Department of

# Mathematics

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

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    December 2016           January 2017          February 2017
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3    1  2  3  4  5  6  7             1  2  3  4
4  5  6  7  8  9 10    8  9 10 11 12 13 14    5  6  7  8  9 10 11
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25 26 27 28 29 30 31   29 30 31               26 27 28



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

Thursday, April 6, 2017

1:00 pm in 7 Illini Hall,Thursday, April 6, 2017

#### Baire measurable colorings of group actions

###### Anton Bernshteyn (UIUC Math)

Abstract: Suppose that a countable group $\Gamma$ acts continuously on a Polish space $X$ and denote this action by $\alpha$. Does there exist a Baire measurable coloring $f \colon X \to \mathbb{N}$ satisfying certain local constraints? Or, better to say, can we characterize the coloring problems which admit Baire measurable solutions over $\alpha$? We will show that, on the one hand, there is no such Borel characterization—the problem is complete analytic. On the other hand, when $\alpha$ is the shift action, we prove that, roughly speaking, a Baire measurable coloring exists if and only if it can be found by a greedy algorithm.

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.

Friday, April 14, 2017

4:00 pm in 345 Altgeld Hall,Friday, April 14, 2017

#### Baire measurable colorings of group actions: Part Ⅱ

###### Anton Bernshteyn (UIUC Math)

Abstract: Suppose that a countable group $\Gamma$ acts continuously on a Polish space $X$ and denote this action by $\alpha$. Does there exist a Baire measurable coloring $f \colon X \to \mathbb{N}$ satisfying certain local constraints? Or, better to say, can we characterize the coloring problems which admit Baire measurable solutions over $\alpha$? We will show that, on the one hand, there is no such Borel characterization—the problem is complete analytic. On the other hand, when $\alpha$ is the shift action, we prove that, roughly speaking, a Baire measurable coloring exists if and only if it can be found by a greedy algorithm.

Friday, April 21, 2017

2:00 pm in 243 Altgeld Hall,Friday, April 21, 2017

#### Baire measurable colorings of group actions: Part Ⅲ

###### Anton Bernshteyn (UIUC Math)

Abstract: Suppose that a countable group $\Gamma$ acts continuously on a Polish space $X$ and denote this action by $\alpha$. Does there exist a Baire measurable coloring $f \colon X \to \mathbb{N}$ satisfying certain local constraints? Or, better to say, can we characterize the coloring problems which admit Baire measurable solutions over $\alpha$? We will show that, on the one hand, there is no such Borel characterization—the problem is complete analytic. On the other hand, when $\alpha$ is the shift action, we prove that, roughly speaking, a Baire measurable coloring exists if and only if it can be found by a greedy algorithm.

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.

Thursday, May 11, 2017

3:00 pm in 243 Altgeld,Thursday, May 11, 2017

#### The Complexity of Classifying Unitaries

###### Aristotelis Panagiotopoulos   [email] (UIUC Math)

Abstract: Classification problems occur in all areas of mathematics. Descriptive set theory provides methods for measuring the complexity of such problems. For example, using a technique developed by Hjorth, Kechris and Sofronidis proved that the problem of classifying all unitary operators of an infinite dimensional Hilbert space up to unitary equivalence is strictly more difficult than classifying graph structures on domain N up to isomorphism. In this talk I will review the basics from descriptive set theory and explain why the problem of classifying unitaries is so hard. Part of my talk will be based on recent joint work with Martino Lupini.

Friday, September 1, 2017

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

#### Organizational meeting

Friday, September 8, 2017

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

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 $f := \mathbb{1}_A - \mathbb{1}_B$.

Tuesday, October 10, 2017

4:00 pm in 314 Altgeld Hall,Tuesday, October 10, 2017

#### Lecture I. Set theory and trigonometric series

###### Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017. A reception will follow this first lecture from 5-6 pm in 239 Altgeld Hall. Alexander Kechris will present: "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.

Wednesday, October 11, 2017

4:00 pm in 245 Altgeld Hall,Wednesday, October 11, 2017

#### Lecture II. The complexity of classification problems in ergodic theory

###### Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017.Alexander Kechris will present "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.

Thursday, October 12, 2017

4:00 pm in 245 Altgeld Hall,Thursday, October 12, 2017

#### Lecture III: Descriptive graph combinatorics

###### Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017. Alexander Kechris will present "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.

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.