Logic Seminar

The University of Maryland Logic Seminar will meet on Tuesdays from 3:30-5:00 pm in Math 1311.

If you would like to present a talk at the seminar, please contact Koushik Pal.

Current Schedule:

Spring 2012:

Tuesday, May 8, Alexei Kolesnikov, Towson University
Title: Saturated pairs spectrum
Abstract: The notion of a "saturated pairs" spectrum was introduced by Shelah in his study of simple theories. The spectrum is a certain set of pairs of cardinals attached to a first-order theory; it is a large set for stable theories and is the smallest possible set for theories that are not simple. It was shown by Shelah that, consistently, the spectrum of any simple theory is the same as the spectrum of the theory of a random graph. It was later shown by Hrushovski that in some models of ZFC it is possible to find a simple theory whose spectrum is strictly smaller than that of the theory of a random graph. I will describe the notion and outline the arguments for the statements above.

Tuesday, April 24, Koushik Pal, UMCP
Title: Transseries - continued...

Tuesday, April 17, Koushik Pal, UMCP
Title: Transseries - a modest beginning

Tuesday, April 10, Richard Rast, UMCP
Title: Conservative Class Theories
Abstract: We axiomatize a theory of classes (NBG) and show that it proves no new theorems about sets. We also show how this can help phrase and prove some things more elegantly, without making these proofs less rigorous.

Tuesday, March 27, Alice Medvedev, UC Berkeley
Title: Unions of chains of signatures
Abstract: We meditate on a particularly naive notion of a limit of a sequence of theories: a union of conservative expansions. That is, we consider a sequence of nested signatures $L_1 \subset L_2 \subset \ldots$, each one a subsignature of the next, and a sequence of $L_i$-theories $T_i$ where each $T_i$ is precisely the set of $L_i$-consequences of $T_{i+1}$ (and hence is a subset of $T_{i+1}$). It turns out that many model-theoretic properties then pass from all $T_i$ to their union $T$; these include consistency, completeness, quantifier elimination, partial quantifier elimination such a model-completeness, elimination of imaginaries, stable embeddedness of some definable set, characterization of algebraic closure; stability, simplicity, rosiness, dependence. Our motivating example is the theory $T$ of fields with an action by $(\mathbb{Q}, +)$, seen as a limit of (theories of) fields with $(\mathbb{Z}, +)$-actions.

Tuesday, March 13, Tim Mercure, UMCP
Title: Proof of completeness and quantifier elimination for RCF

Tuesday, March 6, Maxx Cho, UMCP
Title: The Rasiowa-Sikorski Lemma and Completeness
Abstract: I will show how the completeness theorem depends on the Baire Category Theorem and the Rasiowa Sikorski Lemma in Boolean algebras.

Tuesday, February 28, Koushik Pal, UMCP
Title: Defining groups in o-minimal structures
Abstract: I will continue the story and the enthusiasm from last week about groups interpretable in certain theories, but this time in the unstable context. More precisely, I will state a trichotomy theorem for o-minimal structures and show how to interpret a group under appropriate conditions. This talk will be totally based on the paper "A trichotomy theorem for o-minimal structures" by Ya'acov Peterzil and Sergei Starchenko.

Tuesday, February 21, Chris Laskowski, UMCP
Title: Defining automorphism groups in stable theories

Tuesday, February 14, Eric Pacuit, UMCP
Title: Modeling beliefs in games
Abstract: The central thesis of the so-called epistemic program in game theory is that the basic mathematical model of a game should include an explicit parameter describing the players' informational attitudes. In particular, higher-order information (belief about beliefs, etc.) are key components of an informational context of a game: Witness the following quote from a recent paper on the epistemic foundations of iterated admissibility (A. Brandenburger and A. Friedenberg, Self-Admissible Sets, Journal of Economic Theory, 2010, pp. 785 - 811)

Tuesday, February 7, Eric Pacuit, UMCP
Title: An overview of the Model Theory of Modal Logic
Abstract: I will introduce and discuss the main results about the model theory of propositional modal logic. This includes semantic completeness results and results precisely characterizing the expressivity of the basic modal language. We will also look at the model theory of weak systems of modal logic (eg., "non-normal" modal logics). Time permitting, we will try to understand to what extent the key results can be extended to first-order modal logic.

