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Abstract

We give an overview of some recent results and open problems in the area of multilinear singular integrals and discuss their connection with questions on patterns in large subsets of the Euclidean space.

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The Bergman projection, or the orthogonal projection from L^2 to the Bergman space of square-integrable holomorphic functions on a given domain, is a fundamental operator in complex analysis. Although the Bergman projection is automatically bounded on L^2, it is non-trivial whether it extends to a bounded operator on L^p for 1<p<\infty, or on L^p spaces with respect to different measures (i.e. weighted inequalities). On the other hand, sparse domination is a recently developed powerful technique in harmonic analysis that has been useful in proving weighted inequalities with sharp constants. In this talk, we will sketch the ideas of how sparse domination-like ideas can be used to prove weighted inequalities for the Bergman projection on the unit ball.

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The inequality of Berwald is a reverse-Hölder like inequality for the p-th average of a concave function over a convex body in R^n . We prove Berwald’s inequality for averages of concave functions with respect to measures that have some concavity conditions, e.g. s-concave measures, s ∈ [−∞, 1/n]. As applications, we apply shown results to generalizations of the concepts of radial means bodies and the projection body of a convex body.

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Consider a section of an n-dimensional cube of unit volume by an (n-d)-dimensional affine hyperplane. If the distance from the hyperplane to the center of the cube is greater than 1/2, then the section can be empty. We will show that if this distance is 1/2 or less, then the volume of the section is uniformly bounded below by a constant independent of the dimension. This means that the minimal volume of a section undergoes a phase transition as the distance to the center of the cube increases, dropping from a constant level to zero. If time allows, we will discuss a similar phenomenon for sections by subspaces of smaller dimensions. Joint work with Hermann Koenig.

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We show a probabilistic extension of the Oleszkiewicz-Pełczyński polydisc slicing result. The Haagerup-type phase transition occurs exactly when the p-norm recovers volume, in contrast to the real case. Based on joint work with Chasapis and Singh.

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We discuss a randomized construction of a point configuration, which gives a bound for the minimal dispersion on the cube. The bound is close to optimal, and in some regime it is optimal for the Poisson point process. Joint work with Alexander Litvak.

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I will discuss a family of affine isoperimetric inequalities for bodies that interpolate between intersection bodies and dual Lp centroid bodies. The focus will be a common framework for the Busemann intersection inequality and the Lutwak--Zhang inequality. The approach depends on new empirical versions of these inequalities. Based on joint work with R. Adamczak, G. Paouris and P. Simanjuntak.

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We show that in Euclidean 3-space any closed curve which contains the unit sphere in its convex hull has length at least 4pi, and characterize the case of equality, which settles a conjecture of Zalgaller. Furthermore, we establish an estimate for the higher dimensional version of this problem by Nazarov, which is sharp up to a multiplicative constant. Finally we discuss connections with sphere packing problems, and other optimization questions for convex hull of space curves. This is joint work with James Wenk.

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Dual curvature measures for convex bodies were introduced by Huang-Lutwak-Yang-Zhang in 2016. In this talk, we will discuss how this can be naturally extended to the set of log-concave functions. We will also discuss the Minkowski problem for these measures. This is joint work with Yong Huang, Jiaqian Liu, and Dongmeng Xi.

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Photoacoustic tomography (PAT) is a non-invasive imaging modality that requires recovering the initial data of the wave equation from certain measurements of the solution outside the object. In the standard PAT, the measured data consist of time-dependent signals measured on an observation surface. In contrast, the measured data from the recently invented full-field detection technique provide the solution of the wave equation on a spatial domain at a single instant in time. While reconstruction using classical PAT data has been extensively studied, not much is known for the full field PAT problem. I will discuss the mathematical foundations of the latter problem for variable sound speed and its uniqueness, stability, and exact inversion method using time-reversal. Our results demonstrate the suitability of both the full field approach and the proposed time-reversal technique for high resolution photoacoustic imaging.

