Department of

Mathematics


Seminar Calendar
for Descriptive Set Theory Seminar events the year of Wednesday, September 18, 2019.

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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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Wednesday, January 30, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, January 30, 2019

The cube problem and Schroeder-Bernstein problem for linear orders

Garret Ervin (Carnegie Mellon)

Abstract: We sketch proofs of solutions to two old problems posed by Sierpinski concerning products of linear orders. The first problem asks whether there exists a linear order $X$ that is isomorphic to its lexicographic cube but not to its square; the second, whether there are two non-isomorphic orders $Y$ and $Z$ that divide each other on both the left and right side. For other classes of structures, the corresponding questions are usually either both positive or both negative, but for linear orders the answers diverge: there is no such $X$, but there are such $Y$ and $Z$.

Wednesday, February 6, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, February 6, 2019

"Generic representations of abelian groups and extreme amenability" by J. Melleray and T. Tsankov (Part 1)

Dakota Ihli (UIUC)

Abstract: In this series of talks, we discuss the paper in the title [arXiv link].

Friday, February 8, 2019

4:00 pm in 345 Altgeld Hall,Friday, February 8, 2019

"The complexity of topological group isomorphism" by A. Kechris, A. Nies, and K. Tent (Part 1)

Anush Tserunyan (UIUC)

Abstract: This will be the introductory talk of the series on the paper in the title [arXiv link], which deals with the classification of some natural classes of non-Archimedean groups (= closed subgroups of S) up to topological group isomorphism. It gives a general criterion for a class of non-Archimedean groups to show that the topological group isomorphism on it is Borel-classifiable by countable structures. This criterion is satisfied by the classes of profinite groups, locally compact non-Archimedean groups, and oligomorphic groups.

Wednesday, February 13, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, February 13, 2019

Introduction to well and even better quasi-orders

Raphaël Carroy (Gödel Research Center for Math. Logic, Univ. of Vienna)

Abstract: Well-quasi-orders, or wqos, generalize well-orders in the context of partial orders. They appear naturally in various domains of mathematics, and have been frequently rediscovered. I'll briefly explain why, and what we can do with them. I'll then talk about their limitations and why it's hard to prove that non-trivial quasi-orders are wqo. I will also show how trying to fix these problems leads to the definition of a smaller class of quasi-orders: better-quasi-orders, or bqos. If time allows, I'll get a bit into bqo theory.

Friday, February 15, 2019

3:00 pm in 341 Altgeld Hall ,Friday, February 15, 2019

Note the time and room change!

"The complexity of topological group isomorphism" by A. Kechris, A. Nies, and K. Tent (Part 2)

Jenna Zomback (UIUC)

Abstract: This will be the second talk of the series on the paper in the title [arXiv link], which deals with the classification of some natural classes of non-Archimedean groups (= closed subgroups of S) up to topological group isomorphism. It gives a general criterion for a class of non-Archimedean groups to show that the topological group isomorphism on it is Borel-classifiable by countable structures. This criterion is satisfied by the classes of profinite groups, locally compact non-Archimedean groups, and oligomorphic groups. In this talk, we will fill in some proofs left out last time and prove this general criterion.

Monday, February 18, 2019

3:00 pm in 341 Altgeld Hall,Monday, February 18, 2019

Dense orbits in the space of subequivalence relations

Forte Shinko (Caltech)

Abstract: Given a measure-preserving countable Borel equivalence relation $E$, there is a Polish space $S(E)$ of subequivalence relations, which admits a natural action of the full group $[E]$. One can ask the following natural question: does $S(E)$ have a dense orbit? We will present results due to François Le Maître which show that the answer is yes when $E$ is the hyperfinite ergodic equivalence relation, and that the answer is no when $E$ is induced by a measure-preserving action of a property (T) group.

Friday, February 22, 2019

4:00 pm in 345 Altgeld Hall ,Friday, February 22, 2019

Cancelled

Wednesday, February 27, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, February 27, 2019

"Generic representations of abelian groups and extreme amenability" by J. Melleray and T. Tsankov (Part 2)

Dakota Ihli (UIUC)

Abstract: In this series of talks, we discuss the paper in the title [arXiv link].

Friday, March 1, 2019

4:00 pm in 345 Altgeld Hall ,Friday, March 1, 2019

"The complexity of topological group isomorphism" by A. Kechris, A. Nies, and K. Tent (Part 3)

Mary Angelica Gramcko-Tursi (UIUC)

Abstract: This will be the third talk of the series on the paper in the title [arXiv link], which deals with the classification of some natural classes of non-Archimedean groups (= closed subgroups of S) up to topological group isomorphism. It gives a general criterion for a class of non-Archimedean groups to show that the topological group isomorphism on it is Borel-classifiable by countable structures. This criterion is satisfied by the classes of profinite groups, locally compact non-Archimedean groups, and oligomorphic groups. In this talk, we will show that one or two of the aforementioned classes satisfy this criterion.