Tuesday, January 31, Organizational Meeting.

Previous Logic Seminar Meetings:

Fall 2011:

Tuesday, December 13, Chris Laskowski, UMCP
Title: The lay of the land: Shelah's taxonomy of theories

Tuesday, December 6, Joshua Himmelsbach, UMCP
Title: Geometric Representations in Pseudofinite Fields
Abstract: Following a recent paper of Hrushovski and Beyarslan, I will discuss the concept of geometric representation of groups within theories, with an example from the theory of pseudofinite fields.

Tuesday, November 1, Chris Laskowski, UMCP
Title: Honest definitions and UDTFS
Abstract: I will present recent arguments of Pierre Simon and Artem Chernikov, who prove that if T is a NIP theory, then every formula has UDTFS. I will also mention how this global result might be localized.

Tuesday, October 25, Alexei Kolesnikov, Towson University
Title: Objects witnessing failure of n-uniqueness
Abstract: I will start with a brief summary of known results connecting failure of 3-uniqueness to the existence of definable groupoids in stable theories. Similar characterization of the failure of n-uniqueness for n>3 remains an open question (originally posed by Hrushovski). One of the problems is that it is not clear what type of objects witness the failure of n-uniqueness for n>3. In this talk, I will give an example of an object (whose first-order theory is totally categorical and fails 4-uniqueness) that could be a prototypical witness of the failure of 4-uniqueness.

Tuesday, October 18, David Kueker, UMCP
Title: Ramsey Quantifiers and the Finite Cover Property

Tuesday, October 11, Chris Laskowski, UMCP
Title: Mutually Algebraic Structures and Theories

Tuesday, October 4, James Freitag, UIC
Title: Differential Algebraic Groups as Superstable Groups
Abstract: We will talk about how to treat strongly connected differential algebraic groups as superstable groups in a nontraditional way. Specifically, rather than working with Lascar rank or Morley rank, we will consider two differential birational invariants, differential type and typical differential dimension. We will show how, at least in the case of strongly connected groups, typical differential dimension can be used in similar ways to how Morley rank is used in groups of finite Morley rank. We will discuss several problems on strongly connected groups posed by M. Singer and P. Cassidy and show how this approach has been used to solve some of them. Analogous problems in difference fields will be mentioned as well.

Tuesday, September 20, Koushik Pal, UMCP
Title: Existence and Non-Existence of Model Companions II
Abstract: I will continue with last week's discussion about the non-existence of model companion of certain theories when an automorphism is added to the language. I will also talk about what happens when one adds a predicate to the language instead.

Tuesday, September 13, Koushik Pal, UMCP
Title: Existence and Non-Existence of Model Companions I
Abstract: The existence of model companion of a first-order theory T is a well-desired property because it gives a first-order axiomatization of the class of existentially closed models of T. Unfortunately model companions do not always exist. In this talk, I will give a few examples of theories for which model companions exist, and also a few examples for which model companions do not exist.

Tuesday, September 6, Koushik Pal, UMCP
Title: Model Theory of Multiplicative Valued Difference Fields
Abstract: A valued difference field is a valued field together with an automorphism. To understand the theory of such a structure, we need to specify how the valuation function interacts with the automorphism. One such nice class of structures, where an Ax-Kochen principle holds, is the class of multiplicative valued difference fields. In this talk, I will introduce this class, give an axiomatization for it and show an Ax-Kochen kind of result for this class.

Spring, 2011:

Tuesday, January 25, Organizational meeting.

Tuesday, February 8, Chris Laskowski, UMCP
Title: Preserving stability when adding a predicate
Abstract: Suppose we are given a structure M with a stable theory. What conditions are needed to ensure that (M,A), the expansion of M by adding a unary predicate with interpretation A, remains stable? In the course of working toward a sufficient condition given by Ziegler and Casanovas, I will introduce the Order Property, the Dimensional Order Property (DOP) and the finite cover property.

Tuesday, February 15, Chris Laskowski, UMCP
Title: Mutually algebraic formulas and expansions by predicates

Tuesday, February 22, Vincent Guingona, UMCP
Title: Finite Non-Splitting
Abstract: One problem with the notion of uniform definability of types over finite sets (UDTFS) is that it is not necessarily closed under reducts. In this talk, we investigate a way around this using the notion of non-Δ-splitting for a particular set of formulas, Δ.

