Transport maps serve as a powerful tool to transfer information from source to target measures. However, this transfer of information is possible only if the transport map is sufficiently regular, which is often difficult to show. I will explain how taking the source measure to be an infinite-dimensional measure, and building transport maps based on stochastic processes, solves some of these challenges both in the continuous and discrete settings.

Let K be an origin-symmetric convex body in dimension 3 or higher. For each unit vector u, consider the cone whose base is the projection of K on the hyperplane perpendicular to u, and whose height is the value of the radial function of K in the direction u. If the volume of this cone is constant independent of u, must K be an ellipsoid? This question is dual to the statement of Busemann-Petty Problem 5. We obtain an affirmative result for bodies that are sufficiently close to the Euclidean ball in the Banach-Mazur distance. The proof is more involved that the local affirmative answer to Busemann-Petty Problem 5. This is joint work with F. Nazarov, D. Ryabogin and V. Yaskin.

A recent breakthrough in learning low-degree Boolean functions by Eskenazis and Ivanisvili employs a family of dimension-free polynomial inequalities originating from Littlewood's 1930 work, known as the Bohnenblust–Hille inequalities. In this talk, I will review some recent results that extend these inequalities from discrete hypercubes to qubit systems, along with a reduction of harmonic analysis problems on quantum systems to their classical analogs on cyclic groups. This is based on joint work with Alexander Volberg and Joseph Slote.

What global properties of a function can we infer from local information? Bernstein-type discretization inequalities offer one answer: they show the supremum norm of a polynomial can be controlled by its absolute supremum on a small, discrete subset of the domain. While such inequalities enjoy widespread use in analysis and approximation theory, their multivariate versions are often limited by a strong dependence on dimension or the need for very many test points. In this talk we show how to get a dimension-free discretization with few points, leading to sharpenings of several inequalities in harmonic analysis over cyclic groups. Along the way we develop a probabilistic technique for iterating one-dimensional inequalities without paying a dimension-dependent price. Based on joint work with Lars Becker, Ohad Klein, Alexander Volberg, and Haonan Zhang.

We will construct counterexamples to previously conjectured metric formulations of Kwapień's theorem from Banach space theory. The key new ingredient in our construction is a novel rigidity theorem for metric transforms of nonpositively curved Alexandrov spaces. Time permitting, further geometric and algorithmic applications of this rigidity will also be discussed. The talk is based on joint work with M. Mendel (Open University of Israel) and A. Naor (Princeton).

We extend Bobkov and Chistyakov’s upper bounds on concentration functions of sums of independent random variables to a multivariate entropic setting. The approach is based on pointwise estimates on densities of sums of independent random vectors uniform on centred Euclidean balls. In this vein, we also obtain sharp bounds on volumes of noncentral sections of isotropic convex bodies. This is joint work with James Melbourne and Tomasz Tkocz.

The dual quermassintegrals of a star body are defined as the average volume of sections of the body by hyperplanes of fixed dimension. These quantities, along with general dual mixed volumes, see striking parallels with their counterparts in Brunn-Minkowksi theory, especially for their respective extremal inequalities. In this talk, we show how empirical methods provide a way to establish dual inequalities which are based on new local stochastic features within dual Brunn-Minkowski theory. Joint work with G. Paouris and P. Pivovarov.

This talks concerns Busemann-Petty type problems for the spatial and spherical Radon transforms inspired by the first Busemann-Petty problem. Based on a joint with with A. Koldobsky and A. Zvavitch

Spatio-frequency limiting operators are compact, self-adjoint, and positive definite operators defined on the space of square-integrable functions, with countable and monotonic eigenvalues lying within the interval [0,1]. These operators play a critical role in signal processing and the construction of functions with high concentration in both spatial and frequency domains. They address a paradox arising in pure mathematics (concerning the uncertainty principle) and wireless communication (relating to the existence of functions that achieve simultaneous limitations in both spatial and frequency domains) In this talk, we will present our results on the non-asymptotic distribution of the eigenvalues of space-frequency limiting operators above a threshold ϵ>0 and between two thresholds ϵ and 1−ϵ (with ϵ>1/2). We will demonstrate how these results depend on the choice of spatial and frequency domains used in constructing the operators. If time permits, we will discuss a connection between these operators, the Widom Conjecture, and pseudodifferential operators with discontinuous symbols as a potential direction for future work. The results are based on two joint works with Arie Israel and Kevin Hughes.

