Logic Archives for Fall 2015 to Spring 2016
Organizational Meeting
When: Tue, September 2, 2014 - 3:30pmWhere: Math 1311
Speaker:
Using forcing to prove theorems in ZFC
When: Tue, September 9, 2014 - 3:30pmWhere: Math 1311
Speaker: Chris Laskowski (UMCP) -
Dividing lines for classes of atomic models
When: Tue, September 23, 2014 - 3:30pmWhere: Math 1311
Speaker: Chris Laskowski (UMCP) -
Abstract: We begin the study of the class of atomic models of a complete theory in a countable language. Specifically, we offer two properties and prove: (1) If an atomic class fails either of these properties, then there are many atomic models of size aleph1; and (2) If an atomic class has both of these properties, then there is an atomic model of size continuum. As a corollary to these results, if an atomic class characterizes aleph_alpha for some positive alpha, then the class has many atomic models of size aleph1.
The Hanf number for amalgamation
When: Tue, September 30, 2014 - 3:30pmWhere: Math 1311
Speaker: Alexei Kolesnikov (Towson University) -
The Hanf number for amalgamation, Part II
When: Tue, October 7, 2014 - 3:30pmWhere: Math 1311
Speaker: Alexei Kolesnikov (Towson University) -
Cancelled - On dense/codense subsets of geometric structures
When: Tue, October 14, 2014 - 3:30pmWhere: Math 1311
Speaker: Yevgeniy Vasilyev (Christopher Newport Univeristy and Memorial University of Newfoundland) -
A New Look at the Covering Theorem
When: Tue, October 21, 2014 - 3:30pmWhere: Math 1311
Speaker: David W. Kueker (UMCP) -
Must a unique atomic model be constructible?
When: Tue, October 28, 2014 - 3:30pmWhere: Math 1311
Speaker: Douglas Ulrich (UMCP) -
Completeness and Categoricity (in power): Formalization without Foundationalism
When: Tue, November 4, 2014 - 3:30pmWhere: Math 1311
Speaker: John Baldwin (University of Illinois, Chicago) -
Abstract: Formalization has three roles: 1) a foundation for an area (perhaps all) of math- ematics, 2) a resource for investigating problems in ‘normal’ mathematics, 3) a tool to organize various mathematical areas so as to emphasize commonalities and differences. We focus on the use of theories and syntactical properties of theories in roles 2) and 3). Formal methods enter both into the classification of theories and the study of definable set of a particular model. We regard a property of a theory (in first or second order logic) as virtuous if the property has mathematical consequences for the theory or for models of the theory. We rehearse some results of Marek Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial. But ‘categoricity in power’ illustrates the sort of mathematical consequences we mean. One can lay out a schema with a few parameters (depending on the theory) which describes the structure of any model of any theory categorical in uncountable power. Similar schema for the decomposition of models apply to other theories according to properties defining the stability hierarchy. We describe arguments using properties, which essentially involve formalizing mathematics, to obtain results in ‘mainstream’ mathematics. We consider discussions on method by Kashdan, and Bourbaki as well as such logicians as Hrushovski and Shelah.
Independence in tame abstract elementary classes
When: Tue, November 11, 2014 - 3:30pmWhere: Math 1311
Speaker: Sebastien Vasey (Carnegie Mellon University) -
Abstract: Good frames are one of the main notions in Shelah's classification theory for abstract elementary classes. Roughly speaking, a good frame describes a local forking-like notion for the class. In Shelah's book, the theory of good frames is developped over hundreds of pages, and many results rely on GCH-like hypotheses and sophisticated combinatorial set theory.
In this talk, I will argue that dealing with good frames is much easier if one makes the global assumption of tameness (a locality condition introduced by Grossberg and VanDieren). I will outline a proof of the following result: Assume K is a tame abstract elementary class which has amalgamation, no maximal models, and is categorical in a cardinal of cofinality greater than the tameness cardinal. Then K is stable everywhere and has a good frame.
Countable model theory and the complexity of isomorphism
When: Tue, November 18, 2014 - 3:30pmWhere: Math 1311
Speaker: Richard Rast (UMCP) -
Abstract: We discuss the Borel complexity of the isomorphism relation (for countable models of a first order theory) as the “right” generalization of the model counting problem. In this light we present recent results of Dave Sahota and the speaker which completely characterize the complexity of isomorphism for o-minimal theories, as well as recent work of Laskowski and Shelah which give a partial answer for omega-stable theories. Along the way, we introduce a few open problems and barriers to generalizing the existing results.
