Abstract: We’ll talk about two notions of square function inequality, related to a sequence of functions, which we’ll call direct and converse inequalities. In many cases the direct inequality can be proved by verifying a type of 2r-superorthogonality, that is, proving that the integral of certain 2r-tuples of functions selected from the sequence vanishes. We will demonstrate a hierarchy of “types” of superorthogonality for which this deduction can be carried out quite formally, and meanwhile illustrate a wide variety of specific settings. In particular, we will show that two famous results from number theory, in the setting of bounding character sums, fit neatly into this framework.

Abstract: We will follow up with Luz Roncal's talk, where she presented a directional embedding theorem for Carleson sequences which was in turn used to obtain bounds for directional Rubio de Francia type square functions. We will see how we can extend this result to the weighted setting, from where we can deduce some weighted estimates for the directional maximal function and directional singular integrals.

This is part of joint work with F. Di Plinio, P. Hagelstein, I. Parissis and L. Roncal.

Abstract: Fourier dimension estimates are a growing topic of interest in harmonic analysis, geometric measure theory, and metric Diophantine approximation. In a joint work with Reuben Wheeler, we obtain some lower estimates on the Fourier dimension of Bugeaud’s set of numbers approximable to some exact order ψ.

Abstract: Given a set E of Hausdorff dimension s>d/2 in R^d , Falconer conjectured that its distance set \Delta(E)=\{ |x-y|: x, y \in E\} should have positive Lebesgue measure. When d is even, we show that dim_H E>d/2+1/4 implies |\Delta(E)|>0. This improves on the work of Wolff, Erdogan, Du-Zhang, etc. Our tools include Orponen's radial projection theorem and refined decoupling estimates.

This is joint work with Guth, Iosevich, and Ou and with Du, Iosevich, Ou, and Zhang.

Abstract: We generalize the Guth-Katz joints theorem from lines to varieties. A special case of our result says that N planes (2-flats) in 6 dimensions (over any field) have O(N^{3/2}) joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery's conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects.

Joint work with Jonathan Tidor and Hung-Hsun Hans Yu

Abstract: The Favard length of a subset of the plane is defined as the average length of its orthogonal projections. This quantity is related to the probabilistic Buffon needle problem, which considers the probability that a needle or a line that is dropped at random near a given set will intersect the set. We consider the geometric and probabilistic consequences that arise upon replacing linear projections by more general families of projection-type mappings. In particular, we find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set. Beyond the four-corner set, we also show that if a subset E has finite length in the sense of Hausdorff and is nearly purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then its “curve” projections have very small measure.

Abstract: Charles Fefferman's counterexample for the ball multiplier is intimately linked to square function estimates for directional singular integrals along all possible directions. Quantification of such a failure of the boundedness of the ball multiplier is measured, for instance, through L^p-bounds for the N-gon multiplier which provide information in terms of N.

We present a general approach, based on a directional embedding theorem for Carleson sequences, to study time-frequency model square functions associated to conical or directional Fourier multipliers. The estimates obtained for these square functions are applied to obtain sharp or quantified bounds for directional Rubio de Francia type square functions. In particular, a precise logarithmic bound for the polygon multiplier is shown, improving previous results.

This is joint work with Natalia Accomazzo, Francesco Di Plinio, Paul Hagelstein, and Ioannis Parissis.

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.

Abstract: In the late 80s and early 90s, Bourgain proved pointwise convergence results for polynomial ergodic averages applied to a single function. In this talk I will discuss joint work with Mariusz Mirek and Terence Tao on bilinear analogues of Bourgain's work.

Abstract: The talk will be about a recent joint work with Michael Christ and Polona Durcik on a variant of the triangular Hilbert transform involving curvature. Our results unify various previously known results such as bounds for a bilinear Hilbert transform with curvature and a maximally modulated singular integral of Stein-Wainger type, and Bourgain's non-linear Roth theorem in the reals.

Abstract: Multilinear functionals, and inequalities governing them, arise in various contexts in harmonic analysis (in connection with Fourier restriction), in partial differential equations (nonlinear interactions) and in additive combinatorics (existence of certain patterns in sets of appropriately bounded density). This talk will focus on an inequality that quantifies a weak convergence theorem of Joly, Metivier, and Rauch (1995) concerning threefold products, and on related inequalities for trilinear expressions involving highly oscillatory factors. Sublevel set inequalities, which quantify the impossibility of exactly solving certain systems of linear functional equations (the frustration of the title), are a central element of the analysis.

Abstract: We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This is joint work with Marianna Csörnyei.