Department of

# Mathematics

Seminar Calendar
for events the day of Wednesday, November 6, 2019.

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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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Wednesday, November 6, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, November 6, 2019

#### A pointwise ergodic theorem for hyperfinite equivalence relations

###### Jenna Zomback (UIUC Math)

Abstract: An equivalence relation on a standard Borel space is called hyperfinite if it is a countable increasing union of Borel equivalence relations whose classes are finite. We will prove the natural pointwise ergodic theorem for probability measure preserving hyperfinite equivalence relations, as recorded by Miller and Tserunyan in [MTs17]. If time permits, Anush Tserunyan will continue with a prelude to the next talk on proving the existence of a uniformly ergodic hyperfinite subequivalence relation of a given pmp countable Borel equivalence relation.

4:00 pm in 245 Altgeld Hall,Wednesday, November 6, 2019

#### Framed cobordism and the $J$-homomorphism

###### Ningchuan Zhang

Abstract: One of the main goals in homotopy theory is to compute the stable homotopy groups of spheres. Geometrically, these groups classify cobordism classes of framed submanifolds of spheres. This is a generalization of degrees of maps between oriented manifolds. From this perspective, I will explain in this talk how to compute the first stable homotopy group of spheres. This computation leads to the $J$-homomorphism, whose image is the most well-understood part in the stable homotopy groups of spheres. I will define the $J$-homomorphism using framed cobordism. If time allows, I will also give some computational results of its image, which are related to the Bernoulli numbers.