Midwest Model Theory Day is a long-running conference series, which in recent times has been organized most semesters by James Freitag at UIC. During Spring 2021, we are meeting online every other week on Tuesday afternoon.
This seminar is organizd by James Freitag and Maryanthe Malliaris. If you are interested in details about how to attend, please write us a quick email to be put on the mailing list - we send a reminder email each Monday before the corresponding Tuesday seminar. jfreitag@uic.edu or mem@math.uchicago.edu
January 26 Gabriel Conant (Cambridge) - NIP approximate groups and arithmetic regularity
February 9 Rizos Sklinos (Stevens) - Fields interpretable in the free group
February 23 Leonardo Nagami Coregliano (Chicago) - Continuous combinatorics and natural quasirandomness
March 9 Natasha Dobrinen (Denver) - Fraisse classes with simply characterized big Ramsey degrees
March 23 Marcos Mazari Armida (Carnegie Mellon) - Stability and superstability in classes of modules
April 6 Gregory Cherlin (Rutgers) - Metrically Homogeneous Graphs and Distance Semigroups
April 20 Uri Andrews (Wisconsin) - Effective non-local-finiteness in flat strongly minimal theories (Work joint with Omer Mermelstein)
May 4 (Tentative) Margaret Thomas (Purdue)
Abstracts:
Effective non-local-finiteness in flat strongly minimal theories: I'll talk about a connection between model theory and recursive model theory. In answering Zilber's trichotomy conjecture, Hrushovski built a strongly minimal theory with a geometric property called flatness. Flatness precludes the existence of a definable group, but it does much more. I'll discuss what the assumption of flatness tells us about being able to build recursive models of strongly minimal theories.
Metrically Homogeneous Graphs and Distance Semigroups: A connected graph is metrically homogeneous if it is homogeneous in the sense of Fra\"iss\'e (or of Urysohn, 1924) when equipped with the shortest path metric: in other words, the notions of congruence in the senses of Euclid and Klein agree. Moss and Cameron asked for the classification of these graphs. I conjectured an answer in 2011, after constructing a 6-parameter family of (mostly) new examples. Inspired by the influential work of Kechris/Pestov/Todorcevic, a cabal of combinatorialists studied, and mostly solved, several combinatorial problems connected with the topological dynamics of the automorphism groups of these graphs. Their (or its) main technique is most easily understood as involving generalized metric spaces with values in a partially ordered semigroup (cf. Braunfeld, Conant, Sauer). Specifically, one studies shortest path metrics in partial generalized metric spaces, in the manner of Nesetril, 2007. It now appears that the classification conjecture can be proved by direct but lengthy methods (in progress, with Daniela Amato). One useful ingredient, which we call ``restricted amalgamation,'' is suggested by the structure of the associated semigroups. Various questions arise. In particular, it seems fair to say that the definition of the term``generalized metric space with values in a partially ordered semigroup'' is not yet settled. These matters have been explained to me by Hubicka and Konecny, who are wholly responsible for any errors I may make.
Stability and superstability in classes of modules: Fisher and Baur showed in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In this talk we will explore if the same holds for any abstract elementary class of modules, i.e., for any AEC (K, <_p) such that K is a class of R-modules for a fixed ring R and <_p is the pure submodule relation. In particular, using that the class of p-groups with pure embeddings is a stable AEC, I will present a solution to a problem of L. Fuchs. Moreover, we will study the notion of superstability and show how superstability can be used to give new characterizations of some classical rings.