Tuesday, March 1, Chris Laskowski, UMCP
Title: Borel Completeness of Some Omega Stable Theories

Tuesday, March 8, Lynn Scow, UIC
Title: The Full Binary Tree, SOP₂, and a Limited Modeling Property
Abstract: It can often be useful to find "very indiscernible" witnesses to formula-based properties of theories. For example, in a recent paper of B. Kim and J. Kim, a kind of tree-indexed indiscernible is used to establish the claim that TP1 is equivalent to k-TP1, for a theory. In this talk, we wish to investigate a notion of indiscernibility for the full binary tree that may capture the property SOP₂ , and as weakly as possible. Some different approaches will be discussed, and a limited modeling property result will be presented for one of the approaches.

Tuesday, March 29, Pierre Simon, Paris-Sud
Title: Weak Stable Embeddability

Tuesday, April 5, Cameron Hill, University of Notre Dame
Title: Classification theory in computational complexity
Abstract: Since a number of audience members attended the ASL meeting in Berkeley, where I discussed the main result of my thesis, I will guide a cook's tour of several applications of first-order model theory – that descended from Shelah's classification theory – to computational complexity. I will begin with a discussion of different ways to mine the notion of a "good structure theory" for use around classes K of finite structures, sticking on "detecting non/isomorphism" and "building models efficiently/systematically from fragments" as particularly useful. From there, I will discuss the following two topics with an eye to pointing out how well-known model theoretic concepts arise rather naturally both in the design and analysis of efficient algorithms and in characterizing the hardness of (certain) computational problems. 1. Coordinatization and model building – the classic notion of a good structure theory in model theory often amounts to identifying definable pre-geometries and establishing that these primitive blocks (their kinds and dimensions) control the isomorphism types of models. As a segue, I will discuss (briefly) how such coordinatizations in stable classes can be used both for statistical learning and for recovering models from induced substructures of models. 2. Statistical learning (up to isomorphism) – here, I will discuss how one can associate a family of invariant Keisler measures to a randomized learning algorithm; these can be used to rewrite a given learning algorithm (up to acceptable efficiency losses) in way that is both clearer to the human eye and more clearly model-theoretic

Tuesday, April 12, Valentina Harizanov, George Washington University
Title: Groups, orders, trees, and paths
Abstract: An order on a group is a linear ordering of its elements, which is both left-invariant and right-invariant. We investigate the correspondence of the orders on a computable group to the infinite paths through a computable binary tree, which preserves Turing degrees. The infinite paths through a computable binary tree form an effectively closed set under the usual tree topology. Computability theoretic properties of effectively closed sets have been extensively studied. For example, such a set does not necessarily contain a computable member, but it always contains a member of low Turing degree.

Tuesday, April 19, John Goodrick, University of the Andes
Title: Automorphisms of abelian perfect Polish groups and the Schröder-Bernstein property
Abstract: We study the question of when two bi-embeddable structures are necessarily isomorphic. In joint work with Laskowski, we found a criterion for when this is true of the models of a weakly minimal theory. In the present talk, we discuss how this property for weakly minimal groups can be reduced to a problem about automorphisms of abelian perfect Polish groups which might be of independent (non-logical) interest. The key idea is to use a Dushnik-Miller construction to build bi-embeddable, non-isomorphic dense subgroups in the presence of a certain kind of "generic" automorphism.

Tuesday, April 26,Vincent Guingona, UMCP
Title: Finite Partial Orders
Abstract: I will discuss my recent work on indiscernibility over finite partial orders. In particular, I will present a new characterization of dependence in terms of this generalized indiscernibility.

Tuesday, May 3, Kevin McGoff, UMCP
Title: Subshifts, tilings, and recursion theory
Note: Joint with the Student Dynamics Seminar.

Tuesday, May 10, John Baldwin, UIC
Title: Calculating Hanf Numbers

Fall, 2010:

Tuesday, August 31, Organizational meeting.

Tuesday, September 7, Vincent Guingona, UMCP
Title: On VC-minimal theories and variants
Abstract: I discuss VC-minimal theories and various related notions, including convexly orderable structures and weakly VC-minimal theories. I present new results relating these various notions to other model theoretic notions.