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We consider a generalization of the classical Whitney extension problem. Let $E\subset \mathbb{R}^n$ be a compact set and let $K(x) \subset \mathbb{R}^d$ be a convex set for each $x \in E$. I will describe a procedure to determine whether or not there exists a $C^m$ selection of $K$, i.e., if there exists $\phi \in C^m(\mathbb{R}^n, \mathbb{R}^d)$ such that $\phi(x)\in K(x)$ for every $x \in E$. This is based on the joint work with Kevin Luli and Kevin O'Neill.

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The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a measure theoretic Brunn-Minkowski theory, we prove the Minkowski existence theorem for a large class of Borel measures with density, denoted by $\Lambda^\prime$: for $\nu$ a finite, even Borel measure on the unit sphere and $\mu\in\Lambda^\prime$, there exists a symmetric convex body $K$ such that $$d\nu(u)=c_{\mu,K}dS_{\mu,K}(u),$$ where $c_{\mu,K}$ is a quantity that depends on $\mu$ and $K$ and $dS_{\mu,K}(u)$ is the surface area-measure of $K$ with respect to $\mu$. Examples of measures in $\Lambda^\prime$ are homogeneous measures (with $c_{\mu,K}=1$) and probability measures with continuous densities (e.g. the Gaussian measure).

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An active area of research is to determine when a set of sufficient Hausdorff dimension contains finite point configurations of some geometric type. In this talk, I will discuss how techniques from harmonic analysis are used to study such problems.

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We provide a Homogenous decomposition Theorem for continuous and translation invariant valuations on convex functions with compact domain. As a consequence of an extension argument, these valuations are the same for super coercive convex functions, a case settled by Colesanti, Ludwig and Mussnig. Joint work with Jonas Knoerr.

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We provide an overview of projection bodies in Gaussian probability space for sets of finite Gaussian perimeter and their corresponding applications in functions of Bounded variation space. On the one hand, we study the properties of Gaussian projection bodies for sets of finite Gaussian perimeter under Ehrhard symmetrization and establish a Gaussian projection type inequality. The inequality concludes that Ehrhard symmetrization contracts the Minkowski sum of the Gaussian projection bodies for set of finite Gaussian perimeter $E$ and its reflection $E^v$. On the other hand, we investigate the functional ``lifting" of Ehrhard symmetrization and establish the affine Gaussian P\'olya-Szeg\"o type inequalities in terms of the functional Ehrhard symmetrization. This is based on a joint work with Prof. Youjiang Lin.

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Section and projection functions of convex bodies are not arbitrary functions; in fact, other than in dimension and codimension one, they span a rather small subspace of all functions on the grassmannian, which exhibits a quasianalytic-type property. This phenomenon holds for a class of integral operators on grassmannians, and more generally for certain representations of the general linear group. As corollaries, we will see sharper versions of Alexandrov's projections theorem, Funk's sections theorem, and Klain's injectivity theorem for even valuations.

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A fundamental result in differential geometry states that if a smooth hypersurface in a Euclidean space encloses a bounded domain and one of its mean curvature functions is constant, then it is a Euclidean sphere. This statement has been referred to as the soap bubble theorem. Major contributions are due to Alexandrov (1958) and Korevaar--Ros (1988). While the smoothness assumption is seemingly natural at first thought, based on the notion of curvatures measures of convex bodies Schneider (1979) established a characterization of Euclidean spheres among general convex bodies by requiring that one of the curvature measures is proportional to the boundary measure. We describe extensions in two directions: (1) The role of the Euclidean ball is taken by a nice gauge body (Wulff shape) and (2) the problem is studied in a non-convex and non-smooth setting. Thus we obtain characterization results for finite unions of Wulff shapes (bubbling) within the class of mean-convex sets or even for general sets with positive reach. Several related results are established. They include the extension of the classical Steiner--Weyl tube formula to arbitrary closed sets in a uniformly convex normed vector space, formulas for the derivative of the localized volume function of a compact set and general versions of the Heintze--Karcher inequality. (Based on joint work with Mario Santilli)

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The L^p dual curvature measure was introduced by Lutwak, Yang, and Zhang in 2018. The associated Minkowski problem, known as the L^p dual Minkowski problem, asks about the existence of a convex body with prescribed L^p dual curvature measure. This question unifies the previously disjoint L^p Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this paper, we prove the existence of a solution to the L^p dual Minkowski problem for the case of q < p + 1, −1 < p < 0, and p≠q for even measures.