Fraisse classes with simply characterized big Ramsey degrees: Analogues of the infinite Ramsey theorem to infinite structures have been studied since the 1930's, when Sierpinski gave a coloring of pairs of rationals into two colors such that, in any subset of the rationals forming a dense linear order, both colors persist. Such a coloring is called "unavoidable" since both colors persist in any infinite substructure isomorphic to the original (in this case the rationals as a linear order). In the 1970's Galvin showed that two is the optimum number for pairs of rationals, while Erdos, Hajnal and Posa extended Sierpinski's result to colorings of edges in the Rado graph. These results instigated a steady stream of results for the next several decades, a pinnacle of which was the work of Laflamme, Sauer, and Vuksanovic finding the exact number of colors for unavoidable colorings of finite graphs inside the Rado graph, as well as other Fraisse structures with finitely many binary relations, including the generic tournament. This exact number is called the "big Ramsey degree", a term coined by Kechris, Pestov, and Todorcevic. In this talk, we will provide a brief overview of the area of big Ramsey degrees on Fraisse limits. Then we will present recent joint work with Rebecca Coulson and Rehana Patel characterizing the big Ramsey degrees for some seemingly disparate Fraisse classes. We formulate an amalgamation property, which we call the Substructure Free Amalgamation Property, and show that every Fraisse relational class with finitely many relations satisfying SFAP has big Ramsey degrees which are characterized in a manner as simply as those of the Rado graph. A more general property for disjoint amalgamation classes, which we call SDAP^+, also ensures the same simple characterization of big Ramsey degrees. One of the novelties of our approach is that we build trees of quantifier-free 1-types with special nodes coding the vertices in a given enumerated Fraisse limit. Then we use the method of forcing to do an unbounded search for a finite object, which produces in ZFC the exact big Ramsey degrees for these structure. SDAP^+ holds for unrestricted relational structures, relational structures with forbidden 3-irreducible substructures, and others, producing new lines of results while recovering in a streamlined manner several previous results, including those of Laflamme, Sauer, and Vuksanovic.
Continuous combinatorics and natural quasirandomness: The theory of graph quasirandomness studies several asymptotic properties of the random graph that are equivalent when stated as properties of a deterministic graph sequence and was one of the main motivations for the theory of dense graph limits, also known as theory of graphons. Since the theory of graphons can itself be used to study graph quasirandomness and can be generalized to a theory of dense limits of models of a universal first-order theory, a natural question is whether a general theory of quasirandomness is possible. In this talk, I will briefly introduce the general theory of dense limits of combinatorial objects (often associated with the name continuous combinatorics) and talk about the notion of natural quasirandomness, a generalization of quasirandomness to the same general setting of universal first-order theories. The main concept explored by our quasirandomness properties is that of unique coupleability that roughly means that any alignment of two limit objects on the same ground set "looks random". This talk is based on joint work with Alexander A. Razborov.
Fields interpretable in the free group: In this talk I will show that no infinite field is interpretable in the first-order theory of nonabelian free groups.
NIP approximate groups and arithmetic regularity: I will speak about joint work with Anand Pillay on the structure of finite approximate groups satisfying a local NIP assumption. Our results can be seen as a unification of the model-theoretic study of "tame" arithmetic regularity with work of Hrushovski and Breuillard, Green, and Tao on the structure theory of approximate groups.
Midwest Model Theory Day is a long-running conference series, which in recent times has been organized most semesters by James Freitag at UIC. During Fall 2020, we are meeting online every other week on Tuesday afternoons, alternate with the Midwest Computability Seminar.
This seminar is organizd by James Freitag and Maryanthe Malliaris. If you are interested in details about how to attend, please write us a quick email to be put on the mailing list - we send a reminder email each Monday before the corresponding Tuesday seminar.
jfreitag@uic.edu or mem@math.uchicago.edu
August 25 Nick Ramsey (UCLA) - Model-theoretic tree properties
September 8 Jana Marikova (KGRC, Vienna) - Quantifier elimination for o-minimal groups expanded by a valuational cut
September 22 Sebastian Eterovic (Berkeley) - Model Theory of exp and j
October 6 Sarah Peluse (IAS, Princeton) - The polynomial Szemeredi theorem in finite fields
October 20 Jerry Keisler (University of Wisconsin) - Using Ultraproducts to Compare Continuous Structures
November 17 Lynn Scow (Cal State San Bernidino) - Applications of generalized indiscernible sequences
December 1 Cameron Hill (Wesleyan) - Remarks on countably categorical almost-sure theories
Abstracts:
Model-theoretic tree properties: The first model-theoretic tree properties were introduced by Shelah as a by-product of his analysis of forking in stable theories. Since then, other tree properties have appeared and, together, these combinatorial dividing lines (TP, TP_1/SOP_2, TP_2, SOP_1,...) serve as the basis for a growing body of research in model theory. I'll survey the work done in this area (and try to justify the idea that it can be understood as an area) by explaining three of the core ingredients in the theory developed so far: generalized indiscernibles, dividing at a generic scale, and amalgamation.
