- 1/30/2018 4PM,
Wim Ruitenburg,
Marquette University, Wisconsin

Title: Another proof interpretation for predicate logicThe Brouwer-Heyting-Kolmogorov (BHK) proof interpretation for predicate logic is at least an informal explanation by which to justify or explain the rules of intuitionistic logic. In this talk, rather than going from intuitionistic logic to a BHK style interpretation, we offer a BHK style interpretation and get a logic. - 2/13/2018 4PM,
Matthew Harrison-Trainor, University of Waterloo, Ontario,
Canada

Title: TBATBA

Dinner: TBA at 6PM - 3/13/2018 4PM,
Dan Turetsky, University of Notre Dame, Indiana

Title: TBATBA

Dinner: TBA at 6PM - 3/20/2018 4PM,
Linda Brown Westrick, University of Connecticut-Storrs

Title: TBATBA

Dinner: TBA at 6PM - 4/19/2018 4PM,
Peter Cholak, University
of Notre Dame, Indiana

Title: TBA

Dinner: TBA at 6PM TBA - 4/24/2018 4PM,
Tamvana
Makuluni, UW

Title: TBA (thesis defense)TBA - 5/1/2018 4PM,
Ethan
McCarthy, UW

Title: TBA (thesis defense)TBA - 5/8/2018(?) 4PM,
Dilip Raghavan,
National University of Singapore

Title: TBATBA

Dinner: TBA at 6PM

- 9/5/2017 4PM,
Dino Rossegger,
Vienna University of Technology, Austria

Title: Bi-embeddability spectra of structuresOne aim in computable structure theory is to measure the computational complexity of countable structures. The main property to investigate in this line of research is the degree spectrum of a structure, the set of Turing degrees of its isomorphic copies.In joint work with Ekaterina Fokina and Luca San Mauro we investigate a related notion, bi-embeddability spectra. Two structures are bi-embeddable if each is embeddable in the other and the bi-embeddability spectrum of a structure is the set of Turing degrees of its bi-embeddable copies. We give general results on bi-embeddability spectra and investigate bi-embeddability spectra of strongly locally finite graphs.

Dinner: Great Dane Pub (123 E. Doty St.) at 6PM - 9/12/2017 4PM,
Steffen Lempp,
UW

Title: Numberings of finite families of c.e. setsI will discuss an old open problem posed by Ershov in the 1970's: The Rogers semilattice of a family*F*of c.e. sets is the collection of computable numberings (i.e., uniform enumerations) of*F*, identifying numberings which can be effectively reduced to each other, and partially ordered by this reducibility (i.e., a numbering μ reduces to a numbering ν if there is a computable function f taking a μ-index n to a ν-index f(n) of the same set). It is not hard to check that a Rogers semilattice always forms a distributive upper semilattice.

Ershov asked for a characterization of the finite families*F*and*G*of c.e. sets such that their Rogers semilattices are isomorphic. In the late 1970's, two results were shown: Khutoretskii proved that the Rogers semilattice of any family of c.e. sets must be infinite or of size at most 1. Denissov showed that for any n, the Rogers semilattices of {∅, {0}} and {∅, {0}, {1}, ..., {n}} are isomorphic. (Note that for finite families of c.e. sets, the isomorphism type of the Rogers semilattice is completely determined by the isomorphism type of*F*under set inclusion.) Denissov's proof is quite delicate, by a non-effective back-and-forth argument. In 2003, Ershov showed a partial generalization of Denissov's result: If the Rogers semilattices of the finite families*F*and*G*are isomorphic, then the collections of nonmaximal elements of*F*and*G*(under set inclusion) must be isomorphic; the converse is open and was conjectured by Kudinov to give the desired characterization.

After some background, I will sketch Denissov's and Ershov's proofs and point out the difficulties in proving Kudinov's conjecture. My talk will be based on extensive discussions with Badaev, Ng and Wu. - 9/19/2017 4PM,
Jim Freitag,
University of Illinois at Chicago

Title: Model theory and the combinatorics of banned sequencesIn this talk, we will introduce the combinatorial notion of thicket density. Roughly, the notion sits in relation to Shelah 2-rank as VC-density is to VC-dimension. Earlier this year, Bhaskar established a growth dichotomy theorem in this setting along the lines of the Sauer-Shelah Lemma. In this talk, we will explain how Bhaskar's result, the Sauer-Shelah Lemma, and several other model-theoretic results all come from one unifying combinatorial result. If time permits, we will explain several new results in the style of the Sauer-Shelah Lemma in various model-theoretic settings and we will explain how our combinatorial principle can be used to improve bounds of Malliaris-Terry. - 9/26/2017 4PM,
John Baldwin,
University of Illinois at Chicago

