Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, November 19, 2019.

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Tuesday, November 19, 2019

11:00 am in 347 Altgeld Hall,Tuesday, November 19, 2019

Two theories of real cyclotomic spectra

Jay Shah

Abstract: The topological Hochschild homology $THH(R)$ constitutes a powerful and well-studied invariant of an associative ring $R$. As originally shown by Bokstedt, Hsiang and Madsen, $THH(R)$ admits the elaborate structure of a cyclotomic spectrum, whose formulation depends upon equivariant stable homotopy theory. More recently, inspired by considerations in p-adic Hodge theory, Nikolaus and Scholze demonstrated (under a bounded-below assumption) that the data of a cyclotomic spectrum is entirely captured by a system of circle-equivariant Frobenius maps, one for each prime p. They also give a formula for the topological cyclic homology $TC(R)$ directly from these maps. The purpose of this talk is to extend the work of Nikolaus and Scholze in order to accommodate the study of real topological Hochschild homology $THR$, which is a $C_2$-equivariant refinement of $THH$ defined for an associative ring with an anti-involution, or more generally an $E_\sigma$-algebra in $C_2$-spectra. The key idea is to make use of the $C_2$-parametrized Tate construction. This is joint work with J.D. Quigley and is based on the arXiv preprint 1909.03920.

1:00 pm in 347 Altgeld Hall,Tuesday, November 19, 2019

Global Strichartz estimates for the semiperiodic Schrodinger equation.

Alex Barron (illinois Math)

Abstract: We will discuss some recent results related to space-time estimates for solutions to the linear Schrodinger equation on manifolds which are products of tori and Euclidean space (e.g. a cylinder embedded in R^3 ). On these manifolds it is possible to prove certain analogues of the classical Euclidean Strichartz estimates which are scale-invariant and global-in-time. These estimates are strong enough to prove small-data scattering for solutions to the critical quintic NLS on R T and the critical cubic NLS on R^2 T (where T is the one-dimensional torus).

2:00 pm in 243 Altgeld Hall,Tuesday, November 19, 2019

An overview of Erdős-Rothschild problems and their rainbow variants

Lina Li (Illinois Math)

Abstract: In 1974, Erdős and Rothschild conjectured that the complete bipartite graph has the maximum number of two-edge-colorings without monochromatic triangles over all n-vertex graphs. Since then, a new class of colored extremal problems has been extensively studied by many researchers on various discrete structures, such as graphs, hypergraphs, Boolean lattices and sets.

In this talk, I will first give an overview of some previous results on this topic. The second half of this talk is to explore the rainbow variants of the Erdős-Rothschild problem. With Jozsef Balogh, we confirm conjectures of Benevides, Hoppen and Sampaio, and Hoppen, Lefmann, and Odermann, and completes the characterization of the extremal graphs for the edge-colorings without rainbow triangles. Next, we study a similar question on sum-free sets, where we describe the extremal configurations for integer colorings with forbidden rainbow sums. The latter is joint work with Yangyang Cheng, Yifan Jing, Wenling Zhou and Guanghui Wang.

2:00 pm in 347 Altgeld Hall,Tuesday, November 19, 2019

Absolute continuity and singularity for probability measures induced by a drift transform of the independent sum of Brownian motion and symmetric stable process

Ruili Song (Nanjing University of Finance and Economics, and UIUC Math)

Abstract: We consider a Levy process $X$ which is the independent sum of a Brownian motion and a symmetric $\alpha$-stable process in $R^d$. The probability measure $P^b$ is induced by the drift transform of $X$ via the vector valued function $b$. We study mutual absolute continuity and singularity of $P$ and $P^b$ on the path space. We also investigate the problem of finiteness of the relative entropy of these measures on $R^d$ ($d\ge 3$).