Quantifier elimination for o-minimal groups expanded by a valuational cut: We let R be an o-minimal expansion of a group in a language in which Th(R) eliminates quantifiers, and we let C be a valuational cut in R. We show that if nonforking in certain Morley sequences is symmetric, then the theory of R expanded by a predicate for C and a small number of constants eliminates quantifiers. This is a generalization of results on o-minimal fields with convex subrings satisfying some extra conditions such as T-convexity or o-minimality of the residue field. This is joint work with C. F. Ealy.
Model Theory of exp and j In this talk I will describe some questions that arise from studying the model theory of the complex exponential function, and the work that it has inspired for other functions; in particular, I will present some of the results that have been obtained for the modular j function.
The polynomial Szemeredi theorem in finite fields: Szemer\'edi's theorem on arithmetic progressions states that any subset of the integers with positive upper density contains arbitrarily long arithmetic progressions x,x+y,...,x+my with y nonzero. Bergelson and Leibman proved that the statement of Szemer\'edi's theorem still holds with more general polynomial progressions x,x+P_1(y),...,x+P_m(y) in place of arithmetic progressions. While there are now many approaches to Szemer\'edi's theorem, including Szemer\'edi's original proof using the regularity lemma, Furstenberg's proof using ergodic theory, Gowers's proof using higher order Fourier analysis, and a couple of hypergraph regularity proofs, the only proof of the polynomial Szemer\'edi theorem in full generality is via ergodic theory. In this talk I will discuss some recent different approaches to the polynomial Szemer\'edi theorem, focusing on the finite field setting.
Using Ultraproducts to Compare Continuous Structures: We revisit two research programs that were proposed in the 1960's, remained largely dormant for five decades, and then become hot areas of research in the last decade. The monograph ``Continuous Model Theory'' by Chang and Keisler, Annals of Mathematics Studies (1966), studied structures with truth values in [0,1], with formulas that had continuous functions as connectives, sup and inf as quantifiers, and equality. In 2008, Ben Yaacov, Bernstein, Henson, and Usvyatsev introduced the model theory of metric structures, where equality is replaced by a metric, and all functions and predicates are required to be uniformly continuous. This has led to an explosion of research with results that closely parallel first order model theory, with many applications to analysis. In my forthcoming paper ``Model Theory for Real-valued Structures'', the "Expansion Theorem" allows one to extend many model-theoretic results about metric structures to general [0,1]-valued structures--the structures in the 1966 monograph but without equality. My paper ``Ultrapowers Which are Not Saturated'', J. Symbolic Logic 32 (1967), 23-46, introduced a pre-ordering $\mathcal M\trianglelefteq\mathcal N$ on all first-order structures, that holds if every regular ultrafilter that saturates $\mathcal N$ saturates $\mathcal M$, and suggested using it to classify structures. In the last decade, in a remarkable series of papers, Malliaris and Shelah showed that that pre-ordering gives a rich classification of simple first-order structures. Here, we lay the groundwork for using the analogous pre-ordering to classify [0,1]-valued and metric structures.
Applications of generalized indiscernible sequences: We survey some of the applications of generalized indiscernible sequences, both in model theory and in structural Ramsey theory. We will attempt to explain and draw relationships between: Ehrenfeucht-Mostowski types, the modeling property, and dividing lines in model theory.
Remarks on countably categorical almost-sure theories: I will survey what is known (at least, what I know) about 01-laws for Fraisse classes with countably categorical almost-sure theories. Specifically, I will discuss (1) why invariant measures are probably the only fruitful setting to study, (2) why only rank-1 super-simple theories are part of this story, and (3) the several incorrect conjectures that I’ve made in the past and how to improve them (and some theorems you can prove while being wrong for a while).