Title: What is Euclid's continuum problem?We explore the changing conceptions of the*geometric*continuum in Euclid, Archimedes, Descartes, Hilbert, and Tarski and argue: Hilbert's first-order axioms account for the bulk of Euclid; he adopted an equivalent of Dedekind's axiom to ground analysis, not geometry. Tarski's first-order complete first-order axiomatization provides a finitistically justifiable basis for Descartes' geometry (which is not the modern notion of `Cartesian geometry'). Using o-minimality, we extend this theory to provide a first-order account of the formulas for area and circumference of a circle. Finally, we provide L_{ω1,ω}-categorical axiomatizations of various geometries that we find more fruitful than the traditional second-order versions. This analysis demonstrates the importance of logical formalization for both philosophy and mathematics.

Dinner: Kabul Restaurant (541 State St.) at 6PM - 10/3/2017
Midwest
Model Theory Day, University of Illinois at Chicago

Lunch: in SEO 636 at noon

Speakers:- 1PM, Caroline
Terry, University of Maryland, College Park

Title: A stable arithmetic regularity lemmaIn this talk we present a stable version of the arithmetic regularity lemma for vector spaces over fields of prime order. The arithmetic regularity lemma for F^{n}_{p}(first proved by Green in 2005) states that given A ⊆ F^{n}_{p}, there exists H ≤ F^{n}_{p}of bounded index such that A is Fourier-uniform with respect to almost all cosets of H. In general, the growth of the index of H is required to be of tower type depending on the degree of uniformity, and must also allow for a small number of non-uniform elements. Our main result is that, under a natural stability theoretic assumption, the bad bounds and non-uniform elements are not necessary. Specifically, we present an arithmetic regularity lemma for k-stable sets A ⊆ F^{n}_{p}, where the bound on the index of the subspace is only polynomial in the degree of uniformity, and where there are no non-uniform elements. This result is a natural extension to the arithmetic setting of the work on stable graph regularity lemmas initiated by Malliaris and Shelah.

- 2:30PM, John Goodrick, Universidad de los Andes, Bogotá,
Colombia

Title: Model theory of groups of finite dp-rank and finite burdenWe will present several recent results concerning the model theory of groups whose theories are of finite rank, where "rank" means either dp-rank or burden (inp-rank). First, we review some results (joint with A. Dolich) about ordered Abelian groups of finite burden, where "burden" or "inp-rank" is a generalization of weight which is useful in unstable theories; in this context, we can show that unary definable sets satisfy various desirable properties. Next, we present more recent results (joint with J. Dobrowolski) showing that inp-minimal groups with an ordering invariant under left translations are Abelian, and also showing that finite weight stable groups cannot be too far from being Abelian. Finally, we will present some open questions and possible future directions for research.

- 4PM, Joel
Nagloo, Bronx Community College, CUNY, New York

Title: Towards strong minimality and the Fuchsian triangle groupsFrom the work of Freitag and Scanlon, we have that the ODEs satisfied by the Hauptmoduls of arithmetic subgroups of SL2(Z) are strongly minimal and geometrically trivial. A challenge is to now show that same is true of ODEs satisfied by the Hauptmoduls of all (remaining) Fuchsian triangle groups. The aim of this talk is to both explain why this an interesting/important problem and also to discuss some of the progress made so far.

- 1PM, Caroline
Terry, University of Maryland, College Park
- 10/10/2017 4PM,
Steffen Lempp,
UW

Title: Computable linear orders and productsThere are quite a few results in the study of computable linear orders of the form "τ·L is computable iff L is**0**^{(n)}-computable". The goal of joint work with Frolov, Ng and Wu is to classify the order types τ for which such statements are true. We will concentrate on the case n=0, where we have an exact classification: An order type τ has the property that "τ·L is computable iff L is computable" iff τ is finite and nonempty. I will conclude with some general comments. - 10/17/2017 4PM,
Noah Schweber,
UW

Title: Banach-Mazur games in computability theoryI'll talk about computability-theoretic aspects of Banach-Mazur games. I'll talk about the reals which can be coded into games, and analyze the strength of Banach-Mazur determinacy principles for levels of the Borel hierarchy. Time permitting, I'll also show some interactions between one of the forcings used in studying reals coded into games and general set-theoretic questions. - 10/24/2017 1PM,
Midwest Computability Seminar, University of Chicago

