Logic Seminar for 2008 - 2009:

The University of Maryland Logic Seminar will meet on Tuesdays from 3:00-4:20pm in Math 1311.

If you would like to present a talk, please contact John Goodrick.

Fall, 2009

Tuesday, October 27, Pedro Zambrano, Universidad Nacional de Colombia
Title: Around Metric Abstract Elementary Classes
Abstract: In this talk, we will give some basic facts about metric abstract elementary classes. Also, we will give some basic facts relative to a notion of independence which we will use for understanding a version of superstability (uniqueness of limit models) in this setting.

Tuesday, November 3, Pedro Zambrano, Universidad Nacional de Colombia
Title: Towards uniqueness of limit models in metric abstract elementary classes
Abstract: In this talk, we will prove that the completion of unions of an increasing and continuous chain of reduced towers is reduced and also the density of reduced towers, in the setting of metric abstract elementary classes. These are important steps towards proving the uniqueness of limit models in the setting of abstract elementary classes.

Tuesday, November 10, Lynn Scow, University of California at Berkeley
Title: Generalized indiscernible sequences in stable and NIP theories
Abstract: In the 1970's, S. Shelah gave the following characterization of stable theories: a theory is stable if and only if any indiscernible sequence in a model of the theory is an indiscernible set. I will present a similar characterization of NIP theories, as theories in which any random ordered graph-indiscernible in a model of the theory remains indiscernible strictly with respect to the order. In this talk I will explain what I mean by a random ordered graph-indiscernible and I will indicate how the result is proved using the Nešetřil-Rödl Theorem. If time permits, I will discuss an additional example of a characterization of stable theories by generalized indiscernibles that generalizes more faithfully than Shelah's.

Tuesday, November 17, Michael Lieberman, University of Pennsylvania
Title: AECs and accessible categories: connections and implications
Abstract: We present a family of rank functions -- complete with topological motivation -- for use in the analysis of stability in abstract elementary classes with amalgamation, and derive a partial stability spectrum result for tame classes that generalizes a result of the seminal paper of Baldwin, Kueker, and van Dieren. We also extract a partial spectrum result for weakly tame AECs, thanks to the surprise appearance of a notion from the theory of accessible categories. We highlight the connections between these two fields (whose deep affinities have yet to be fully appreciated) and distill AECs down to their category-theoretic essence. Once we begin looking at things through the eyes of a category theorist, some very surprising results appear out of nowhere. In particular, using nothing more than the Yoneda embedding, we obtain a peculiar structure theorem for categorical AECs, an equivalence of categories that identifies the large structures in a kappa-categorical AEC with sets equipped with an action of the monoid of endomorphisms of the unique structure of cardinality kappa.

Tuesday, November 24: No seminar this week. Happy Thanksgiving!

Spring, 2009

Special Meeting Friday, January 9, 2 - 4 p.m., MATH 2400

Tuesday, January 27, Organizational Meeting.

Tuesday, February 3, Chris Laskowski, UMCP
Title: An overview of NIP theories

Tuesday, February 10, NO MEETING

Tuesday, February 17, John Goodrick, UMCP
Title: Type amalgamation properties in unstable theories

Tuesday, February 24, Vincent Guingona, UMCP
Title: Kueker's Conjecture
Abstract: Kueker's Conjecture is that if T is a complete (countable?) theory all of whose uncountable models are omega-saturated then T must be categorical in some infinite cardinality. We will discuss Hrushovski's proof of this conjecture for stable T and T with Skolem functions, then talk about partial results for UDTFS theories (a subclass of NIP theories).

Tuesday, March 3, Alfred Dolich
Title: Dense pairs

Tuesday, March 10, Alfred Dolich
Title: Dp-minimality

Tuesday, March 17, NO MEETING (Spring Break)

Special Meeting Monday, March 23, 4-5pm, Math 1308

Tuesday, March 31, NO MEETING

Tuesday, April 7, John Goodrick, UMCP
Title: Types and models in superstable nonmultidimensional theories
Abstract: The nicest kinds of formulas in a stable theory are the weakly minimal ones. One can define an ordinal-ranked hierarchy of formulas which are almost this nice: let Θ₀ be the family of all weakly minimal formulas, and let Θ_α be the family of all weakly Θ_{< α}-minimal formulas. Pillay noticed that in a superstable n.m.d. theory, any type is nonorthogonal to a regular type that is a generic extension of a formula in Θ_α (for some α). Chowdhury, Loveys, and Tanovic used this to construct a definable continuous rank in such theories which witnesses forking.
We will explain what these (relatively old) results mean, give concrete examples, and (time permitting) try to explain how this is useful in building models of such theories.

Tuesday, April 14, John Goodrick, UMCP
Title: Types and models in superstable nonmultidimensional theories, take 2
Abstract: See last week's abstract -- I cancelled my talk last week because many people couldn't make it to seminar, so I will try again this week.

