compression schemes for the model theorist
Tuesday, October 2, NO MEETING
Tuesday, October 9, Alexei Kolesnikov, Towson
Title: What can go wrong with stability and tameness?
Tuesday, October 16, John Goodrick, UMCP
Title: Introduction to dependent theories
Tuesday, October 23, Chris Shaw, UMCP
Title: Being inspired by expanding the real line
Tuesday, October 30, John Goodrick, UMCP
Title: Stable types in dependent theories
Tuesday, November 6, Chris Laskowski, UMCP
Title: Borel complexity of some omega-stable theories
Tuesday, November 13, Justin Brody, UMCP
Title: Axioms for Hrushovski constructions
Tuesday, January 29, Vincent Guingona, UMCP
Title: One-based groups
Tuesday, February 5, John Goodrick, UMCP
Title: Dp-minimality, inp-minimality, and ordered groups
Abstract:
Dp-minimal theories are a subclass of dependent theories that generalize
weakly minimal theories in the stable context, arising from Shelah's
"dp-ranks," and inp-minimality is an even more general condition that does
not imply the dependence property. Inp-minimality can be defined quite
simply, and in the context of ordered structures it generalizes weak
o-minimality. In a divisible abelian ordered group, inp-minimality is not
the same as weak o-minimality, but we show that a slight weakening of the
Monotonicity Theorem holds: any definable unary function is a union of
finitely many continuous, locally monotonic (partial) functions.
Tuesday, February 12, Alexei Kolesnikov, Towson University
Title: Disjoint amalgamation spectrum
Abstract:
We address the following question: is it possible for
the disjoint amalgamation property in an abstract elementary class
to hold for (many) small cardinals, but fail eventually?
The answer is yes; in this talk, we present a family of examples.
Tuesday, February 19, Alexei Kolesnikov, Towson University
Title: Disjoint amalgamation spectrum, Part II.
Abstract:
We address the following question: is it possible for disjoint amalgamation
property in an abstract elementary class to hold for (many) small cardinals,
but fail eventually? In this talk, we show that, consistently with ZFC, for
every countable ordinal α there is an AEC in a countable language that
has disjoint amalgamation up to בα, but failing amalgamation eventually.
Tuesday, February 26, Greggo Johnson, UMCP
Title: Approximations in AECs with finite character
Tuesday, March 4, John Goodrick, UMCP
Title: Definable groupoids and internal covers
Abstract:
We discuss a generalization of the construction of binding groups in stable theories, which works in any first-order theory and gives "*-definable groupoids." If there is time, we will disuss connections with generalized imaginaries and amalgamation. (This is a presentation of some material from Hrushovski's "Groupoids, imaginaries, and internal covers.")
Tuesday, March 11: NO MEETING
Tuesday, March 18: NO MEETING (Spring Break)
Tuesday, March 25, Justin Brody, UMCP
Title: Conjectures and refutations
Tuesday, April 1, Vincent Guingona, UMCP
Title: Eliminating quantifiers in theories of modules
Tuesday, April 8, Valentina Harizanov, The George Washington University
Title: Effective categoricity of equivalence structures
Abstract: We investigate algorithmic categoricity of computable equivalence structures. A computable structure A is computably categorical if for every computable isomorphic copy B of A, there is a computable isomorphism from A onto B. We establish that a computable equivalence structure A is computably categorical if and only if A has only finitely many finite equivalence classes, or A has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k.
A computable structure A is relatively computably categorical if for every B isomorphic to A, there is an isomorphism that is computable relative to the atomic diagram of B. It is known that A is relativley computably categorical if and only if A has a computably enumarable Scott family of computable existential formulas. We show that all computably categorical equivalence structures are relatively computably categorical. We further investigate categoricity and relatvie categoricity of computable equivalence structures at highter levels of the arithmetical hierarchy.
This is joint work with W. Calvert, D. Cenzer, and A. Morozov.
Tuesday, April 15, Chris Shaw, UMCP
Title: Weakly o-minimal structures: Foundations and fun
Abstract: We discuss some essential results in weakly o-minimal structures, particularly ordered groups. We shall define and explore the concept of valuational and nonvaluational cuts, and how the presence of one or the other impacts the cellular decomposition of a weakly o-minimal group.
Also integral to the study of weakly o-minimal structures is the relationship between a weakly o-minimal structure and its Dedekind completion. We discuss how an elusive result of Poizat leads to a nice property about o-minimal structures.
Finally, we discuss preliminary results on the presence of Skolem functions, and the tricks that went into finding these results.
Tuesday, April 22, Martin Koerwien, University of Illinois at Chicago
Title: Omega stability and Borel reducibility
Tuesday, April 29, David Lippel, University of Notre Dame
Title: Positive elimination in valued fields
Abstract: A "positive elimination theorem" is a statement that certain positive existential formulas are equivalent to positive quantifier-free
formulas. For example, the main theorem of classical elimination theory can be interpreted as a positive elimination result. (Let X be a coordinate projection of a Zariski-closed subset of complex projective space; then, X has a positive existential definition. Elimination theory says that X is Zariski-closed, so in fact, X also has a positive quantifier-free definition.)
Prestel has proved some positive elimination results for valued fields, working in a one-sorted language. I will discuss some generalizations to two-sorted languages; I will show how these can be used to re-prove some basic facts in tropical geometry. This is joint work with Matthias Aschenbrenner and Sergei Starchenko.
Tuesday, May 6, Alfred Dolich, Chicago State University
Title: Intersections of o-minimal theories
Abstract: Availible here.
Tuesday, May 13, Krzysztof Krupinski, University of Illinois at Urbana-Champaign
Title: Getting fields in rosy theories
Abstract: After a brief introduction about rosy theories, I will talk about some results yielding infinite interpretable fields in rosy groups of finite thorn U-rank. These results generalize some theorems from the finite Morley rank case and from o-minimal structures. I will prove the existence of such fields in the presence of certain V-definable rings (generalizing a result by Peterzil and Starchenko for o-minimal structures) and in a situation when an infinite, definable abelian group acts definably as a group of automorphisms on a definable abelian group. The interesting fact is that the lack of most of the tools (such as the uniform chain condition on intersections of uniformly definable subgroups or Zilber's Indecomposables theorem) has forced me to use completely fundamental tools (such as the compactness theorem and basic properties of dimension), and as a result I have obtained simpler proofs than those in the finite Morley rank case or o-minimal structures. Using these results, I have proved the existence of an infinite interpretable field in any solvable-by-finite but not nilpotent-by-finite group of finite thorn U-rank satisfying NIP.