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We show isoperimetric inequalities for weighted L_p affine surface areas which appear in the recently established L_p Steiner formula of the L_p Brunn Minkowski theory. We show that they are related to f-divergences of the cone measures of the convex body and its polar, namely the Kullback-Leibler divergence and the Renyi-divergence. Based on joint work with Kateryna Tatarko.

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We will talk about close (say in Hausdorff metric) convex bodies constructions for which the homothety implies an ellipsoid or a ball. (joint work in progress)

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The Gauss Image problem is a generalization to the question originally posed by Aleksandrov who studied the existence of the convex body with the prescribed Aleksandrov's integral curvature. A simple discrete case of the Gauss Image Problem can be formulated as follows: given a finite set of directions in Euclidian space and the same number of unit vectors, does there exist a convex polytope in this space containing the origin in its interior with vertices at given directions such that each normal cone at the vertex contains exactly one of the given vectors. In this talk, we are going to discuss the discrete Gauss Image Problem, and its relation to other Minkowski-type problems. Two different proofs of the problem are going to be addressed: A smooth proof based on transportation polytopes and a discrete proof based on Helly’s theorem. This work is based on the recent results of the author.

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The presence of log-concave sequences is prevalent in diverse areas of mathematics ranging from geometry to combinatorics. The ubiquity of such sequences is not completely understood but the last decade has witnessed major progress towards this goal. However, we know very little about the extremals of such sequences: If we have equality somewhere along the sequence, what can be said about the sequence itself? This question is related to optimal structures (e.g. the ball in the isoperimetric inequality) and it is a necessary step towards the improvement and stability of the inequalities themselves. I will talk about the extremals of such sequences coming from the theory of partially ordered sets (posets). R. Stanley showed in the 80's how to associate polytopes to posets and, using this correspondence (via the Alexandrov-Fenchel inequality), he proved that sequences which count the number of linear extensions of posets are log-concave. The extremals of these sequences were unknown however, with even conjectures lacking. I will explain the resolution of this problem and the complete characterization of the extremals. The extremals turn out to be complicated and rich structures which exhibit new phenomena depending on the geometry of the associated polytopes. Towards the resolution of this problem we developed new tools that shed brighter light on the relation between the geometry of polytopes and the combinatorics of partially ordered sets. Joint work with Zhao Yu Ma.

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Affine fractional isoperimetric inequalities are established that are stronger (and directly imply) the Euclidean fractional isoperimetric inequalities. These inequalities are fractional versions of the Petty projection inequality. Using the functional version of fractional polar projection bodies, affine fractional Sobolev inequalities are established that are stronger that the fractional Sobolev inequalities of Almgren and Lieb and imply (in the limit) the affine Sobolev inequality by Gaoyong Zhang. Joint work with Julián Haddad (Universidade Federal de Minas Gerais)

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Symmetric decreasing rearrangement (when applicable) can reduce a geometric variational problem to a radial problem, where the unknown functions depend on the single variable |x|. Classical inequalities for perimeter, gradient norms, and convolution integrals indicate that symmetric decreasing rearrangement reduces the overall oscillation of functions. Less is known about its effect on the mean oscillation of a function. I will discuss recent result (w. Galia Dafni and Ryan Gibara) on inequalities and continuity properties. The question of sharp inequalities remains open.

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We consider functional intrinsic volumes on convex functions. In many ways these objects behave similarly to the classical intrinsic volumes on convex bodies. However, we will also show where analogies fail. The presented results include characterizations, representations, integral geometry and inequalities and we will see that some classical results can be retrieved from the new ones. Joint work with Andrea Colesanti, Monika Ludwig and Jacopo Ulivelli.

