Seminar Abstracts

• October 23rd, 3pm EDT
Speaker: Theresa Anderson, Purdue University
Title: Dyadic analysis meets number theory
Abstract: We explore one of many ways that analysis and number theory interact by showing what goes on in the background to construct a measure that is $p$-adic doubling for any finite set of primes $p$ yet not doubling.  This is recent work joint with Bingyang Hu.  I will follow this by a short talk on the adjacent topic of “Complete classification of adjacent dyadic systems”.  There will be ample question time and questions pertaining to professional development and careers in the mathematical sciences are especially welcome.

• October 30th, 3pm EDT
Speaker: Kornélia Héra, University of Chicago
Title: Hausdorff dimension of Furstenberg-type sets
Abstract: We say that a planar set F is a (t,s)-Furstenberg set, if there exists an s-dimensional family of lines in the plane such that each line of this family intersects F in an at least t-dimensional set. We present Hausdorff dimension estimates for (t,s)-Furstenberg sets and for more general Furstenberg type sets in higher dimensions.
The talk is based on joint work with Tamás Keleti and András Máthé, and with Pablo Shmerkin and Alexia Yavicoli.

• November 6th, 3pm EST (Note: Daylight Savings Time ends November 1st)
Speaker: Kyle Hambrook, San Jose State University
Slides: Click Here for Slides
Title: Explicit Salem Sets of Arbitrary Dimension in Euclidean Space
Abstract: A set in $R^n$ is called Salem if it supports a probability measure whose Fourier transform decays as fast as the Hausdorff dimension of the set will allow. We construct the first explicit (i.e., non-random) examples of Salem sets in $R^n$ of arbitrary prescribed Hausdorff dimension. This completely resolves a problem proposed by Kahane more than 60 years ago. The construction is based on a form of Diophantine approximation in number fields. This is joint work with Robert Fraser.

• November 13th, 3pm EST
Speaker: Polona Durcik, Chapman University
Title: A triangular Hilbert transform with curvature
Abstract: The triangular Hilbert transform is a two-dimensional bilinear singular integral originating in time-frequency analysis. No Lebesgue space bounds are currently known for this operator. In this talk, we discuss recent joint work with Michael Christ and Joris Roos on a variant of the triangular Hilbert transform involving curvature. As an application, we also discuss a quantitative nonlinear Roth type theorem on patterns in the Euclidean plane.

• November 20th, 3pm EST
Speaker: Alex Barron, University of Illinois Urbana-Champaign
Title: A sharp global Strichartz estimate for the Schrodinger equation on the cylinder
Abstract: The classical Strichartz estimates show that a solution to the linear Schrodinger equation on Euclidean space is in certain Lebesgue spaces globally in time provided the initial data is in L^2. On compact manifolds one can no longer have global control, and some loss of derivatives is necessary in interesting cases (meaning the initial data needs to be in a Sobolev space rather than L^2). On non-compact manifolds it is a challenging problem to understand when one can have good space-time estimates with no loss of derivatives.

In this talk we discuss a global-in-time Strichartz-type estimate for the linear Schrodinger equation on the infinite cylinder. Our estimate is sharp, scale-invariant, and requires only L^2 data. Joint work with M. Christ and B. Pausader.
• December 4th, 3:30pm EST
Speaker: Bruno Poggi, University of Minnesota
Title: Theory of $A_{\infty}$ weights for elliptic measures and generalized Carleson perturbations for elliptic operators
Abstract: We present Carleson perturbations for elliptic operators on domains for which there exists a robust elliptic PDE theory. Such domains include, in particular, (a) 1-sided NTA domains satisfying the capacity density condition (thus we extend some recent results of Akman-Hofmann-Martell-Toro), (b) domains with low-dimensional Ahlfors-David regular boundaries, and (c) certain domains with boundaries with pieces of distinct dimensions. Our Carleson perturbations are generalized in the sense that, in addition to the classical additive perturbations, we allow for scalar-multiplicative perturbations, which admit non-trivial differences on the boundary between the perturbed matrix and the original matrix.  Finally, we investigate corollaries of our techniques, with implications to free boundary problems an a characterization of  $A_{\infty}$ among elliptic measures. This is joint work with Joseph Feneuil.

• December 4th, 4pm EST
Speaker: Kan Jiang, Ningbo University
Title: Some thickness theorems and their applications
Abstract: In this talk, I will introduce some old and new thickness theorems. In terms of these results, we are able to give many interesting applications.

• December 11th, 3pm EST
Speaker: Tyler Bongers, Harvard University
Title: Energy techniques for nonlinear projections and Favard curve lengths
Abstract: There are many classical results relating the structure, dimension, and measure of a set to the structure of its orthogonal projections, including theorems of Marstrand and Besicovitch. It turns out that many nonlinear projection-type operators also have special geometry that allow us to build similar relationships between a set and its “projections,” just as in the linear setting. In this work, we will show how energy techniques of Mattila can be strengthened and generalized to projection-type operators satisfying a transversality condition, and apply these results to study visibility and Favard curve lengths of sets. This work is joint with Krystal Taylor.