Department of

# Mathematics

Seminar Calendar
for Model Theory and Descriptive Set Theory Seminar events the year of Tuesday, March 13, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, January 26, 2018

4:00 pm in 345 Altgeld Hall,Friday, January 26, 2018

#### First order expansions of the ordered real additive group — Part II

###### Philipp Hieronymi (UIUC Math)

Abstract: This is a (hopefully self-contained) continuation of Erik's talk from Tuesday. I will discuss in more detail expansions of the ordered real additive group that are of type B (ie expansions of $(\mathbb{R},<,+)$ that define an order $(D,\prec)$ of order-type $\omega$ whose underlying set $D$ is somewhere dense in $\mathbb R$). In many (all?) type B structures 0-definability is equivalent to recognizablility by a Buechi automaton. Therefore results about the definable sets in type B structures imply results about sets recognizable by such automata. In this talk I will focus on our current research on continuous definable functions in type B structures, and I will discuss how it connects to results in computer science by Chaudhuri, Sankaranarayanan and Vardi.

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

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, 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.

Friday, May 4, 2018

4:00 pm in 345 Altgeld Hall,Friday, May 4, 2018

#### Analytic hypergraphs and proper forcing

###### Jindrich Zapletal (University of Florida Math)

Abstract: Given a countable family $G$ of analytic hypergraphs on a Polish space $X$, one can consider the quotient poset $P(G)$ of Borel sets modulo the ideal generated by Borel sets which are anticliques in at least one hypergraph in $G$. This is a broad family of forcings, many of them proper. The family is closed under such operations as countable support iteration of (in some cases) product. Most traditional fusion arguments disappear in this class of forcing. They are replaced by simple combinatorial considerations about the edges of the generating hypergraphs.