The Asymptotic Couple of the Field of Logarithmic Transseries
When: Tue, November 25, 2014 - 3:30pmWhere: Math 1311
Speaker: Allen Gehret (UIUC) -
Abstract: We will define the differential field of logarithmic transseries and discuss its value group $\Gamma$. The value group $\Gamma$ can be given the additional structure of a map $\psi:\Gamma\to\Gamma$ which is induced by the field derivation. The structure $(\Gamma,\psi)$ is the asymptotic couple of the field of logarithmic transseries. We will discuss properties of abstract asymptotic couples (i.e., ordered abelian groups with an additional map that satisfies certain axioms). We will present a quantifier elimination result for the theory of the asymptotic couple $(\Gamma,\psi)$ in an appropriate first-order language and discuss various other things (definable functions on a certain discrete set, a stable embedding result, and NIP).
On dense/codense subsets of geometric structures
When: Tue, December 2, 2014 - 3:30pmWhere: Math 1311
Speaker: Yevgeniy Vasilyev (Christopher Newport Univeristy and Memorial University of Newfoundland) -
Abstract: A theory $T$ is called geometric if in models of $T$, algebraic closure satisfies the exchange property and $T$ eliminates the $\exists^\infty$ quantifier. Examples include strongly minimal and o-minimal structures.
We say that a subset $P$ of a model $M$ of $T$ is dense/codense if any nonalgebraic 1-type over a finite dimensional subset of $M$ has a realization in $P$ and a realization "generic" over $P$.
Requiring that $P$ is algebraically independent or algebraically closed gives rise to two kinds of well-behaved unary predicate expansions of $T$. We will focus on the latter (known as lovely pair expansion). In particular, we will look at the properties of three closure operators: $acl$ in $T$, $acl$ in the expansion $T_P$, and the "small closure" operator associated with the pair $(M,P)$. This is a joint work with A. Berenstein.
Organizational Meeting
When: Tue, January 27, 2015 - 3:30pmWhere: Math 1311
Speaker: Organizational Meeting () -
Constructing customized models of size continuum
When: Tue, February 17, 2015 - 3:30pmWhere: Math 1311
Speaker: Chris Laskowski (UMCP) -
Polygroupoids 2.0
When: Tue, February 24, 2015 - 3:30pmWhere: Math 1311
Speaker: Alexei Kolesnikov (Towson University) -
Abstract: I will talk about the objects that characterize the failure of generalized amalgamation properties in stable theories. It was established by Hrushovski that the failure of 3-uniqueness in stable theories is characterized by the presence of definable groupoids, but it was not clear what definable objects characterize the failure of n-uniqueness for n greater than 3. John Goodrick, Byunghan Kim, and I were working to address this problem.
Two years ago, I discussed mathematical structures, called n-polygroupoids, the first order theory of which is stable, but fails (n+1)-uniqueness. It was not clear, however, whether such structures could be recovered from an arbitrary stable theory that fails (n+1)-uniqueness. The new and improved version of n-polygroupoids that I will describe fits perfectly into the puzzle.
Reducts of Homogeneous Structures
When: Tue, March 3, 2015 - 3:30pmWhere: Math 1311
Speaker: Amy Lu (Kutztown University) -
Abstract: Simon Thomas conjectured that every countable homogeneous structure with a finite relational language has only finitely many inequivalent reducts in 1991. Apart from being true for some fundamental homogeneous structures, we know very little about this conjecture. In this talk, I will present some those homogeneous structures including the rationals ( Q, < ), the random graph, the random tournament, the expansion of (Q, < ) by a constant, the random partial order, and the random ordered graph. Furthermore, I will talk about our research on the reducts of the random graph.
Building Borel models of size continuum
When: Tue, March 24, 2015 - 3:30pmWhere: Math 1311
Speaker: Chris Laskowski (UMCP) -
Reducts of Homogeneous Structures
When: Tue, April 7, 2015 - 3:30pmWhere: Math 1311
Speaker: Amy Lu (Kutztown University) -
Abstract: Simon Thomas conjectured that every countable homogeneous structure with a finite relational language has only finitely many inequivalent reducts in 1991. Apart from being true for some fundamental homogeneous structures, we know very little about this conjecture. In this talk, I will present some those homogeneous structures including the rationals ( Q, < ), the random graph, the random tournament, the expansion of (Q, < ) by a constant, the random partial order, and the random ordered graph. Furthermore, I will talk about our research on the reducts of the random graph.
The Complexity of Isomorphism for Linear Orders
When: Tue, April 14, 2015 - 3:30pmWhere: Math 1311
Speaker: Richard Rast (UMCP) -
Excellent exuberance -- The rise and fall of locally finite AEC's
When: Tue, April 28, 2015 - 3:30pmWhere: Math 1311
Speaker: Chris Laskowski (UMCP) -
Polygroupoids 2.0, continued
When: Tue, May 5, 2015 - 3:30pmWhere: Math 1311
Speaker: Alexei Kolesnikov (Towson University) -
Abstract: I will continue talking about the objects that characterize the failure of generalized amalgamation properties in stable theories. I will outline how to recover these objects in a stable theory that fails (n+1)-uniqueness.