Department of

Mathematics


Seminar Calendar
for Model Theory events the next 12 months of Sunday, January 1, 2017.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, January 27, 2017

4:00 pm in 345 Altgeld Hall,Friday, January 27, 2017

On "Structurable equivalence relations" by R. Chen and A. Kechris: Introduction

Anush Tserunyan (UIUC Math)

Abstract: For a class $\mathcal{K}$ of countable relational structures, a countable Borel equivalence relation $E$ is said to be $\mathcal{K}$-structurable if there is a Borel way to put a structure from $\mathcal{K}$ on each $E$-equivalence class. The paper of Chen and Kechris [arXiv link] studies the global structure (including Borel homomorphisms and reductions) of the classes of $\mathcal{K}$-structurable equivalence relations for various $\mathcal{K}$. In this introductory talk, we will give some background and survey the main results of the paper.

Friday, February 3, 2017

4:00 pm in 345 Altgeld Hall,Friday, February 3, 2017

On "Structurable equivalence relations" by R. Chen and A. Kechris: Universal equivalence relations (2nd talk)

Anush Tserunyan (UIUC Math)

Abstract: In our previous talk, we stated the first main result of the paper: a characterization of the elementary classes of countable equivalence relations. In this second talk, we prove that every elementary class admits a $\sqsubseteq_B^i$-universal equivalence relation. This implies one direction of the aforementioned characterization.

Friday, February 10, 2017

4:00 pm in 345 Altgeld Hall,Friday, February 10, 2017

On "Structurable equivalence relations" by R. Chen and A. Kechris: Characterization of elementary classes (3rd talk)

Anush Tserunyan (UIUC Math)

Abstract: In our previous talk, we proved that any elementary class of equivalence relations admits an invariantly injective universal element. This completes one direction of the characterization of elementary classes. In this third talk, we will prove the other direction of the characterization, as well as discuss other results of the paper if time permits.

Friday, February 17, 2017

4:00 pm in 345 Altgeld Hall,Friday, February 17, 2017

"Strong theories of ordered abelian groups" by A. Dolich and J. Goodrick: Introduction

Erik Walsberg (UIUC Math)

Friday, February 24, 2017

4:00 pm in 345 Altgeld Hall,Friday, February 24, 2017

On "Structurable equivalence relations" by R. Chen and A. Kechris: Structurability by structures with TDC (4th talk)

Aristotelis Panagiotopoulos (UIUC Math)

Abstract: In this talk, we prove a theorem of A. Marks included in the current paper. It says that every aperiodic countable Borel equivalence relation can be $\mathcal{A}$-structured for any countable structure $\mathcal{A}$ with trivial definable closure (TDC). Examples include the rationals, the random graph, and the rational Urysohn sphere.

Friday, March 10, 2017

4:00 pm in 345 Altgeld Hall,Friday, March 10, 2017

Strong Theories of Ordered Abelian groups

Travis Nell (UIUC Math)

Abstract: We will continue discussing the paper "Strong Theories of Ordered Abelian Groups" by A. Dolich and J. Goodrick

Tuesday, March 14, 2017

1:00 pm in Altgeld Hall,Tuesday, March 14, 2017

Model theory and Painleve equations

James Freitag (UIC Math)

Abstract: We will discuss how to use model theory to prove some transcendence results for solutions of Painleve equations.

Tuesday, April 4, 2017

1:00 pm in UIC SEO 636,Tuesday, April 4, 2017

MidWest Model Theory Day at UIC

Friday, April 7, 2017

4:00 pm in 245 Altgeld Hall,Friday, April 7, 2017

Model Theory as a Geography of Mathematics

Lou van den Dries (UIUC)

Abstract: This is a dry run for the first talk in the Tarski lectures I am giving the week after in Berkeley. This first talk is for a rather general audience of mathematicians, logicians, and philosophers. I like to think of model theory as a {\em geography of mathematics \}, especially of its ``tame'' side. Here {\em tame\/} roughly corresponds to {\em geometric\/} as opposed to {\em combinatorial-arithmetic}. In this connection I will discuss Tarski's work on the real field, and the notion of o-minimality that it suggested. A structure $M$ carries its own mathematical territory with it, via interpretability: its own posets, groups, fields,and so on. Understanding this ``world according to $M$'' can be rewarding. Stability-like properties of $M$ forbid certain combinatorial patterns, thus providing highly intrinsic and robust information about this world.

Wednesday, April 12, 2017

3:00 pm in 243 Altgeld Hall,Wednesday, April 12, 2017

First order logic and Sub-riemannian spheres.

Erik Walsberg (UIUC)

Abstract: I will discuss a connection between sub-riemannian geometry and first order model theory. Researchers in sub-riemannian geometry have essentially been working on the following: are sub-riemannian spheres definable in an o-minimal expansion of the ordered field of real numbers? (O-minimality is an important and popular topic in model theory developed in large part by UIUC's own Lou van den Dries). It also seems that some model theorists have been trying to construct the kind of structure that the geometers are looking for. As far as I can tell neither side was really aware of what the other was doing until now. I will try to explain some of this. No knowledge of logic or sub-riemannian geometry is necessary.

Friday, April 21, 2017

4:00 pm in 345 Altgeld Hall,Friday, April 21, 2017

The logical complexity of finitely generated commutative rings

Matthias Aschenbrenner (UCLA)

Abstract: Since the work of G\"odel we know that the theory of the ring $\mathbb Z$ of integers is very complicated. Using the coding techniques introduced by him, every finitely generated commutative ring can be interpreted in $\mathbb Z$ and therefore has a theory which is no more complicated than that of $\mathbb Z$. It has also been long known that conversely, every infinite finitely generated commutative ring interprets the integers, and hence its theory is at least as complex as that of $\mathbb Z$. However, this mutual interpretability does not fully describe the class of definable sets in such rings. The correct point of view is provided by the concept of bi-interpretability, an equivalence relation on the class of first-order structures which captures what it means for two structures to essentially have the same categories of definable sets and maps. We characterize algebraically those finitely generated rings which are bi-interpretable with $\mathbb Z$. (Joint work with Anatole Kh\'elif, Eudes Naziazeno, and Thomas Scanlon.)

