Department of

# Mathematics

Seminar Calendar
for events the day of Monday, September 25, 2017.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, September 25, 2017

3:00 pm in 141 Altgeld Hall,Monday, September 25, 2017

#### An Invitation to Motivic Homotopy Theory

###### Daniel Carmody   [email] (UIUC)

Abstract: In this talk I’ll introduce some of the basic constructions in motivic homotopy theory while trying to give motivations for some of the more complex definitions. This will be largely based on Dan Dugger’s $Universal$ $Homotopy$ $Theories$ paper.

4:00 pm in 245 Altgeld Hall,Monday, September 25, 2017

#### Rigidity in orbit equivalence via cost

###### Anush Tserunyan   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: The talk is tailored around the following basic question: Let $\mathbb{F}_n$↷$[0,1]$ and $\mathbb{F}_m$↷$[0,1]$ be free actions of the free groups on $n$ and $m$ generators and assume that these actions preserve the Lebesgue measure and are ergodic (a bunch of words you're not supposed to know—ignore). If these actions produce the same orbits (i.e. their orbit equivalence relations are equal), must it be that $n = m$? This is an instance of the more general question: how much of the group is "remembered" by the orbit equivalence relations of its free actions? The phenomenon of a weaker notion "remembering" a stronger one is referred to as rigidity. We will describe the answer to the initial question (due to D. Gaboriau) by introducing an invariant called cost, which is tied to measurable graphs and combinatorics, and even ideas from homology theory. I will only assume familiarity with the words graph and measure.