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At the beginning of the talk we say a few words about two distinguished mathematicians, Y. Gordon and N. Tomczak-Jaegermann, who passed away in June 2022. Then we review several results related to the volume ratio and the modified Banach-Mazur distance between convex bodies. Finally, we show that for every convex body $K$ there exists a symmetric convex body $L$ such that for any two projections $P$ and $Q$ of rank $k>\sqrt{n \ln n}$ the volume ratio between $PK$ and $QL$ is large. This is a joint work with D.Galicer, M.Merzbacher, and D.Pinasco.

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This talk will be about a type of discrete isoperimetric inequality and uses projections of polytopes in a fundamental way. A conjecture of Milena Mihail and Umesh Vazirani states that the edge expansion of the graph of every 0/1 polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a 0/1 polytope in R^d is greater than 1 over some polynomial function of d. This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a random 0/1 polytope in R^d is at least 1/12d with high probability. This is joint work with Brett Leroux.

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In this joint work with Philippe Jaming and Abderrazek Karoui our aim is to investigate the quality of approximation of almost time- and almost band-limited functions by its expansion in two classical orthogonal polynomials bases: the Hermite basis and the ultraspherical polynomials bases (which include Legendre and Chebyshev bases as particular cases). This allows us to obtain the quality of approximation in the $L^2$ Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of the Legendre series expansion of the prolate spheroidal wave functions.

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In this talk we will describe how one can detect regularity in Jordan curves through analysis of associated geometric square functions. We will particularly focus on the resolution of a conjecture of L. Carleson. Joint work with Xavier Tolsa and Michele Villa.

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In 2011, Lutwak, Yang and Zhang extended the definition of the Lp-Minkowski convex combination (p ≥ 1) from convex bodies containing the origin in their interiors to all measurable subsets in R n , and as a consequence, extended the Lp-Brunn-Minkowski inequality to the setting of all measurable sets. In this talk, I will present a functional extension of their Lp-Minkowski convex combination— the Lp,s–supremal convolution and the Lp-BMI for measurable sets to the class of Borel measures on R n having 1 s -concave densities, with s ≥ 0. Moreover, the Lp-BMI for product measures with quasi-concave densities, Lp isoperimetric inequalities for general measures, etc, will also be provided under this new definition. This talk is based on a joint work with Dr. Michael Roysdon.

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The utility of log-concavity in asymptotic geometric analyis is well-known. One very fruitful perspective on this condition is provided by the formalism of Γ-calculus due to Bakry and Émery, according to which log-concave measures are simply measures with "nonnegative curvature." In this talk, we will explain this formalism and propose a new method for extending it to the setting of graphs, which yields a replacement for the notion of log-concavity on graphs. As an application, we show that the Poincaré constant of a log-concave sequence decreases along the heat flow, which is a discrete variant of a previous result of Klartag and the speaker.

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We present a Johnson-Lindenstrauss-type dimension reduction algorithm with additive error for incompressible subsets of $\ell_p$. The proof relies on a derandomized version of Maurey’s empirical method and a combinatorial idea of Ball.

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The connections between functionals in the calculus of variations and Minkowksi problems have been first investigated in the '90s by Jerison, and later by Colesanti for the particular case of the torsional rigidity. In this talk we show how the same approach could be employed in the context of the Log-Minkowski problem, and propose how to attack a fractional version of the same problem.

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Functions on the Hamming cube {-1,1}^n can be written as Fourier-Walsh expansions. In this talk, we study an Lp-inequality of Naor relating certain truncations of said Fourier-Walsh expansions, which happen to be conditional expectations, and discrete derivatives. The above result has deep connections with the theory of Lipschitz inclusions between Banach spaces, and it is proven using harmonic analysis tools. We shall investigate Lp-estimates for balanced averages of Fourier truncations in other group algebras, in terms of differential operators acting on them. Our prime example is the free group Fn. Our main inequality relates norms in Lp(LFn), the noncommutative Lp space associated with the group von Neumann algebra of Fn. For our balanced Fourier truncations, we will explore two natural options: conditional expectations and Hilbert transforms. We shall also discuss the right notion of discrete derivative in our group theoretic setting.

