Department of

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for events the day of Tuesday, September 15, 2020.

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Tuesday, September 15, 2020

11:00 am in On-line : Zoom,Tuesday, September 15, 2020

Large class groups

Jesse Thorner (UIUC Math)

Abstract: For a number field F of degree d \geq 2 over the rationals, let D_F be the absolute discriminant. In 1956, Ankeny, Brauer, and Chowla proved that for a given degree d, there exist infinitely many number fields of degree d such that for any fixed \epsilon > 0, the class group of F has size at least (D_F)^{1/2-\epsilon}. This was conditionally refined by Duke in 2003: assuming Artin's holomorphy conjecture and the generalized Riemann hypothesis, there exist infinitely many number fields F of degree d such that the class group of F has size \asymp (D_F)^{1/2} (\log\log D_F / \log D_F)^{d-1}. In particular, given d \geq 2, there are (conditionally) infinitely many number fields of degree d whose class group has maximal asymptotic order. In 2014, Cho showed that Artin's holomorphy conjecture and the generalized Riemann hypothesis can be replaced with the single assumption that Artin representations are automorphic (which implies Artin's holomorphy conjecture), unconditionally establishing Duke's conclusion for d \leq 5. I will discuss joint work with Robert Lemke Oliver and Asif Zaman in which we unconditionally establish Duke's conclusion for all d \geq 2 (among many other things).

11:00 am in Zoom,Tuesday, September 15, 2020

How do field theories detect the torsion in topological modular forms?

Dan Berwick-Evans   [email] (UIUC)

Abstract: Since the mid 1980s, there have been hints of a connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) for which cocycles are 2-dimensional (supersymmetric, Euclidean) field theories. Properties of these field theories lead naturally to the expected integrality and modularity properties of classes in TMF. However, the abundant torsion in TMF has always been mysterious from for the supersymmetric sigma model with target determines a cocycle representative of the generator of pi_3(TMF) = Z/24. (Contact Jeremiah Heller for Zoom info: jbheller@illinois.edu)

2:00 pm in Zoom,Tuesday, September 15, 2020

Dirac-type theorems for random hypergraphs

Asaf Ferber (Univeristy of California - Irvine)

Abstract: For positive integers $d < k$ and $n$ divisible by $k$, let $m_{d}\left(k,n\right)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says that $m_{1}\left(2,n\right)=\lceil n/2 \rceil$. However, in general, our understanding of the values of $m_{d}\left(k,n\right)$ is still very limited, and it is an active topic of research to determine or approximate these values. In this talk we discuss a ``transference'' theorem for Dirac-type results relative to random hypergraphs. Specifically, for any $d< k$, and any "not too small" $p$, we prove that a random $k$-uniform hypergraph $G$ with $n$ vertices and edge probability $p$ typically has the property that every spanning subgraph of $G$ with minimum $d$-degree at least $\left(1+o(1)\right)m_{d}\left(k,n\right)p$ has a perfect matching. Observe that this result holds even in cases that $m_{d}\left(k,n\right)$ is still unknown.

Joint work with Matthew Kwan.

Please contact Sean at SEnglish (at) illinois (dot) edu for the Zoom ID.