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for events the day of Tuesday, November 5, 2019.

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Tuesday, November 5, 2019

11:00 am in 347 Altgeld Hall,Tuesday, November 5, 2019

Loop space constructions of elliptic cohomology

Matthew Spong

Abstract: Elliptic cohomology is a type of generalised cohomology theory related to elliptic curves which was introduced in the late 1980s. An important motivation for its introduction, which came from physics, was to help understand index theory for families of differential operators over free loop spaces. Yet for a long time, the only known constructions of elliptic cohomology were purely algebraic, and the precise connection to free loop spaces remained obscure. In this talk, I will summarise two constructions of complex analytic, equivariant elliptic cohomology: one from the K-theory of free loop spaces, and one from the ordinary cohomology of double free loop spaces. If time permits, I will also describe the construction of a Chern character-type map from the former to the latter.

1:00 pm in 347 Altgeld Hall,Tuesday, November 5, 2019

A bilinear proof of decoupling for the quartic moment curve

Zane Li (Indiana Math)

Abstract: Using a bilinear method inspired from Wooley's nested efficient congruencing method, we prove a sharp $l^2 L^{20}$ decoupling inequality for the moment curve in $\mathbb{R}^4$. This is joint work with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich.

1:00 pm in 345 Altgeld Hall,Tuesday, November 5, 2019

Deciding the theories of expansions of the real ordered group via Ostrowski numeration

Christian Schulz (UIUC Math)

Abstract: For which irrational numbers $\alpha$ does the theory of $(\mathbb{R}, <, +, \mathbb{Z}, \alpha \mathbb{Z})$ have a decision algorithm? Previously, this was known for quadratic $\alpha$ thanks to work by Hieronymi. In this talk, we present the progress so far on generalizing this result to non-quadratic $\alpha$. We also discuss applications of this work to the study of combinatorics on words using automated theorem-proving software. We end with a discussion of potential future work and goals.

2:00 pm in 243 Altgeld Hall,Tuesday, November 5, 2019

Super-pancyclic hypergraphs and bipartite graphs

Dara Zirlin (Illinois Math)

Abstract: We find Dirac-type sharp sufficient conditions for a hypergraph $H$ with few edges to have a hamiltonian Berge cycle. Furthermore, these conditions yield that $H$ is super-pancyclic, i.e., for each $A \subseteq V(H)$ with $|A| \ge 3$, $H$ contains a Berge cycle with vertex set $A$. To do this, we exploit the language of bipartite graphs. In particular, we extend some results of Jackson on the existence of long cycles in bipartite graphs where the vertices in one part have high degrees, and prove his conjecture from 1981 on the topic.

This is joint work with Alexandr Kostochka and Ruth Luo.

2:00 pm in 347 Altgeld Hall,Tuesday, November 5, 2019

A Probabilistic Intuitive Boundary Construction

Peter Loeb (Illinois Math)

Abstract: Robert Martin’s 1941 generalization of the boundary of the unit disk is now a fundamental tool in potential theory and probability theory. After an introduction for the non-specialist, I will present an alternative approach to Martin’s construction and integral representation. The approach looks inside a domain using Brownian paths starting at nonstandard points that merge to form boundary points.

3:00 pm in 243 Altgeld Hall,Tuesday, November 5, 2019

Rational points and derived equivalence

Ben Antieau (UIC)

Abstract: Suppose that X and Y are smooth projective varieties over a field k and suppose that X and Y have equivalent derived categories of sheaves. If X has a rational point, does Y have a rational point? This question was asked 10 years ago by Esnault. I will report on joint work with Addington, Frei, and Honigs which shows that, in general, the answer is ‘no’, in contrast to what happens for curves (Antieau—Krashen—Ward) or in dimension at most 3 over finite fields (Honigs).