We consider the conditions required on a set that guarantee it contains arithmetic progressions. Szemeredi proved the existence of arithmetic progressions in subsets of the natural numbers with positive upper density. In the fractal setting, it is known by Maga and Keleti that full Hausdorff dimension is not enough to guarantee the existence of a 3-term arithmetic progression in subsets of d-dimensional Euclidean space; however, it turns out that Fourier decay coupled with nearly full Hausdorff dimension is sufficient for the existence of arithmetic progressions, as shown by Laba and Pramanik. In this talk, we consider another notion of size: Newhouse thickness. It is known that thickness larger than 1 is enough in the real line to guarantee the existence of a 3-term arithmetic progression. In higher dimensions, Yavicoli showed that it takes thickness larger than 10^8, along with some additional assumptions, to guarantee a 3-point configuration. We give the first result in higher dimensions showing the existence of 3-term arithmetic progressions in sets of thickness larger than 2/(1-2r), where r is a constant dependent on the set.

In this talk, we revisit classical fractional Sobolev embedding theorems and the algebra property of the fractional Sobolev space $H^s(\mathbb{R})$ from a dyadic point of view. Inspired by previous work due to Aimar and coauthors we use Haar functions and dyadic decompositions to develop alternative proofs of these classic results. In particular, in low regularities we provide explicit examples for the failure of the algebra property without using the Fourier transform.

We examine the problem of minimizing the continuum nonlocal, nonconvex variational problem that arises from modeling a large number of pairwise interacting particles in the presence of thermal noise (i.e., molecular dynamics). Determining global minima (ground states) to these functionals is important as they characterize the structure of matter, self-assembly, and phase transitions. Determining global minima is, however, in general difficult. We will derive linear programming lower (LP) in the spirit of Cohn and Kumar via a dual approach as convex relaxations over closed subsets of probability measures. We will then present solutions to the LP bound for several interaction kernels inspired by molecular dynamics – the Morse potential and Onsagar potentials from liquid crystals. We will also discuss several discrete cases where the minimizers are provable sharp and counter examples where the LP bound fails to be a tight lower bound.

A remarkable result of Davies shows that an arbitrary measurable set in the plane can be covered by lines in such a way that the union of the lines minus the original set has measure zero. This theorem has an equivalent dual formulation which says that one can find a single set in the plane with given "prescribed" projections in almost every direction, up to measure zero errors. We extend these results to a non-linear setting and prove that a set in the plane can be covered efficiently by translates of a single curve satisfying a mild curvature assumption.

The floating area was previously investigated as a natural extension of classical affine surface area to non-Euclidean convex bodies in spaces of constant positive curvature. We introduce the family of $L_p$-floating areas for spherical convex bodies, as an analog to $L_p$-affine surface area measures from Euclidean geometry. We investigate a duality formula, monotonicity and isoperimetric inequalities for this new family of curvature measures on spherical convex bodies. Furthermore, using the $L_p$-floating area, we introduce a new entropy functional for spherical convex bodies and a dual isoperimetric inequality is established. Based on joint works with Florian Besau.

We will show that the homothety conjecture holds true for plane centrally symmetric convex bodies near the disk. Joint work with M. Alfonseca, F. Nazarov, D. Ryabogin and V. Yaskin.

We will show that the homothety conjecture is false for non-symmetric plane convex bodies. Joint work with M. Alfonseca, F. Nazarov, A. Stancu and V. Yaskin.

It is well known that among all convex bodies in R^n with a given surface area, the Euclidean ball has the largest inradius. We will show that this result can be reversed in the class of convex bodies with curvature at each point of their boundary bounded below by some positive constant λ (λ-convex bodies). In particular, we show that among λ-convex bodies of a given surface area, the λ-convex lens (the intersection of two balls of radius 1/λ) minimizes the inradius.