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The surface area measure of a convex body controls the way its volume changes under infinitesimal perturbations. In this talk we will discuss a functional analogue of this construction, i.e. measures that control how the integral of a log-concave function changes under perturbations. These were first studied by Colesanti and Fragalà and (implicitly) by Cordero-Erausquin and Klartag. We will show that these measures are well defined for all log-concave functions with no regularity assumptions, and explain how this fact is related to the classical co-area formula. We will also use this result to discuss a functional version of the Minkowski problem.

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Given a multi-valued function defined on a closed subset or Rn with closed convex sets in Rd as values, I would like to describe a procedure to determine if we can smoothly sample this function, i.e., a smooth selection. In the spirit of Fefferman's solution to the Whitney Extension Problem, we turn to a more difficult problem: given a bundle where each fiber is a closed convex set in some polynomial ring, how to tell if this bundle admits a smooth section? It turns out that this generalization has implications in the study of constrained extension and of solving linear systems on varieties. I hope to discuss two key ingredients in the solution -- the Brudnyi-Shvartsman Finiteness Principle and the Glaeser refinement, and the not-so-well-understood notion of Whitney convexity, which measures how a convex set of polynomials deviates from a polynomial ideal. If time permits, I will say a word about the technicality induced by the presence of boundaries in a convex set.

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We discuss upper bounds for the Fisher information in high dimensions in terms of the total variation and norms in Sobolev spaces.

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Finding a simple description of a convex hull of a set K in n-dimensional Euclidean space is a basic problem in mathematics. When K has some additional geometric structures one may hope to give an explicit construction of its convex hull. A good starting point is when K is a space curve. In this talk I will describe convex hulls of space curves which have a "very" positive torsion. In particular, we obtain a parametric representation of the boundary of the convex hull, different formulas for their Euclidean volumes and the surface areas, and the solution to a general moment problem corresponding to such curves. This is joint work with Jaume de Dios Pont and Jose Ramon Madrid Padilla.

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I will discuss several conjectures about level sets of eigenfunctions in convex domains.

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While there is extensive literature on approximation, deterministic as well as random, of general convex bodies in the symmetric difference metric, or other metrics coming from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance. For a polygon Q in the plane, the convex hull of n points chosen at random on the boundary of Q gives a random polygon Q_n. We determine the exact limiting behavior of the expected Hausdorff distance between Q and a random polygon Q_n as the number n of points chosen on the boundary of Q goes to infinity. Based on joint work with J. Prochno, C. Schuett and M. Sonnleitner.

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Generalized Probabilistic Theories (GPTs) are theories of nature that have random features. A GPT must specify the set of states purporting to represent the physical reality, the allowable measurements, the rules for outcome statistics of the latter, and the composition rules describing what happens when we merge subsystems and create a larger system. Examples include classical probability and quantum theory. The composition rules alluded to above usually involve tensor products, including tensor products of normed spaces, convex sets and of cones. Among tensor products that have operational meaning in the GPT context, the projective and the injective product are the extreme ones, which leads to the natural question "How much do they differ?" considered already by Grothendieck and Pisier (in the 1950s and 1980s). We report on quantitative results concerning projective/injective discrepancy for finite-dimensional normed spaces. Some of the results are essentially optimal, but others can be likely improved. The methods involve a wide range of techniques from geometry of Banach spaces and random matrices. We also report on parallel results in the context of cones. Finally, we will encourage a more systematic study of convex bodies with the allowed morphisms being projective transformations. Joint work with G. Aubrun, L. Lami, C. Palazuelos, A. Winter (and a parallel work by a subset of co-authors and M. Plavala).