Tuesday, September 14, Hunter Johnson, John Jay College
Title: Investigations of UDTFS
Abstract: I will explore a sufficient condition for UDTFS, called coherence, and discuss possibilities for extending some recent work of Guingona.

Tuesday, September 21, Alexei Kolesnikov, Towson University
Title: The prism lemma
Abstract: In a joint work with Goodrick and Kim, we construct a family homology groups in a general model-theoretic setting. The purpose of the construction is to measure whether (and, if yes, how badly) does generalized amalgamation fail in a first-order theory. The goal of this talk is to present a proof of a key lemma to compute the homology groups. The speaker will make an honest effort to keep the argument as non-technical as possible.

Tuesday, September 28, Joseph Flenner, University of Notre Dame
Title: Relative definability in henselian valued fields
Abstract: While model theory has produced many results about the p-adics, among thema decision procedure due to Cohen and quantifier elimination by Macintyre, the general theory of henselian valued fields presents an inherent difficulty: they are built on structures of arbitrary complexity in the residue field and value group. Ax-Kochen and Ershov, however, proved their completeness result for some henselian valued fields relative to the theories of the residue field and value group, and more recently, there have been some relative quantifier elimination theorems by Kuhlmann. In this spirit, we describe a structure of leading terms associated to a valued field, and outline a proof of decidability and quantifier elimination for henselian valued fields of characteristic 0 relative to the leading term structures.

Tuesday, October 5, Alexei Kolesnikov, Towson University
Title: Homology groups in model theory
Abstract: This is a continuation of the talk from the seminar on September 21. The goal of the series of talks is to present a construction of homology groups in a model-theoretic context and to give a proof of a key lemma to compute the homology groups. This is a joint work with Goodrick and Kim. The speaker will continue making an honest effort to keep the arguments as non-technical as possible.

Tuesday, October 12, Alexei Kolesnikov, Towson University
Title: Homology groups in model theory, Part II
Abstract:This is the next installment in the series of talks. The goal of the series is to present a construction of homology groups in a model-theoretic context and to give a proof of a key lemma to compute the homology groups. This is a joint work with Goodrick and Kim. The speaker will continue making an honest effort to keep the arguments as non-technical as possible.

Tuesday, October 19, Alexei Kolesnikov, Towson University
Title: Homology groups in model theory, Part III
Abstract: The goal of the series is to present a construction of homology groups in a model-theoretic context. This is a joint work with Goodrick and Kim. This talk will be dedicated to computing homology groups in particular examples. The speaker will continue making an honest effort to keep the arguments as non-technical as possible.

Tuesday, October 26, Vincent Guingona, UMCP
Title: On VC-minimal theories and variants, Part 2
Abstract: I will continue my talk from September 7th. In particular, I will present new results about VC-minimal theories, including methods of showing that some formulas have VC-density one. I will conclude with a proof that VC-minimal theories satisfy the Kueker Conjecture.

Tuesday, November 2, Alfred Dolich, East Stroudsburg University
Title: Ordered Structures of Finite Dp-rank and related concepts.
Abstract: In this talk I will discuss recent progress in the study of ordered structures (generally expansions of divisible ordered Abelian groups) under the assumption that they have finite dp-rank as defined by Shelah. We begin by considering the rank 1 (dp-minimal) case and construct new examples of dp-minimal ordered structures. We then go on, building on recent work of Simon, to consider a very broad general class of structures, those in which infinite definable subsets have interior, which includes the dp-minimal structures. We investigate the structure of definable sets under this assumption and construct a host of examples. Finally we consider the general finite rank case and discuss some new structure results as well as several examples.

Tuesday, November 16, Justin Brody, Franklin and Marshall College
Title: Stability in Generic Graphs with Weakly Non-algebraic Extensions
Abstract: The Hrushovski construction proceeds by amalgamating a class of finite structures to obtain a generic structure. Associated with the construction is a closure operator, whose properties determine the resulting generic. In the more studied examples of the construction, closed extensions of a finite substructure of the generic are algebraic over their base, which leads to a stable generic. Restricting our attention to classes of graphs, we will examine amalgamations in which closed extensions need not be algebraic. We will discuss conditions on the closure operator under which the generic is stable, and also conditions under which the generic is essentially undecidable.

