College of Liberal Arts and Sciences

# Department of Mathematics

## Analysis and Probability Seminar

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Title: Embeddings of the Heisenberg group and the Sparsest Cut problem
Speaker: Robert Young (Courant Institute)
Time: Friday, January 19, 2018 at 1:30 pm
Place: MONT 414Abstract: The Heisenberg group $\mathbb{H}$ is a sub-Riemannian manifold that is hard to embed in $\mathbb{R}^n$. Cheeger and Kleiner introduced a new notion of differentiation that they used to show that it does not embed nicely into $L_1$. This notion is based on surfaces in $\mathbb{H}$, and in this talk, we will describe new techniques that let us quantify the "roughness" of such surfaces, find sharp bounds on the distortion of embeddings of $\mathbb{H}$, and estimate the accuracy of an approximate algorithm for the Sparsest Cut Problem. (Joint work with Assaf Naor)

Title: Spectral heat content for Levy processes
Speaker: Hyunchul Park (SUNY at New Paltz)
Time: Friday, February 16, 2018 at 1:30 pm
Place: MONT 414Abstract: In this talk, we study a short time asymptotic behavior of spectral heat content for Levy processes. The spectral heat content of a domain $D$ can be interpreted as the amount of heat if the initial temperature on $D$ is 1 and temperature outside $D$ is identically 0 and the motion of heat particle is governed by underlying Levy processes. We study spectral heat content for arbitrary open sets with finite Lebesgue measure in a real line under some growth condition on the characteristic exponents of the Levy processes. We observe that the behavior is very different from the classical heat content for Brownian motions. We also study the spectral heat content in general dimensions when the processes are of bounded variation. Finally we prove that asymptotic expansion of spectral heat content is stable under integrable perturbation when heat loss is sufficiently large. This is joint work with Renming Song and Tomasz Grzywny.

Title: TBA
Speaker: Simon Bortz (University of Minnesota)
Time: Friday, March 9, 2018 at 1:30 pm
Place: MONT 414

Title: Densely defined non-closable curl on the Mackay-Tyson-Wildrick carpets
Speaker: Alexander Teplyaev (University of Connecticut)
Time: Friday, March 23, 2018 at 1:30 pm
Place: MONT 414Abstract: The talk will discuss the possibly degenerate behavior of the exterior derivative operator defined on 1-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they have positive two-dimensional Lebesgue measure and carry nontrivial 2-forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1-forms) is not closable, and that its adjoint operator has a trivial domain. This is a joint work with Michael Hinz.

Title: TBA
Speaker: Phanuel Mariano (University of Connecticut)
Time: Friday, March 30, 2018 at 1:30 pm
Place: MONT 313Abstract: TBA

Title: TBA
Speaker: Silvia Ghinassi (Stony Brook University)
Time: Friday, April 6, 2018 at 1:30 pm
Place: MONT 414

Title: TBA
Speaker: Tapio Rajala (University of Jyvaskyla)
Time: Friday, April 13, 2018 at 1:30 pm
Place: MONT 414

Title: TBA
Speaker: Jeremy Tyson (University of Illinois at Urbana-Champaign)
Time: Friday, April 20, 2018 at 1:30 pm
Place: MONT 414

Organizer: Vasileios Chousionis and Oleksii Mostovyi