Tuesday, April 21, John Goodrick, UMCP
Title: More on types and models in superstable nonmultidimensional theories

Tuesday, April 28, Justin Brody, UMCP
Title: TBA

Tuesday, May 5, Yun Lu, Kutztown University
Title: Reducts of countably categorical graphs
Abstract: Let M be a countably categorical structure, homogeneous for a finite relational language. A reduct of M corresponds, up to bi-interpretability, to a closed subgroup of Sym(M) containing Aut(M). In this talk, I will describe classifications of reducts given by Higman, Thomas and Bennett. I will also present my own results classifying reducts of the random bipartite graph and the random bipartite graph having more than two cross types.

Tuesday, May 12, Ahuva Shkop, University of Illinois at Chicago
Title: Rotund varieties and exponential polynomials: a proof of the Henson-Rubel theorem for pseudoexponentiation
Abstract: In 1984, Henson and Rubel proved the Schanuel nullstellensatz, i.e. the only exponential polynomials with no zeros in the complex field are of the form e^g for some exponential polynomial g. We will prove the analogue for Zilber's field directly from the axioms. Furthermore, this proof relies on the exponential closedness axiom without any reference to Schanuel's conjecture.

Fall, 2008

Tuesday, September 2, John Goodrick, UMCP
Title: Internality, definable groupoids, and amalgamation in stable theories, Part I
Abstract: Building on Hrushovski's intriguing preprint "Groupoids, imaginaries, and internal covers," we present some results on connections between the following things: definable groupoids (i.e. categories in which every morphism has an inverse), finite internal covers (or "generalized imaginary sorts"), and amalgamation properties (spefically, 3-uniqueness and 4-existence). This is ongoing joint work with Alexei Kolesnikov.

Tuesday, September 9, Vincent Guingona, UMCP
Title: The Mordell-Lang Conjecture
Abstract: This talk will be about Ehud Hrushovki's famous model-theoretic proof of the Mordell-Lang conjecture for function fields (see Hrushovski, "The Mordell-Lang conjecture for function fields," Journal of the AMS, vol. 9 (1996), pp. 667-690).

Tuesday, September 16, John Goodrick, UMCP
Title: Internality, definable groupoids, and amalgamation in stable theories, Part II
Abstract: Continuing the talk from September 2, we will quickly review the basic definitions, then describe some interesting examples of failures of 3-uniqueness and of non-retractable groupoids. If we have time, we will say a little more about the theory of non-retractable groupoids and what they have to do with amalgamation.

Tuesday, September 23, Alexei Kolesnikov, Towson University
Title: Internality, definable groupoids, and amalgamation in stable theories, Part III

Tuesday, September 30, Justin Brody, UMCP
Title: Model theory of random graphs
Abstract: The random graph G(n, n^-a) is formed by taking n vertices and randomly assigning an edge to each pair of vertices with probability n^-a. Spencer and Shelah showed that if a is irrational in (0,1) then any sentence in the language of graphs will hold with probability 0 or 1 in the limit. Baldwin and Shelah later showed that the almost-sure theory thereby obtained has models which are the generics obtained by Hrushovski's amalgamation construction. We examine some analogues of this construction in the case that a is rational.

Tuesday, October 7, Moshe Kamensky, University of Waterloo
Title: Model theory of the Tannakian formalism
Abstract: A Tannakian category is an axiomatic description of the category of representations of an affine algebraic group. I will explain the axioms and how they can be viewed within model theory, and will show how the fundamental theorem becomes an instance of the Galois group obtained from internality in a theory associated with the category.

Tuesday, October 14, NO MEETING

Tuesday, October 21, Jennifer Chubb, The George Washington University
Title: Computability theoretic properties of relations on computable structures
Abstract: We consider algorithmic properties of additional relations definable on computable structures. For example, for a computable linear ordering we may consider the successor relation, which does not have to be computable. I will discuss some general results in the literature, and present some examples from my recent collaborative projects. We will see that for a large class of linear orderings, the Turing degree spectra of the successor relation is closed upward in the c.e. degrees. Then, we will use algorithmic information theory to analyze the strong degree spectra of the ω-type initial segment of computable linear orderings of type ω + ω^*, and compare it to the Turing degree spectra of this relation.

Tuesday, October 28, Alexei Kolesnikov, Towson University
Title: TBA

Tuesday, November 4, Alexei Kolesnikov, Towson University
Title: Finite covers, continued
Abstract: Continuing last week's talk, the finite covers as defined in David Evans' "Finite covers with finite kernels" (APAL 88 (1997) 109--147) will be discussed, and this will be compared with Hrushovski's notion of finite internal covers.

Tuesday, November 11, John Goodrick, UMCP
Title: TBA

Tuesday, November 18, Valentina Harizanov, The George Washington University
Title: Effective Scott families of computable abelian p-groups
Abstract: We will discuss effective categoricity and effective Scott families of computable structures. We will focus on abelian p-groups and on characterizing those with computably enumerable Scott families of existential formulas, as well as those with computably enumerable Scott families of computable infinitary Sigma-2 formulas. While the first class coincides with computably categorical abelian p-groups, it is not known whether the second class includes all limit computably categorical abelian p-groups.

Tuesday, November 25, Vincent Guingona, UMCP
Title: Uniform Definability of Types over Finite Sets

Tuesday, December 2, NO MEETING

Tuesday, December 9, NO MEETING

Logic Seminar for 2007 - 2008:

Fall, 2007

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

Spring, 2008

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 ב_α, 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.