Among all convex bodies, sections of ellipsoids and bodies of revolution exhibit particular symmetry. Namely, all hyperplanar sections of an ellipsoid are centrally symmetric and have an axis of symmetry, whereas all hyperplanar sections of a body of revolution have an axis of revolution. H. Brunn proved in 1889 that the central symmetry of all the sections characterizes ellipsoids. Much later, in 1965, C. A. Rogers observed that it is enough to consider only sections passing through a fixed point. Regarding axial symmetry, in 1999, K. Bezdek posed his celebrated conjecture that the axial symmetry of all the sections characterizes bodies of revolution in 3-dimensional space. Now, it is natural to formulate higher-dimensional analogs of Bezdek's conjecture, and there are many ways to do it. Our main result is a variant of Bezdek's conjecture in arbitrary dimension n≥3, where we assume that all the sections passing through a fixed point have an axis of symmetry, satisfying certain alignment condition. Further, if we weaken the hypothesis and consider only a 1-codimensional family of hyperplanes, we obtain a similar characterization of axially symmetric bodies. For each of these problems, we show both the orthogonal and the affine variant. Interestingly, in 3-dimensional space the proof is essentially different and touches on the theory of floating bodies. The talk is based on a joint work in progress with M. Angeles Alfonseca.

We obtain the optimal estimate for the probability that an n by n random matrix with i.i.d. entries has rank at most n-k, for any k which is less than n^{1/2}. This estimate allows to prove a conjecture of Feige and Lellouche from computer science.

We construct a convex body K in dimensions 5 and higher, with the property that there is exactly one hyperplane section of K through its centroid, such that the centroid of this section coincides with the centroid of K. This provides answers to questions of Gruenbaum and Loewner for dimensions starting from 5. The proof is based on the existence of non-intersection bodies in these dimensions. Joint work with S. Myroshnychenko and K. Tatarko.

The interplay between convexity and Euclidean harmonic analysis has been very fruitful. While it is less clear that notions of convexity should even appear in abstract harmonic analysis, Stanley observed in the 1980’s that representation theory can be powerfully used to prove unimodality of certain combinatorial sequences, and Okounkov observed in the 1990’s that notions of convexity and log-concavity also arise naturally in the representation theory of various Lie groups. Inspired by, but completely unrelated to these historical bridges, we show that natural discrete probability measures that arise in harmonic analysis of the symmetric group (to be precise, the distributions of row lengths of Young diagrams drawn from the “poissonized" Plancherel measure) are log-concave. This has several significant implications, including the verification of a variant of a 2008 conjecture of W.Y.C.Chen about the longest increasing subsequence of a random permutation, and the fact that the Tracy-Widom laws arising in probability and mathematical physics are log-concave. The talk is based on joint work with Jnaneshwar Baslingker and Manjunath Krishnapur.

In this talk, we will discuss a dyadic variant of the Hilbert transform, which is a useful model of its continuous counterpart and the prototypical example of a so-called "Haar shift". After discussing some background and motivation in the Lebesgue measure case, we will turn to the situation where the L^2 Haar functions are defined with respect to a locally finite Borel measure μ, which may not satisfy the dyadic doubling condition. In this more general setting, Lopez-Sanchez, Martell, and Parcet identified a weak regularity condition on the measure μ which characterizes weak-type and L^p estimates for this dyadic Hilbert transform. I then will discuss joint work with Jose Conde Alonso and Jill Pipher, where we obtain a domination of the dyadic Hilbert transform (and more generally, Haar shifts) by a modified sparse form. As an application, we characterize the class of weights where the dyadic Hilbert transform and related operators are bounded. A surprising novelty is that the usual (dyadic) Muckenhoupt A_2 condition is necessary, but no longer sufficient in the non-doubling setting, and our modified weight condition reflects the "complexity" of the underlying Haar shift. Finally, we will examine a different dyadic Haar shift model of the Hilbert transform and its relationship to BMO (bounded mean oscillation) functions via commutators in the non-doubling setting (joint with Tainara Borges, Jose Conde Alonso, and Jill Pipher).