Abstract: We discuss the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, outline a proof of the conjecture (conditional in some cases, unconditional in others), and demonstrate how these estimates can be applied to obtain sharp restriction inequalities on some degenerate hypersurfaces. This is joint work with Jeremy Schwend.

Abstract: Following the introduction of techniques from additive combinatorics to some problems in Banach spaces by Wojciechowski, we discuss the Balog-Szemerédi-Gowers Lemma and how it can be used to tackle some questions about approximate homomorphisms and a conjecture of Pełczyński.

Abstract: In this talk we sketch a proof of the Carleson ε^2-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson ε^2-square function. This is a joint work with Ben Jaye and Xavier Tolsa.

Abstract: The space BV of functions of bounded variation is the space of integrable functions whose gradient is a Radon measure. Extending this definition to the trendy A-free measures, I will define the space BV^A of functions of bounded A-variation: functions such that A(D)u is a measure, where A(D) is a linear elliptic operator with constant coefficients. I will introduce general aspects of this theory, share a few recent results, and some difficult open problems:

1. L1-estimates -> life without Calderón-Zygmund

2. Slicing, geometry of A-bounded measures -> life without co-area formula

3. Continuity properties, #2^ndHardestProblemCalcVar -> life without co-area formula, again.

Interestingly, these measure theoretic properties were solved/require Fourier analysis methods.

Abstract: In the last decade a plethora of sharp weighted estimates has been obtained for several different operators. These estimates (sharp in the dependence on the characteristic of the weight) follow from a sparse domination of the operator. Roughly speaking, a sparse domination consists in controlling the operator with a positive dyadic form. It has been shown that Calderón–Zygmund operators and square functions admit such domination even under minimal T1 hypothesis.

In this talk we introduce the concept of sparse domination and we present some ideas that allow to upgrade the classical T1 theorems by David, Christ and Journé to sparse ones.

Abstract: Ten years ago, Bourgain, Clozel & Kahane established a surprising "sign uncertainty principle" (SUP), asserting that if a function and its Fourier transform are nonpositive at the origin and not identically zero, then they cannot both be nonnegative outside an arbitrarily small neighbourhood of the origin. In 2017, Gonçalves & Cohn solved the 12-dimensional SUP via connections to the sphere packing problem, and discovered a complementary SUP. This talk will focus on some new sign uncertainty principles which generalise the developments of Bourgain, Clozel & Kahane and Cohn & Gonçalves. In particular, we will discuss SUPs for Fourier series, the Hilbert transform, spherical harmonics, and Jacobi polynomials. As a by-product, we determine some sharp instances of the spherical SUP via connections to tight spherical designs. Time permitting, we will outline a possible path towards the sharp 1-dimensional SUP. This talk is based on recent joint work with Felipe Gonçalves and João Pedro Ramos.

Abstract: The separation profile of an infinite graph was introduced by Benjamini-Schramm-Timar. It is a function which measures how well-connected the graph is by how hard it is to cut finite subgraphs into small pieces. In earlier joint work with David Hume and Romain Tessera, we introduced Poincaré profiles, generalising this concept by using p-Poincaré inequalities to measure the connectedness of subgraphs. I will discuss these invariants, their applications to coarse embedding problems, and work nearing completion where we find the profiles of all connected unimodular Lie groups. Joint with Hume and Tessera.

Abstract: Let P_1,...,P_m be polynomials with integer coefficients and zero constant term. Bergelson and Leibman’s polynomial generalization of Szemerédi’s theorem states that any subset A of {1,...,N} that contains no nontrivial progressions x, x+P_1(y), ..., x+P_m(y) must satisfy |A|=o(N). In contrast to Szemerédi's theorem, quantitative bounds for Bergelson and Leibman's theorem (i.e., explicit bounds for this o(N) term) are not known except in very few special cases. In this talk, I will discuss recent progress on this problem, focusing on arguments involving Fourier analysis.

Abstract: In 1999 Cheeger gave a far reaching generalisation of Rademacher’s differentiation theorem which replaces the domain by a metric space equipped with a measure that satisfies a version of the Poincare inequality. The first half of this talk will consist of a gentle introduction to this result and some of its consequences. No prior knowledge will be assumed.

The work of Cheeger inspired a large number of new results in the area of analysis on metric spaces. The second half of this talk will present a new, simpler proof of Cheeger’s theorem based on these developments and the multilinear Kakeya inequality for rectifiable curves (in Euclidean space). This is based on joint work with Ilmari Kangasniemi and Tuomas Orponen.

Abstract: Grushin operators are among the simplest examples of subelliptic operators. Due to the lack of ellipticity, standard techniques based on heat kernel estimates yield spectral multiplier theorems that are typically not sharp in terms of the smoothness requirement on the multiplier. We show that, for a large class of Grushin operators on the plane, a sharp multiplier theorem can be proved, with the same smoothness requirement as in the case of the standard Laplacian. Our argument is robust enough to handle nonhomogeneous coefficients vanishing of arbitrarily high order, and hinges on the analysis of one-parameter families of Schroedinger operators.

This is based on joint work with Gian Maria Dall'Ara (Birmingham).