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I will provide a survey of large deviation principles for projections of high-dimensional convex bodies. This is based on various joint works with N. Gantert, S.S. Kim, Y.-T. Liao, P. Lopatto and D. Xie.

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Many of us use smartphones and rely on tools like auto-complete and spelling auto-correct to make using these devices more pleasant, but building these tools presents a challenge. On the one hand, the machine-learning algorithms used to provide these features require data to learn from, but on the other hand, who among us is willing to send a carbon copy of all our text messages to device manufacturers to provide that data? "Local differential privacy" (LDP) and related models have become the gold standard for understanding the tradeoffs possible between utility and privacy loss. In this talk we present a new LDP mechanism for estimating data histograms over large numbers of users, making use of projective geometry together with a dynamic programming based reconstruction algorithm. I will also mention the opportunity for tools from this community to have impact in mobile devices, e.g. the SQKR mechanism of [Chen, Kairouz, Ozgur'20] on private mean estimation using work on Kashin representations by Lyubarskii and Vershynin. This talk is based on joint work with Vitaly Feldman (Apple), Huy Le Nguyen (Northeastern), and Kunal Talwar (Apple).

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Using aspects of Talagrand's generic chaining theory, we show how to construct spectral hypergraph eps-sparsifiers with only O(eps^{-2} log(r) n log n) hyperedges, where n is the number of vertices and r is the rank of the hypergraph.

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A well known conjecture of Burr and Erdos asserts that the Ramsey number of the hypercube on 2^n vertices is of the order O(2^n). Motivated by this problem, we construct randomized embeddings of the hypercube into dense bipartite graphs and, as a corollary, show that the Ramsey number of the hypercube is of order O(2^{2n−cn}) for a universal constant c>0. This improves upon the previous best known bound O(2^{2n}), due to Conlon, Fox and Sudakov.

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We shall discuss results and open questions pertaining to the regularity (and irregularity) of solutions of weighted isoperimetric-type problems over the class of symmetric convex sets. Based on joint work with G. Moschidis (EPFL)

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The Gaussian Elimination with Partial Pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Although in specific cases the loss of precision in GEPP due to roundoff errors can be very significant, empirical evidence strongly suggests that for a typical square coefficient matrix, GEPP is numerically stable. We obtain a (partial) theoretical justification of this phenomenon by showing that, given the random n x n standard Gaussian coefficient matrix A, the growth factor of the Gaussian Elimination with Partial Pivoting is at most polynomially large in n with probability close to one. This implies that with high probability the number of bits of precision sufficient to solve Ax=b to m bits of accuracy using GEPP is m + O(log(n)), which improves an earlier estimate m + O( log^2 n) of Sankar, and which we conjecture to be optimal by the order of magnitude. We further provide tail estimates of the growth factor which can be used to support the empirical observation that GEPP is more stable than the Gaussian Elimination with no pivoting. This talk is based on a joint work with Konstantin Tikhomirov.

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Let $X=(\xi_1, ..., \xi_m)$ be a centered Gaussian random vector, such that the variances of each $\xi_i$ equals to 1. Under what assumptions on the covariance matrix is the expectation of $\min_i |\xi_i|$ minimized? We discuss known results and conjectures related to this question.

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We will consider several classical problems for hypoelliptic diffusions and random walks: the large deviations principle (LDP), the small ball problem (SBP), Chung’s law of iterated logarithm (LIL), and finding the Onsager-Machlup functional. As two very different examples we will look at hypoelliptic Brownian motion and the corresponding random walk on the Heisenberg group, and the Kolmogorov diffusion. We will explore the role of space-time scaling property, Gaussianity, and spectral properties via Dirichlet forms in these settings. The Onsager-Machlup functional is used to describe the dynamics of a continuous stochastic process, and it is closely related to the SBP and LIL, as well as the rate functional in the LDP. Unlike in the elliptic (Riemannian) case we do not rely on the tools from differential geometry such as comparison theorems or curvature bounds as these are not always available in the hypoelliptic (sub-Riemannian) setting. The talk is based on the joint work with Marco Carfagnini, Tai Melcher and Jing Wang.

