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for events the day of Tuesday, March 3, 2020.

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Tuesday, March 3, 2020

11:00 am in 243 Altgeld Hall,Tuesday, March 3, 2020

Approximating higher algebra by derived algebra

William Balderrama

Abstract: Obstruction theories and spectral sequences from the basic computational tools for accessing complex homotopical structure by means of pure algebra. Many of these are constructed via a careful examination of the relevant notion of ``free resolution''; the difficulties in their construction are then in maintaining sufficient control over these resolutions, as well as in the identification of the relevant obstruction groups. I will describe a general conceptual framework for producing these tools, based on a higher categorical variation on the notion of an algebraic theory, which is easily applicable to a wide variety of situations and provides a direct bridge between homotopical structure and algebraic structure.

2:00 pm in 345 Altgeld Hall,Tuesday, March 3, 2020

Heat kernel of fractional Laplacian with Hardy drift via desingularizing weights

Damir Kinzebulatov   [email] (Universite Laval)

Abstract: We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, using the method of desingularizing weights. This is joint work with Yu.A.Semenov (Toronto) and K.Szczypkowski (Wroclaw).

2:00 pm in 243 Altgeld Hall,Tuesday, March 3, 2020

The avoidance density of (k, l)-sum-free sets

Yifan Jing (University of Illinois, Urbana-Champaign)

Abstract: Let $\mathscr{M}_{(2,1)}(N)$ be the infimum of the size of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $\omega(N)\to\infty$ as $N\to\infty$, such that $cN+\omega(N)<\mathscr{M}_{(2,1)}(N)<(c+\varepsilon)N$ for any $\varepsilon>0$. The constant $c(2, 1)$ is recently determined by Eberhard, Green, and Manners, while the existence of $\omega(N)$ is still open. In this talk, we consider the analogue conjecture for $(k,l)$-sum-free sets. We determine the constant $c(k,l)$ for every $(k,l)$, and prove the existence of the function $\omega(N)$ for infinitely many $(k,l)$. The proof uses tools from probabilistic combinatorics, fourier analysis, and nonstandard analysis.

3:00 pm in 243 Altgeld Hall,Tuesday, March 3, 2020

Equivariant elliptic cohomology and 2-dimensional supersymmetric gauge theory

Dan Berwick-Evans (UIUC)

Abstract: Elliptic cohomology can be viewed as a natural generalization of ordinary cohomology and K-theory. However, in contrast to our robust geometric understanding of cohomology and K-theory classes, we do not know of any such description for elliptic cohomology classes. A long-standing conjecture looks to provide such a geometric description using 2-dimensional supersymmetric field theory. I will describe a step forward in this program that relates equivariant elliptic cohomology over the complex numbers to certain supersymmetric gauge theories. This both refines the pre-existing program and sheds light on certain aspects, e.g., the emergence of formal group laws in the language of field theories. I will assume no prior knowledge of either elliptic cohomology or supersymmetric field theory. This is joint work with Arnav Tripathy.