Department of

# Mathematics

Seminar Calendar
for Descriptive Set Theory Seminar events the year of Tuesday, March 26, 2019.

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events for the
events containing

More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    February 2019            March 2019             April 2019
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2                   1  2       1  2  3  4  5  6
3  4  5  6  7  8  9    3  4  5  6  7  8  9    7  8  9 10 11 12 13
10 11 12 13 14 15 16   10 11 12 13 14 15 16   14 15 16 17 18 19 20
17 18 19 20 21 22 23   17 18 19 20 21 22 23   21 22 23 24 25 26 27
24 25 26 27 28         24 25 26 27 28 29 30   28 29 30
31


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.