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Random polynomials have been studied since the early 1900s with notable publications by Erdos, Kac, Littlewood, Offord, and others. In recent years, universality has emerged as a powerful method that systematically tackles long-standing problems on the distribution of roots. In this talk, I will discuss a general framework to establish universality properties. I shall go over the application of this framework to different families of polynomials and how to use these properties to answer classical questions in the field. This is based on joint work with Yen Do, Doron Lubinsky, Hoi Nguyen, Igor Pritsker, and Van Vu.

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A basic problem in probability theory and engineering is finding a way of representing a complicated probability measure as a simpler probability measure under some transformation. A desirable property of such transformations is that it is Lipschitz, since it allows for information from the simpler probability measure to be transferred to the complicated measure. While various transformations (optimal transport, Knothe-Rosenblatt rearrangement) exist, establishing their regularity is a difficult problem. In this talk, I will discuss the Lipschitz properties of the Langevin transport map which is constructed infinitesimally along the Langevin dynamics. I will show that this map is Lipschitz in many settings where no other Lipschitz transport maps are known to exist. I will conclude the talk by introducing a new connection between the Langevin transport map and renormalization groups methods from quantum and statistical field theories.

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The Brunn-Minkowski inequality fails dramatically on the hyperbolic plane. We show that this can be remedied by defining Minkowski summation with horocycles instead of geodesics. Joint with Bo'az Klartag.

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We will show that for any even log-concave measure \mu and any pair of symmetric convex sets K and L, and any t between 0 and 1, one has the inequality: \mu(tK+(1-t)L)^{c(n)}\geq t\mu(K)^{c(n)}+(1-t)\mu(L)^{c(n)}, Where c(n)=n^{-4-o(1)}. This constitutes progress towards the Dimensional Brunn-Minkowski conjecture.

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We discuss the method of bounding suprema of canonical processes based on the inclusion of their index set into a convex hull of a well-controlled set of points. While the upper bound is immediate, the reverse estimate was established to date only for a narrow class of regular stochastic processes (for which the chaining method works). We show that for specific index sets, including arbitrary ellipsoids, regularity assumptions may be substantially weakened.

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A positive correlation inequality is established for circular-invariant plurisubharmonic functions, with respect to complex Gaussian measures. The main ingredients of the proofs are the Ornstein-Uhlenbeck semigroup, and another natural semigroup associated to the Gaussian dbar-Laplacian. Joint work with Franck Barthe.

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Let P be a bounded polyhedron, defined as the intersection of the non-negative orthant in R^n and an affine subspace of codimension m. I present a simple and computationally efficient formula that approximates the volume of P within a factor c^m, where c > 0 is an absolute constant. This is joint work with Mark Rudelson.

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We prove that Bourgain’s hyperplane conjecture and the Kannan-Lovasz-Simonovits isoperimetric conjecture hold true up to a factor that is polylogarithmic in the dimension.

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We refine the recent breakthrough technique of Klartag and Lehec to obtain an improved polylogarithmic bound for the KLS constant.

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We revisit the notion of noise stability in the hypercube and show how one can replace the usual heat semigroup with more general stochastic processes. We will then introduce a re-normalized Brownian motion, embedding the discrete hypercube into the Wiener space, and analyze the noise stability along its paths. Our approach leads to a new quantitative form of the 'Majority is Stablest' theorem from Boolean analysis and to progress on the 'most informative bit' conjecture of Courtade and Kumar.

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This talk will mainly be about applications of complex function theory to inverse spectral problems for canonical systems, which constitute a broad class of second order differential equations. I will start with the basics of Krein-de Branges theory, then present an algorithm for inverse spectral problems developed by Makarov and Poltoratski for locally-finite periodic spectral measures. I will then extend the algorithm to certain classes of non-periodic spectral measures and present several examples. If time permits, I will talk about the connection between inverse spectral problems and nonlinear Fourier transforms. This is joint work with Alexei Poltoratski.