Lunch: Ryerson Hall 352 at noon

Speakers:- 1PM, Noah
Schweber, UW

Title: Computability and Banach-Mazur gamesWe'll look at some questions around Banach-Mazur games. On the pure computability-theoretic side, after establishing the effectiveness of some basic facts about Banach-Mazur games, we classify the functions computable from all winning strategies for some Banach-Mazur game as exactly the hyperarithmetic sets, using an analogue of Hechler forcing for building strategies. On the reverse mathematical side, we parallel this by showing that Borel Banach-Mazur determinacy is equivalent to ATR0, and that this equivalence goes "level-by-level;" by contrast, we also show that there is a Turing ideal satisfying lightface Σ^{1}_{1}-Banach-Mazur determinacy but not containing 0^{(α)}, this time using an analogue of Spector forcing for building strategies.

- 2PM, Don Stull,
Iowa State University, Ames

Title: Effective dimension of points on linesThis talk will cover recent work using Kolmogorov complexity to study the dimension of points on lines in the Euclidean plane and its application to important questions in fractal geometry. In particular, we will show that this work strengthens the lower bounds of the dimension of Furstenberg sets. We will also discuss future research and open problems in this area. This talk is based on joint work with Neil Lutz.

- 3:30, Rose Weisshaar, University of Notre Dame, Indiana

Title: Countable ω-models of KP and paths through computable ω-branching treesIt is well known that the Π^{0}_{1}-class C_{PA}⊆ 2^{ω}of completions of Peano arithmetic is universal among nonempty Π^{0}_{1}-subsets of Cantor space. When we consider Π^{0}_{1}-subsets of Baire space, however, there is no such universal example. In this talk, we consider a Π^{0}_{1}-class C_{KP}⊆ ω^{ω}whose paths compute the complete diagrams of countable ω-models of Kripke-Platek set theory (KP). We develop an analogy between how elements of C_{PA}and C_{KP}try to compute members of nonempty Π^{0}_{1}-subsets of Cantor space and Baire space, respectively, and we examine how this analogy breaks down. This is joint work with Julia Knight and Dan Turetsky.

- 4:10PM, Dan Turetsky, University of Notre Dame, Indiana

Title: C.e. equivalence relations and the linear orders they realizeQuotient structures are well studied. In the case of linear orders, it is known that the order-types realized by c.e. quotient structures are precisely those realized by Δ^{0}_{2}-linear orders. We come at this from a different perspective, by considering, for each c.e. equivalence relation, which order-types can be realized as a quotient by that equivalence relation. We study the relationship between computability-theoretic properties of the equivalence relation and the algebraic properties of the order-types it can realize. We also define a pre-order on equivalence relations by comparing the collection of order-types realized in each.

- 1PM, Noah
Schweber, UW
- 11/2/2017 4PM
**(B231 Van Vleck)**, Anand Pillay, University of Notre Dame, Indiana

Title: Stability, Vapnik-Chervonenkis dimension, groups, and regularity theoremsI will try to give an accessible and not particularly technical talk about dividing lines in model theory, and the connections to other parts of mathematics including combinatorics. I plan to include some current work with Conant and Terry.

Dinner: Maharani Restaurant (380 W. Washington Ave.) at 6PM - 11/7/2017 4PM,
Iván Ongay
Valverde,
UW

Title: Effectivizing Localization numbers: the surviving degreesInspired by the work of Rupprecht and others, we study the effective analogue of k-localization numbers. These, the k-surviving degrees, are the degrees that are capable of computing a function in (k+1)^{ω}that is not a branch of any k-subtree of (k+1)^{<ω}. Interestingly, these degrees can be computably traceable. In this talk I will introduce the necessary concepts to talk about these degrees and their weaker brother, the globally k-surviving degrees which, surprisingly, lead us in a direction to have new results in cardinal invariants of set theory related to evasion and prediction. This is a joint work with Noah Schweber. - 11/14/2017 4PM,
Mariya Soskova,
UW

Title: Characterizing the continuous degreesThe continuous degrees were introduced by J. Miller as a way to capture the effective content of elements of computable metric spaces. They properly extend the Turing degrees and naturally embed into the enumeration degrees. Although nontotal (i.e., non-Turing) continuous degrees exist, they are difficult to construct: every proof we know invokes a nontrivial topological theorem.In 2014, M. Cai, Lempp, J. Miller and Soskova discovered an unusual structural property of the continuous degrees: if we join a continuous degree with a total degree that is not below it then the result is always a total degree. We call degrees with this curious property almost total. We prove that the almost total degrees coincide with the continuous degrees. Since the total degrees are definable in the partial order of the enumeration degrees, this implies that the continuous degrees are also definable. Applying earlier work of J. Miller on the continuous degrees, this shows that the relation "PA above" on the total degrees is definable in the enumeration degrees.

