Department of

Mathematics


Seminar Calendar
for Descriptive Set Theory Seminar events the year of Friday, November 9, 2018.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
     October 2018          November 2018          December 2018    
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     1  2  3  4  5  6                1  2  3                      1
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Friday, February 2, 2018

4:00 pm in 345 Altgeld Hall,Friday, February 2, 2018

On "A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets" by Achille and Berarducci

Elliot Kaplan (UIUC Math)

Abstract: I will discuss the paper "A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets" by Alessandro Achille and Alessandro Berarducci [arXiv]. In this paper, they compare the definable homotopy groups of a space $X$ definable in an o-minimal structure (denoted $\pi_n(X)^\mathrm{def}$) with the (usual) homotopy groups of the quotient $X/E$, where $E$ is a certain kind of type-definable equivalence relation on $X$. Under certain assumptions, these groups are actually isomorphic. These types of quotients were first studied by Pillay in the case that $X$ is actually a definable group. This is the first talk that I will give on this paper.

Friday, February 9, 2018

4:00 pm in 345 Altgeld Hall,Friday, February 9, 2018

More on ``A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets'' by Achille and Berarducci

Elliot Kaplan (UIUC Math)

Abstract: I will continue to discuss the paper ``A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets'' by Alessandro Achille and Alessandro Berarducci [arXiv]. I will sketch the proofs of the two main theorems, in which they show that under certain conditions, the definable homotopy groups of a space $X$ definable in an o-minimal structure are isomorphic to the (usual) homotopy groups of the quotient of $X$ by a certain kind of type-definable equivalence relation $E$. This is the second talk on this paper.

Friday, February 16, 2018

4:00 pm in 345 Altgeld Hall,Friday, February 16, 2018

Canceled

Tuesday, February 20, 2018

4:00 pm in 243 Altgeld Hall,Tuesday, February 20, 2018

A pointwise ergodic theorem for quasi-pmp graphs

Anush Tserunyan (UIUC Math)

Abstract: We prove a pointwise ergodic theorem for quasi-pmp locally countable graphs, which states that the global condition of ergodicity amounts to locally approximating the means of $L^1$-functions via increasing subgraphs with finite connected components. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is constructive and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant and a simple method of exploiting nonamenability. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.

Friday, February 23, 2018

4:00 pm in 345 Altgeld Hall,Friday, February 23, 2018

"On dp-minimal ordered structures" by P. Simon (part 1)

Travis Nell (Illinois Math)

Abstract: This is an introductory talk to "On dp-minimal ordered structures" by P. Simon [arXiv].

Friday, March 2, 2018

4:00 pm in 1 Illini Hall,Friday, March 2, 2018

"On dp-minimal ordered structures" by P. Simon (part 2)

Alexi Block Gorman (Illinois Math)

Abstract: We continue reading through "On dp-minimal ordered structures" by P. Simon [arXiv].

Friday, March 9, 2018

4:00 pm in 345 Altgeld Hall,Friday, March 9, 2018

"On dp-minimal ordered structures" by P. Simon (part 3)

Alexi Block Gorman (Illinois Math)

Abstract: We continue reading through "On dp-minimal ordered structures" by P. Simon [arXiv].

Friday, March 16, 2018

4:00 pm in 345 Altgeld Hall,Friday, March 16, 2018

Cancelled

Friday, March 30, 2018

4:00 pm in 345 Altgeld Hall,Friday, March 30, 2018

Dp-minimal expansions of $(\mathbb{Q},<,+)$

Erik Walsberg (Illinois Math)

Abstract: I will use P. Simon's result to relate dp-minimal expansions of $(\mathbb{Q},<,+)$ to o-minimal expansions of $(\mathbb{R},<,+)$, I will also describe a nice application of the Pettis lemma to o-minimality.

Friday, April 6, 2018

4:00 pm in 345 Altgeld Hall,Friday, April 6, 2018

Dp-minimal expansions of $(\mathbb{Q},<,+)$ (continued)

Erik Walsberg (Illinois Math)

Abstract: I will use P. Simon's result to relate dp-minimal expansions of $(\mathbb{Q},<,+)$ to o-minimal expansions of $(\mathbb{R},<,+)$, I will also describe a nice application of the Pettis lemma to o-minimality.

Tuesday, April 10, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, April 10, 2018

"On dp-minimal ordered structures" by P. Simon (part 4)

Elliot Kaplan (Illinois Math)

Abstract: We continue through "On dp-minimal ordered structures" by P. Simon [arXiv].

Friday, April 20, 2018

4:00 pm in 345 Altgeld Hall,Friday, April 20, 2018

Polishable Borel equivalence relations

Sławomir Solecki (Cornell)

Abstract: We introduce the notion of Polishable equivalence relations. This class of equivalence relations contains all orbit equivalence relations induced by Polish group actions and is contained in the class of idealistic equivalence relations of Kechris and Louveau. We show that each orbit equivalence relation induced by a Polish group action admits a canonical transfinite sequence of Polishable equivalence relations approximating it. The proof involves establishing a lemma, which may be of independent interest, on stabilization of increasing $\omega_1$-sequences of completely metrizable topologies.

Thursday, September 6, 2018

12:00 pm in 241 Altgeld Hall,Thursday, September 6, 2018

On "Closed subgroups generated by generic measure automorphisms" by S. Solecki, Part 1

Dakota Ihli (UIUC Math)

Abstract: We will read through this paper.

