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for events the day of Tuesday, September 10, 2019.

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Tuesday, September 10, 2019

11:00 am in 347 Altgeld Hall,Tuesday, September 10, 2019

Dirichlet character twisted Eisenstein series and $J$-spectra

Ningchuan Zhang

Abstract: Bernoulli numbers show up in both the $q$-expansions of normalized Eisenstein series and the image of the $J$-homomorphism in the stable homotopy groups of spheres. Number theorists have defined generalized Bernoulli numbers and twisted Eisenstein series associated to Dirichlet characters. The goal of this talk is to construct a family of Dirichlet character twisted $J$-spectra and explain the relations between their homotopy groups and congruences of the twisted Eisenstein series. In the course of that, we will generalize Nicholas Katzís algebro-geometric explanation of congruences of the (untwisted) normalized Eisenstein series in his Antwerp notes.

1:00 pm in 345 Altgeld Hall,Tuesday, September 10, 2019

An ergodic advertisement for descriptive graph combinatorics

Anush Tserunyan (UIUC Math)

Abstract: Dating back to Birkhoff, pointwise ergodic theorems for probability measure preserving (pmp) actions of countable groups are bridges between the global condition of ergodicity (measure-theoretic transitivity) and the local combinatorics of the actions. Each such action induces a Borel equivalence relation with countable classes and the study of these equivalence relations is a flourishing subject in modern descriptive set theory. Such an equivalence relation can also be viewed as the connectedness relation of a locally countable Borel graph. These strong connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, probability theory, and descriptive graph combinatorics. I will discuss how descriptive set theoretic thinking combined with combinatorial and measure-theoretic arguments yields a pointwise ergodic theorem for quasi-pmp locally countable graphs. This theorem is a general random version of pointwise ergodic theorems for group actions and is provably the best possible pointwise ergodic result for some of these actions.

2:00 pm in 243 Altgeld Hall,Tuesday, September 10, 2019

The largest projective cube-free subsets of $\mathbb Z_{2^n}$

Adam Zsolt Wagner (ETH Zurich Math)

Abstract: What is the largest subset of $\mathbb Z_{2^n}$ that doesn't contain a projective $d$-cube? In the Boolean lattice, Sperner's, Erdos's, Kleitman's and Samotij's theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of $\mathbb Z_2^n$ we work in $\mathbb Z_{2^n}$, analogous statements hold if one replaces the word $k$-chain by projective cube of dimension $2^{k-1}$. The largest $d$-cube-free subset of $\mathbb Z_{2^n}$, if $d$ is not a power of two, exhibits a much more interesting behaviour.

(Joint work with Jason Long)

3:00 pm in 243 Altgeld Hall,Tuesday, September 10, 2019

Deformation theory and partition Lie algebras

Akhil Mathew (U Chicago)

Abstract: A theorem of Lurie and Pridham states that over a field of characteristic zero, derived "formal moduli problems" (i.e., deformation functors defined on derived Artinian commutative rings), correspond precisely to differential graded Lie algebras. This formalizes a well-known philosophy in deformation theory, and arises from Koszul duality between Lie algebras and commutative algebras. I will report on joint work with Lukas Brantner, which studies the analogous situation for arbitrary fields. The main result is that formal moduli problems are equivalent to a category of "partition Lie algebras"; these are algebraic structures (which agree with DG Lie algebras in characteristic zero) which arise from a monad built from the partition complex.