Wednesday, March 6, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, March 6, 2019

Abstract systems of congruences

Spencer Unger (Tel Aviv University)

Abstract: Abstract systems of congruences provide a different perspective for viewing geometrical paradoxes from the early 20th century. Consider partitioning a space into $n$ pieces $A_1, A_2, \dots, A_n$. An abstract system of congruences is a collection of statements (called congruences), like $A_2 \cup A_6$ is isometric to $A_{17}$. Such a system is satisfied by a particular partition if each congruence is satisfied. We survey some recent results and some open problems.

Friday, March 8, 2019

4:00 pm in 345 Altgeld Hall ,Friday, March 8, 2019

Organizational meeting

Wednesday, March 13, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, March 13, 2019

Hyperfiniteness and descriptive combinatorics: ideas and proofs

Clinton Conley (Carnegie Mellon)

Abstract: We discuss some ideas and proofs behind the results surveyed in the first part of this talk on Tuesday.

Friday, March 15, 2019

4:00 pm in 345 Altgeld Hall ,Friday, March 15, 2019

The theory of addition with predicates for the powers of 2 and 3

Christian Schulz (UIUC Math)

Abstract: This talk concerns the intricate boundary between decidable and undecidable of expansions of Presburger artithmetic, i.e., the structure $(\mathbb{N}, +)$. For a natural number $p \ge 2$, let $p^{\mathbb{N}}$ denote the set of powers of $p$, and let $V_p$ be a predicate that allows us to access the full base-$p$ expansion of a natural number. It is known that the expansion $(\mathbb{N}, +, V_p)$ of Presburger arithmetic retains decidability, but $(\mathbb{N}, +, V_p, q^{\mathbb{N}})$, for $q$ multiplicatively independent from $p$, has an undecidable theory. In this talk, I present a proof that the reduct $(\mathbb{N}, +, p^{\mathbb{N}}, q^{\mathbb{N}})$ also has an undecidable theory, specifically in the case $p = 2$, $q = 3$. I conclude with a note on how the proof extends to other structures, as well as some discussion of directions for further research.

Wednesday, March 27, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, March 27, 2019

"Generic representations of abelian groups and extreme amenability" by J. Melleray and T. Tsankov (Part 3)

Dakota Ihli (UIUC)

Abstract: In this series of talks, we discuss the paper in the title [arXiv link]. In this final talk, we prove that the set of probability measure preserving automorphisms that topologically generate a copy of the group $L_0(\mathbb{T})$ is dense in $\mathrm{Aut}(\mu)$.

Wednesday, April 3, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, April 3, 2019

Hurewicz' theorem (1930) on uncountable sets — a variant approach

Robert Kaufman (UIUC Math)

Abstract: In the theorem below, $C(K)$ is the space of continuous functions on the Cantor space $K$ and $C^*(K) \subseteq C(K)$ is the set of functions with uncountable range.

Theorem. For any analytic set $A$ in a metric space $M$, there is a continuous map $\varphi$ of $M$ into $C(K)$ such that $\varphi^{-1}(C^*(K)) = A$.

The argument uses only classical analysis; an important role is played by the notion of ultrametric space. A few minutes will be devoted to the representation of analytic sets as "projective" sets.

Friday, April 5, 2019

4:00 pm in 345 Altgeld Hall ,Friday, April 5, 2019

Generic derivations on o-minimal structures

Elliot Kaplan (UIUC Math)

Abstract: We study derivations $\delta$ on o-minimal fields $K$. We introduce the notion of a $T$-derivation, which is a derivation which cooperates with the 0-definable $\mathcal{C}^1$-functions on $K$. For example, if $K$ is an elementarily equivalent to the real exponential field, we require that $\delta \exp(a) = \exp(a)\delta a$ for all $a \in K$. Let $T$ be the theory of $K$ in an appropriate language $L$ and let $T^\delta$ be the $L\cup \{\delta\}$ theory stating that $\delta$ is a $T$-derivation. We show that if $T$ has quantifier elimination, then $T^\delta$ has a model completion $T^\delta_G$. The derivation in models $K$ of $T^\delta_G$ behaves "generically," it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $K$. If $T$ is the theory of real closed ordered fields, then $T^\delta_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^\delta_G$ has $T$ as its open core and that $T^\delta_G$ is distal. This is joint work with Antongiulio Fornasiero.

Wednesday, April 10, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, April 10, 2019

Coloring Borel graphs equitably

Anton Bernshteyn (Carnegie Mellon)

Abstract: In this talk I will describe some of the main ideals and tools behind the proofs of the results surveyed in my talk in the Combinatorics and Graph Theory Seminar yesterday (based on joint work with Clinton Conley).