Wednesday, September 12, 2018

3:00 pm in 241 Altgeld Hall,Wednesday, September 12, 2018

On "Closed subgroups generated by generic measure automorphisms" by S. Solecki, Part 2

Dakota Ihli (UIUC Math)

Abstract: We will sketch the proof of the main result in this paper.

Wednesday, September 19, 2018

3:00 pm in 241 Altgeld Hall,Wednesday, September 19, 2018

On "Closed subgroups generated by generic measure automorphisms" by S. Solecki, Part 3

Dakota Ihli (UIUC Math)

Abstract: We will sketch the proof of the main result in this paper.

Wednesday, September 26, 2018

3:00 pm in 241 Altgeld Hall,Wednesday, September 26, 2018

A combinatorial proof of the classical pointwise ergodic theorem

Jenna Zomback (UIUC Math)

Abstract: We will discuss A. Tserunyan's proof of the classical pointwise ergodic theorem for pmp actions of $\mathbb{Z}$ given in [arXiv link], which reduces proving this analytic theorem to solving a finitary tiling problem on the Cayley graph of $\mathbb{Z}$. We hope to use this method in the future to give streamlined proofs of known instances of the pointwise ergodic theorem for other groups, as well as obtain new ones.

Wednesday, October 3, 2018

3:00 pm in 241 Altgeld Hall,Wednesday, October 3, 2018

Measurable matchings and Følner tilings: Part 1

Anush Tserunyan (UIUC Math)

Abstract: We read through "Følner tilings of actions of amenable groups" by C. Conley et al [arXiv]. This first talk will introduce to amenable groups and actions, and analyze Hall's marriage theorem for locally finite bipartite graphs (which involves an ultrafilter or the Axiom of Choice). More importantly, we will discuss an example of a closed graphs on $[0,1]$, which satisfies Hall's condition, hence admits a perfect matching, yet not a measurable one.

Wednesday, October 10, 2018

3:00 pm in 241 Altgeld Hall,Wednesday, October 10, 2018

Measurable matchings and Følner tilings: Part 2

Anush Tserunyan (UIUC Math)

Abstract: We read through "Følner tilings of actions of amenable groups" by C. Conley et al [arXiv]. This second talk will the measurable version of Hall's marriage theorem for expansive bipartite probability measure preserving Borel graphs, due to R. Lyons and F. Nazarov.

Friday, October 12, 2018

9:00 am in 350 Altgeld Hall,Friday, October 12, 2018

Measurable matchings and Følner tilings: Part 3

Anush Tserunyan (UIUC Math)

Abstract: We read through "Følner tilings of actions of amenable groups" by C. Conley et al [arXiv]. In this third talk, I will introduce the notion of Banach density and prove the main theorem about the existence of Følner tilings a.e. using the Banach density version of the Rokhlin lemma (to be proven) and the Lyons–Nazarov matching theorem (proven last time).

4:00 pm in 345 Altgeld Hall,Friday, October 12, 2018

On "A model-theoretic approach to rigidity of strongly ergodic, distal actions" by T. Ibarlucía and T. Tsankov: Part 1

Ruiyuan (Ronnie) Chen (Illinois Math)

Abstract: We begin reading through "A model-theoretic approach to rigidity of strongly ergodic, distal actions" by T. Ibarlucía and T. Tsankov [arXiv].

Monday, October 22, 2018

3:00 pm in 241 Altgeld Hall,Monday, October 22, 2018

Measurable matchings and Følner tilings: Part 4

Ruiyuan (Ronnie) Chen (UIUC Math)

Abstract: We read through "Følner tilings of actions of amenable groups" by C. Conley et al [arXiv]. In this fourth and hopefully last talk, we will sketch the proof of the Banach density version of the Rokhlin lemma.

Wednesday, October 31, 2018

3:00 pm in 241 Altgeld Hall,Wednesday, October 31, 2018

$\ell^2$ Betti numbers of countable Borel equivalence relations: Part 1

Ruiyuan (Ronnie) Chen (UIUC Math)

Abstract: We will discuss Gaboriau's 2002 paper "Invariants $\ell^2$ de relations d'équivalence et de groupes" from a descriptive set-theoretic point of view. This first talk will consist of a survey of the main results, as well as a review of some relevant concepts from (ordinary and $\ell^2$) homology theory.

Friday, November 2, 2018

4:00 pm in 345 Altgeld Hall,Friday, November 2, 2018

A model-theoretic approach to rigidity of strongly ergodic distal actions

Ronnie Chen (UIUC)

Abstract: We will continue reading through "A model-theoretic approach to rigidity of strongly ergodic, distal actions" by T. Ibarlucía and T. Tsankov

Monday, November 5, 2018

2:00 pm in 343 Altgeld Hall,Monday, November 5, 2018

$\ell^2$ Betti numbers of countable Borel equivalence relations: Part 2

Ruiyuan (Ronnie) Chen (UIUC Math)

Abstract: We will discuss Gaboriau's 2002 paper "Invariants $\ell^2$ de relations d'équivalence et de groupes" from a descriptive set-theoretic point of view. In this second talk, we will discuss Borel fields of sets and structures (in particular, simplicial complexes and Hilbert spaces) over a countable Borel equivalence relation, and how to measure their "Borel cardinality" using Tarski's theory of cardinal algebras.

Friday, November 9, 2018

4:00 pm in 345 Altgeld Hall,Friday, November 9, 2018

A model-theoretic approach to rigidity of strongly ergodic distal actions

Dakota Ihli (UIUC)

Abstract: We will continue reading through "A model-theoretic approach to rigidity of strongly ergodic, distal actions" by T. Ibarlucía and T. Tsankov