Tuesday, November 30, Allen Gehret, UMCP
Title: Two ω-Categorical Theories of Triangle-Free Graphs

Spring, 2010:

Tuesday, January 26, Organizational meeting.

Tuesday, February 2, No seminar.

Tuesday, February 9, No seminar (campus closed due to snow).

Tuesday, February 16, Alexei Kolesnikov, Towson University
Title: Groupoids and their symmetric witnesses.
Abstract: In [1], it was shown that the failure of the 3-uniqueness property in a stable theory is witnessed by a certain symmetric configuration of elements in the monster model of that theory. The ''symmetric witness'' is then used to construct a definable groupoid. In this talk, I will disucss extensions (joint work of John Goodrick, Byunghan Kim, and the speaker) of these results in two directions. First, there is a surprising connection between the formula that "binds" the symmetric witness and the composition in the resulting groupoid. Second, I will discuss what happens (or does not happen) in simple theories.

I will attempt to make the talk accessible to a beginning graduate student in model theory and will mention a number of open problems.

[1] John Goodrick and Alexei Kolesnikov, "Groupoids, covers, and 3-uniqueness in stable theories," Journal of Symbolic Logic, to appear.

Tuesday, February 23, John Goodrick, UMCP
Title: Homology theory for types in a simple theory
Abstract: Continuing the theme of applying ideas from algebraic toplogy to model theory from last weeks' talk, we will describe some recent work (joint with Alexei Kolesnikov and Byunghan Kim) to develop a homology theory for simple and stable theories which allows one to compute homology groups for any complete type. These homology groups are related to the usual amalgamation properties for such theories. We will explain some connections and give examples. (No prior knowlege of homology theory or last week's talk will be assumed.)

Tuesday, March 2, Vincent Guingona, UMCP
Title: Dependence, isolated extensions, and definability of types

Tuesday, March 9, Justin Brody, Franklin and Marshall College
Title: TBA

Tuesday, March 16, SPRING BREAK

Tuesday, March 23, Maryanthe Malliaris, The University of Chicago.
Title: Edge distribution and density in the characteristic sequence
Abstract: The characteristic sequence of a given formula phi(x;y) is a countable sequense of hypergraphs defined on the parameter space of phi. We will show how graph-theoretic techniques, notably Szemeredi's celebrated regularity lemma, can be naturally applied to the study of model-theoretic complexity via the characteristic sequence. Specifically, we relate classification-theoretic properties of a formula and its associated hypergraphs to density between components in Szemeredi-regular decompositions of graphs in the characteristic sequence. In addition, we use Szemeredi regularity to calibrate model-theoretic notions of independence; this sheds light on the interplay of independence and order in unstable theories.

Tuesday, April 6, John Goodrick, UMCP
Title: TBA

Tuesday, April 13, Valentina Harizanov, The George Washington University
Title: Orders on groups
Abstract: We investigate properties of left orders and bi-orders on groups. A group is left-orderable if there is a linear ordering of its domain, which is left-invariant with respect to the group operation. If the ordering is also right-invariant, then the group is bi-orderable. There is a natural topology on the set of all left orders of a semigroup (or even magma), and this space is compact. A group is computable if its domain is a computable set and its group operation is computable. A computable orderable group does not necessarily have a computable ordering. For familiar computable groups, we investigate Turing degrees of their orderings.

Tuesday, April 20, Christopher Shaw, UMCP
Title: TBA

Tuesday, April 27, Alexei Kolesnikov, Towson University
Title: The properties of n-existence and n-uniqueness revisited
Abstract: In this talk, we give a category-theoretic definition of the generalized amalgamation properties in the context of a first-order theory and examine the connection between the n-uniqueness and (n+1)-existence properties. The motivation for the results is twofold: on one hand, we are able to paint a more complete and systematic picture about the connection between (n+1)-existence and n-uniqueness; on the other hand, phrasing the notions in a cateogry-theoretic language serves as a step toards developing homology theory assoicated with generalized amalgamation.

Tuesday, May 4, John Goodrick, UMCP
Title: More on n-existence and n-uniqueness for simple theories

Tuesday, May 11, Vincent Guingona, UMCP
Title: UDTFS in dp-minimal theories
Abstract: It holds.

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

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.

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

Page Last Updated: May 10, 2011