Wednesday, April 17, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, April 17, 2019

Introduction to quasi-Polish spaces

Ruiyuan (Ronnie) Chen (UIUC)

Abstract: We give an introduction to de Brecht's quasi-Polish spaces, a possibly non-Hausdorff generalization of Polish spaces sharing most of their descriptive set-theoretic properties while enjoying some additional and highly useful closure properties.

Friday, April 19, 2019

4:00 pm in 345 Altgeld Hall ,Friday, April 19, 2019

Cancelled

(UIUC Math)

Wednesday, May 1, 2019

3:00 pm in 341 Altgeld Hall,Wednesday, May 1, 2019

A combinatorial proof of the pointwise ergodic theorem for actions of amenable groups along Tempelman Følner sequences

Jenna Zomback (UIUC)

Abstract: A pointwise ergodic theorem for the action of a countable group $\Gamma$ on a probability space equates the global ergodicity (atomicity) of the action to its pointwise combinatorics. Our main result is a short, combinatorial proof of the pointwise ergodic theorem for actions of amenable groups along Tempelman Følner sequences, which is a slightly less general version of Lindenstrauss's celebrated theorem. Without assuming any prior knowledge, we will work up to the general idea of the proof, which stems from Tserunyan's proof of the pointwise ergodic theorem for $\mathbb{Z}$ actions. This is joint work with Jon Boretsky.

Tuesday, May 28, 2019

2:00 pm in 243 Altgeld Hall,Tuesday, May 28, 2019

Stallings' foldings and cost of treeable equivalence relations (Part 1)

Anush Tserunyan (UIUC)

Abstract: Introduced by Levitt and extensively developed by Gaboriau, cost is a very useful real-valued invariant for probability measure preserving countable equivalence relations that measures the infimum amount of edges needed to connect a.e. equivalence class in a uniformly Borel fashion. We discuss Gaboriau's original proof of his Fundamental Theorem of Cost, which states that the cost of a treeable equivalence relation is achieved by any Borel treeing of it.

Wednesday, May 29, 2019

2:00 pm in 243 Altgeld Hall,Wednesday, May 29, 2019

Stallings' foldings and cost of treeable equivalence relations (Part 2)

Anush Tserunyan (UIUC)

Abstract: Introduced by Levitt and extensively developed by Gaboriau, cost is a very useful real-valued invariant for probability measure preserving countable equivalence relations that measures the infimum amount of edges needed to connect a.e. equivalence class in a uniformly Borel fashion. We discuss Gaboriau's original proof of his Fundamental Theorem of Cost, which states that the cost of a treeable equivalence relation is achieved by any Borel treeing of it.

Wednesday, August 28, 2019

3:30 pm in 241 Altgeld Hall,Wednesday, August 28, 2019

Organizational meeting

Friday, September 6, 2019

4:00 pm in 345 Altgeld Hall,Friday, September 6, 2019

"On the nonexistence of Følner sets" by Isaac Goldbring

Elliot Kaplan (UIUC Math)

Abstract: This will be the first (and possibly only) talk on the preprint "On the nonexistence of Følner sets" by Isaac Goldbring (https://arxiv.org/abs/1901.02445). I will introduce all of the necessary model-theoretic and group-theoretic background. Time permitting, I may get to the proof of the main result.

Wednesday, September 11, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, September 11, 2019

On spatial actions and whirly actions

Dakota Ihli (UIUC Math)

Abstract: Given a Borel measure-preserving action of a Polish group $G$ on $\left[ 0,1 \right]$, ignoring null sets gives a corresponding action on the measure algebra. If $G$ is locally compact, this can always be reversed: a measure-preserving action on the measure algebra of $\left[ 0,1 \right]$ can always be induced by a (pointwise) action on $\left[ 0,1 \right]$. This fails in the non-locally-compact case. In this talk, I will discuss a dynamical condition on the action ("whirliness") which characterizes when an action on the measure algebra admits a pointwise realization. This talk is based on 'Spatial and non-spatial actions of Polish groups' by E Glasner and B Weiss.

Wednesday, September 18, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, September 18, 2019

Polish groups whose measure preserving actions are whirly

Pavlos Motakis (UIUC Math)

Abstract: Let $\mathrm{MALG}(X)$ denote the measure algebra of a standard probability space $(X,\mu)$. A measure preserving action of a Polish group $G$ on $\mathrm{MALG}(X)$ is called whirly if for any $A, B$ in $\mathrm{MALG}(X)$ with positive measure and for any open neighborhood $U$ of the identity of $G$ there exists $g\in U$ so that $(gA)\cap B$ has positive measure. We follow a paper of Glasner–Tsirelson–Weiss to show that if $G$ is certain type of Polish group, namely a Lévy group, then any non-trivial Borel action on $\mathrm{MALG}(X)$ is whirly. We also show that the Polish group $L_0(\mathbb{T})$ of all measurable functions $[0,1] \to \mathbb{T}$ is Lévy using a suitable concentration inequality.
This is a follow up to the lecture of D. Ihli on 09/11/2019.

