University of Connecticut

Analysis and Probability Seminar


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Title: Two Approaches to Minkowski-type Estimates
Speaker: Sean McCurdy (University of Washington)
Time: Friday, September 14, 2018 at 1:30 pm
Place: MONT 421Abstract: This talk will consider two different techniques--one, a refinement of Almgren stratification due to Cheeger, Naber, and Valtorta, and the other, based upon a substantial improvement due to Naber and Valtorta--in their application to investigating the fine structure of solutions to certain two-sided free boundary problems. While both techniques rely upon monotonicity formulae to build inductively-refined covers, we shall attempt to highlight their relative strengths. The context in which we will apply these techniques builds upon the work of Kenig, Preiss, Toro and Badger, Engelstein, Toro on two-sided free boundary problems under mild assumptions on the geometry of the boundary and regularity of the interior and exterior harmonic measures. Surprisingly, in this context, we are able to prove Minkowski-type bounds not just on the singular strata of the free boundary, but on the critical strata of the Green's functions up to and including the boundary.

Title: The Traveling Salesman Theorem in Carnot groups
Speaker: Scott Zimmerman (University of Connecticut)
Time: Friday, September 21, 2018 at 1:30 pm
Place: MONT 313Abstract: Peter Jones proved his famous Traveling Salesman Theorem in the plane in 1990. His result classified those sets in the plane which are contained in a rectifiable curve via the boundedness of a certain Carleson integral. The methods introduced by Jones have seen applications in harmonic analysis and geometric measure theory, and his theorem has since been generalized to the setting of Hilbert spaces, the Heisenberg group, and the graph inverse limits of Cheeger and Kleiner. I will present recent work with V. Chousionis and S. Li in which we proved one direction of the Traveling Salesman Theorem for rectifiable curves in any Carnot group. A Carnot group is a type of nilpotent Lie group whose abelian members are precisely Euclidean spaces, and these groups have been the focus of much recent study in geometric measure theory. As an application, I will show that this theorem may be used to prove uniform boundedness of the singular integral operator associated with a certain non-negative kernel on any set contained in a rectifiable curve.

Title: A new proof of the concentration-compactness principle of Trudinger-Moser inequality
Speaker: Jungang Li (University of Connecticut)
Time: Friday, September 28, 2018 at 1:30 pm
Place: MONT 313Abstract: The concentration-compactness principle plays a key role in the study of PDEs with critical growth. The classical proof of the concentration-compactness principle of Trudinger-Moser inequality depends on a rearrangement argument. In this talk I will give a rearrangement-free proof of such principle. This proof enables us to extend the related results to non-Euclidean spaces, e.g. Heisenberg Groups and Riemannian manifolds. This is a joint work with Prof. Guozhen Lu and Prof. Maochun Zhu.

Title: A trajectorial interpretation of Doob's martingale inequalities
Speaker: Walter Schachermayer (University of Vienna)
Time: Friday, October 12, 2018 at 1:30 pm
Place: MONT 313Abstract: We present a unified approach to Doob's $L^p$ maximal inequalities for $1\leq p<\infty.$ The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover our deterministic inequalities lead to new versions of Doob's maximal inequalities. These are best possible in the sense that equality is attained by properly chosen martingales. Based on the joint work with Acciaio, Bieglbock, Penkner, and Temme.

Title: The Inverse Function Theorem in the Noncommutative Setting
Speaker: Mark E. Mancuso (Washington University in St. Louis)
Time: Friday, October 19, 2018 at 1:30 pm
Place: MONT 313Abstract: Classically, the inverse function theorem says that a $C^1$ function is locally invertible around a point of nonsingularity of its derivative. In this talk, we introduce the theory of noncommutative functions, which in the matrix case, are functions on a graded domain of tuples of matrices that preserve direct sums and similarities. We present inverse function-type theorems in this setting. Unlike in the classical case, noncommutative inverse function theorems are global: the derivative of a noncommutative function $f$ is invertible everywhere if and only if $f$ is invertible on its domain. We also discuss recent work on the inverse function theorem for operator noncommutative functions defined on domains sitting inside of $B(\mathcal{H})^d,$ for an infinite dimensional Hilbert space $\mathcal{H}$.

Title: TBA
Speaker: Guozhen Lu (University of Connecticut)
Time: Friday, November 9, 2018 at 1:30 pm
Place: MONT 313Abstract: TBA

Title: TBA
Speaker: David Herzog (Iowa State University)
Time: Friday, November 16, 2018 at 1:30 pm
Place: MONT 313Abstract: TBA

Title: TBA
Speaker: Jose Conde Alonso (Brown University)
Time: Friday, November 30, 2018 at 1:30 pm
Place: MONT 313Abstract: TBA

Organizer: Vasileios Chousionis and Oleksii Mostovyi