Friday, April 28, 2017

4:00 pm in 345 Altgeld Hall,Friday, April 28, 2017

On definability and interpretability in model theory

Ward Henson (UIUC, UC Berkeley)

Abstract: This will be an expository talk. Topics covered/mentioned will include: extension by definition and expansion/extension by interpretation (the eq-construction), Beth's Theorem (characterizing definability), and Makkai's Theorem (characterizing the eq-expansion). The setting will be model theory of classical (discrete) structures and then of metric (real-valued) structures. In the metric setting, definability of sets/relations has some subtleties that do not arise in discrete structures.

Friday, May 12, 2017

4:00 pm in 345 Altgeld Hall,Friday, May 12, 2017

O-minimality and the Sub-Riemannian Sphere

Erik Walsberg

Abstract: I will discuss an interesting connection I recently found between Sub-riemannian geometry and o-minimality. In short: sub-riemannian geometers believe that the spheres in real analytic sub-riemannian metrics are definable in some o-minimal expansion of the real field. Furthermore, people in o-minimality are trying to construct the kind of o-minimal expansion that the sub-riemannian geometry are looking for, It seems that the o-minimality community was entirely unaware of this until now. No knowledge of sub-riemannian geometry will be assumed in this talk. Basic knowledge of model theory will be assumed.

Friday, September 1, 2017

4:00 pm in 345 Altgeld Hall,Friday, September 1, 2017

Organizational meeting

Friday, September 8, 2017

1:00 pm in 141 Altgeld Hall,Friday, September 8, 2017

Keisler Measures, ctd.

Travis Nell   [email]

Abstract: We continue reading chapter 7 of Simon's A Guide to NIP Theories.

4:00 pm in 345 Altgeld Hall,Friday, September 8, 2017

"Borel circle squaring" by Marks and Unger: Part 1

Anton Bernshteyn (Illinois Math)

Abstract: This is the first in a series of talks based on a recent paper of A.S. Marks and S.T. Unger. In 1925, Tarski asked if it is possible to decompose a disk in the plane into finitely many pieces and then rearrange them to form a square of the same area. Due to the apparent similarity between this problem and the Banach--Tarski paradox, it might appear that any such "circle squaring" must rely on the Axiom of Choice; and indeed, in 1990 Laczkovich answered Tarski's question in the affirmative using a non-constructive approach. However, Marks and Unger show in their paper that, somewhat surprisingly, it is possible to perform a "circle squaring" using only Borel pieces. In this talk I will go over the history of the problem and sketch some of the main ingredients that go into Marks and Unger's proof.

Friday, September 15, 2017

4:00 pm in 345 Altgeld Hall,Friday, September 15, 2017

"Borel circle squaring" by Marks and Unger: Part 2

Dakota Ihli (Illinois Math)

Abstract: This is the second part in a series of talks based on a recent paper of A.S. Marks and S.T. Unger. The main result of the paper gives sufficient conditions for two Borel subsets of $\mathbb{R}^k$ to be equidecomposable by translations with Borel pieces. In the first part, flows on graphs were introduced as the primary means to prove this result. In this part I will outline in more detail how these flows are constructed.

Friday, September 22, 2017

1:00 pm in 141 Altgeld Hall,Friday, September 22, 2017

Travis Nell

Abstract: We continue reading Simon's "A guide to NIP Theories", chapter 7.

4:00 pm in 345 Altgeld Hall,Friday, September 22, 2017

Constructing Analyzable Types in Differentially Closed Fields with Logarithmic Derivatives

Ruizhang Jin (U Waterloo Math)

Abstract: We generalize the well-known fact that the equation $\delta(\log \delta x) = 0$ is analyzable in but not internal to the constants. We use the logarithmic derivative as a building block to construct analyzable types with a unique analysis of minimal length (up to interalgebraicity). We also look for criteria for a given definable set such that its pre-image under the logarithmic derivative is analyzable in but not internal to the constants.

Friday, September 29, 2017

4:00 pm in 345 Altgeld Hall,Friday, September 29, 2017

"Borel circle squaring" by Marks and Unger: Part 3

Anush Tserunyan (Illinois Math)

Abstract: In the previous talk of this series, we built a real valued Borel flow by taking the average of the "relative" matchings over each connected component. In the current talk, I'll describe how to obtain an integer valued Borel flow out of a real valued one, which is the main difficulty of the whole proof.

Friday, October 6, 2017

4:00 pm in 345 Altgeld Hall,Friday, October 6, 2017

"Borel circle squaring" by Marks and Unger: Part 4 (last)

Anush Tserunyan (Illinois Math)

Abstract: In the previous talk of this series, we discussed how to obtain an integral Borel flow out of a real valued one. In this last talk of the series, we will put all the ingredients together and show how to obtain a Borel matching from an integral Borel $f$-flow for the specific $f := \mathbb{1}_A - \mathbb{1}_B$.

Friday, October 20, 2017

4:00 pm in 345 Altgeld Hall,Friday, October 20, 2017

Distal and non-distal ordered abelian groups

Allen Gehret (UCLA Math)

Abstract: I will discuss various things we know about distal and non-distal ordered abelian groups. This is joint work with Matthias Aschenbrenner and Artem Chernikov.