Friday, September 20, 2019

4:00 pm in 345 Altgeld Hall,Friday, September 20, 2019

A Logician's Introduction to the Problem of P vs. NP

Alexi Block Gorman (UIUC Math)

Abstract: Central to much of computer science, and some areas of mathematics, are questions about various problems' computability and complexity (whether the problem can be solved "algorithmically," and how "hard" it is to do so). In this talk, I will first give an overview of the complexity hierarchy for machines (from finite automata to Turing machines) and the mathematical properties of the space of languages that we associate with them. Next, I will discuss the relationship of deterministic and non-deterministic machines, which will allow us to segue from questions of computability to that of complexity. Finally, I will give a precise formulation of the problem of P vs. NP, and try to illustrate why the problem remains rather elusive. This talk does not require any background in logic or computer science, and should be accessible to all graduate students.

Wednesday, September 25, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, September 25, 2019

Spatial actions of Lévy groups
(90 minute talk)

Pavlos Motakis (UIUC Math)

Abstract: Following up last week's seminar we prove a theorem of Glasner–Tsirelson–Weiss according to which any spacial action of a Lévy group is trivial.

Friday, September 27, 2019

4:00 pm in 345 Altgeld Hall,Friday, September 27, 2019

O-minimal complex analysis according to Peterzil–Starchenko (Part 1)

Lou van den Dries (UIUC)

Abstract: This is the first of two survey talks on the subject of the title. Neer (and others?) will follow up with a more detailed treatment in later talks. O-minimal complex analysis is one way that ideas from o-minimality have been used in recent work in arithmetic algebraic geometry (Pila, Zannier, Tsimerman, Klingler,…), the other one being the Pila–Wilkie theorem. The two topics relate because important objects like the family of Weierstrass p-functions turn out to be "o-minimal".

Wednesday, October 2, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, October 2, 2019

Generic measure-preserving transformations are whirly
(50 minute talk)

Dakota Ihli (UIUC Math)

Abstract: We show that the action of a generic measure-preserving transformation on a standard probability space is whirly. This will be the final part in our series of seminar talks discussing the paper "Spatial and non-spatial actions of Polish groups" by Glasner and Weiss.

Friday, October 4, 2019

4:00 pm in 345 Altgeld Hall,Friday, October 4, 2019

O-minimal complex analysis according to Peterzil–Starchenko (Part 2)

Lou van den Dries (UIUC)

Abstract: This is the first of two survey talks on the subject of the title. Neer (and others?) will follow up with a more detailed treatment in later talks. O-minimal complex analysis is one way that ideas from o-minimality have been used in recent work in arithmetic algebraic geometry (Pila, Zannier, Tsimerman, Klingler,…), the other one being the Pila–Wilkie theorem. The two topics relate because important objects like the family of Weierstrass p-functions turn out to be "o-minimal".

Friday, October 11, 2019

4:00 pm in 345 Altgeld Hall,Friday, October 11, 2019

"Complex-like" analysis in o-minimal structures (Part 3)

Neer Bhardwaj (UIUC)

Abstract: Analogues of many of the basic results in complex analysis can be established over an arbitrary algebraically closed field $K$ of characteristic zero, in the context of an o-minimal expansion of a real closed field $R$, with $K=R[i]$. I will show in particular how one can define winding numbers, and how differentiability begets infinite differentiability in this setting. This expository talk follows the survey given by Lou in the last two weeks and will be based mostly on: "Expansions of algebraically closed fields in o-minimal structures" by Starchenko–Peterzil (https://link.springer.com/article/10.1007/PL00001405)

Wednesday, October 23, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, October 23, 2019

Stormy actions of Bernoulli shifts

Joseph Zielinski

Abstract: An equivalence relation is essentially countable when it is Borel reducible to the orbit equivalence relation of some action of a countable group. In a 2005 paper, G. Hjorth introduced the notion of a stormy Polish G-space and showed that this condition is the fundamental obstruction to essential countability for orbit equivalence relations. Specifically, he proved that if a Polish group, G, acts continuously on a Polish space, X, with a Borel orbit equivalence relation then this relation fails to be essentially countable if and only there is a continuous G-equivariant embedding of a stormy action into X. In the first part of this talk we recall the basic concepts and results in Hjorth's work on stormy actions. In the second part, we apply these to the constructions considered in recent joint work with A.S. Kechris, M. Malicki, and A. Panagiotopoulos for actions of non-Archimedian groups and separable Banach spaces.