In order to prove that every almost total degree is continuous, we pass through another characterization of the continuous degrees that slightly simplifies one of Kihara and Pauly. Like them, we identify our almost total degree as the degree of a point in a computably regular space with a computable dense sequence, and then we apply the effective version of Urysohn's metrization theorem to reveal our space as a computable metric space.

This is joint work with Uri Andrews, Greg Igusa, and Joseph Miller.

- 11/21/2017 4PM,
Meng-Che ("Turbo")
Ho, Purdue University

Title: Finitely generated groups are universalIt is well known that any structure can be coded in a graph. Hirschfeldt, Khoussainov, Shore, and Slinko showed that you can also code any structure in a partial ordering, lattice, integral domain, or 2-step nilpotent group, and showed that certain computability properties are preserved under this coding. Recently, R. Miller, Poonen, Schoutens, and Shlapentokh added fields to this list, and they gave a category-theoretic framework to establish their result. Around the same time, Montalbán introduced the notion of effective interpretation and showed that effectively bi-interpretable structures share many computability-theoretic properties. Thus, using either of these notions, we may define the notions of a class of structures being universal, and Harrison-Trainor, Melnikov, R. Miller, and Montalbán showed that these two notions are indeed equivalent.In joint work with Harrison-Trainor, we use these ideas to define a class of structures being universal within finitely generated structures. We then use small cancelation techniques from group theory to show that the class of finitely generated groups with finitely many constants named is universal within finitely generated structures.

Dinner: Great Dane Pub (123 E. Doty St.) at 6PM - 11/28/2017 4PM,
Ethan McCarthy,
UW

Title: Cototal enumeration degrees and their applications to effective mathematicsA set A ⊆ ω is cototal under enumeration reducibility if A is e-reducible to its complement, in which case we also call the enumeration degree of A cototal. We will look at several classes of mathematical objects whose enumeration degrees are characterized by cototality, including the complements of maximal anti-chains on ω^{<ω}, the enumeration-pointed trees on 2^{<ω}(which are known to code every nontrivial structure spectrum with an F_{σ}base), and the languages of minimal subshifts on 2^{<ω}(which are known to code the Turing degree spectra of minimal subshifts on 2^{<ω}). - 12/5/2017 4PM,
Tamvana Makuluni,
UW

Title: "Local" continuity of functions in ordered convexly orderable theoriesA convexly orderable theory is one in which a linear order can be added in such a way that families of definable sets in the home sort have uniformly finitely many convex components in the added order. Convex orderability is a generalization of VC-minimality, but it is unknown exactly how they are related. I present a result on the behavior of functions definable in certain convexly orderable theories which indicates a potential means of attacking this question, and also hints at a potential classification of definable sets in a convexly orderable theory. - 12/12/2017 4PM,
Denis Hirschfeldt,
University of Chicago, Illinois

Title: The reverse mathematics of model theory and first-order principlesReverse mathematics seeks to classify the relative strength of theorems across mathematics, often in terms of a small number of axiomatic systems with natural computability-theoretic analogs, such as RCA_{0}, which corresponds\ roughly to computable mathematics, and ACA_{0}, which corresponds roughly to arithmetic mathematics. Recently, a great deal of attention has been focused on theorems that fall outside of this neat picture, exhibiting complex reverse-mathematical and computability-theoretic behaviors. Many of these, like Ramsey's Theorem for pairs, have come from combinatorics, but model theory has also been a rich source.I will discuss some results on the computability theory and reverse mathematics of theorems concerning basic model-theoretic notions such as atomic, homogeneous, and saturated models, including connections with combinatorial principles, and with first-order principles such as forms of induction restricted to formulas of low complexity. I will draw in particular from my recent paper "Induction, bounding, weak combinatorial principles, and the Homogeneous Model Theorem" with Karen Lange and Richard Shore.

Dinner: Morris Ramen (106 King St.) at 6PM