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An isoperimetric inequality for the Lp centroid body for p≥1 was first proved by Lutwak, Yang, and Zhang, which extends the inequality for the dual Lp centroid body by Lutwak and Zhang. We show that the volume of the randomized dual Lp centroid body is also maximized by the Euclidean ball for certain values of p<1, which implies the isoperimetric inequality for the deterministic object. We use a probabilistic approach which associates certain random star bodies to the dual Lp centroid body. In this talk, we will discuss the tools used in the randomized framework, focusing on the case p=0.

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A classic problem in geometric measure theory is the Falconer distance problem, which asks how large the Hausdorff dimension of a set must be to ensure that the pairwise distances between points has positive Lebesgue measure. More generally, one may ask how large the Hausdorff dimension must be to generate more complicated geometric patterns. The main framework for studying such problems is to look at the Fourier transforms of measures supported on the set, which reduces the study of geometric configurations to harmonic analysis. I will give a general overview of this class of problems, and prove that sets of large dimension span a positive measure worth of areas.

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In this talk I will review known results about polynomially integrable bodies as well as present new developments. A body $K$ in $\mathbb R^n$ is called polynomially integrable if the Radon transform of its indicator function $R\chi_K(\xi, t)$, where $\xi \in S^{n-1}$ and $t\in \mathbb R$, is a polynomial of $t$ on its support. It was shown some time ago that the only bodies with this property are ellipsoids in odd dimensions. In even dimensions such bodies do not exist. Thus it is natural to ask the following question. What condition do we need to impose on $R\chi_K(\xi, t)$ in order to obtain ellipsoids in even dimensions?

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The Brascamp-Lieb inequality gives a bound for the variance of a function with respect to a log-concave measure, in terms of the gradient of this function, and the (inverse of the) Hessian of the potential of the measure. This beautiful inequality is connected to many isoperimetric-type inequalities in convexity. In this talk, I will first quickly remind the sketch of the L2 proof of this inequality, and then tell several curious facts related to it (some of which are published/posted, others will be posted soon, and some are parts of various early stage on-going projects). Namely, I will explain the equality case characterization for the convex body version of this inequality. I will also explain the novel strengthening of it in the unconditional case. I will point out a curious generalization of it “with a concave function”. I will explain another strengthening (with a dimensional factor) of it in some particular cases. I also explain an interesting strengthening of it in the Gaussian case for linear functions, related to Ehrhard’s inequality, and maybe mention its consequences. Finally, I will sketch some proofs upon the audience’s choice. PS: There is many other inequalities in Analysis under the name Brascamp-Lieb, notable the one which is a version of reverse Holder inequality in which gaussians are extremizers. This inequality is not related to the one we discuss in any way, apart from the fact that the same people proved it, so I hope there will be no false expectations from people interested in this other Brascamp-Lieb inequality:)

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In a celebrated paper in 1973, Lieb proved what we now call Lieb's Concavity Theorem and resolved a conjecture of Wigne, Yanase and Dyson. This result has found many applications in mathematical physics and quantum information theory. In recent years, when studying the data processing inequalities for alpha-z Rényi relative entropies in quantum information, Audenaert and Datta conjectured that certain trace functionals are jointly convex. Later on, a stronger conjecture was made by Carlen-Frank-Lieb when reviewing the related problems. In this talk, I will present a simple variational method to study the concavity/convexity of trace functionals. This allows us to resolve these two conjectures and recover many known results easily. The talk is based on arXiv:1811.01205.

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Two subsets of a Riemannian surface grow with time. The part of the surface conquered by each set remains in the set forever. Will both sets keep growing indefinitely, or will one of them trap the other in a bounded region? We shall discuss the connection between the answer to this question and the geometry of the surface in the case of rotationally symmetric surfaces.

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We consider generalizations of several questions of convex geometry to the setting of arbitrary measures in place of volume.

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Masani and Wiener asked to characterize the regularity of vector stationary stochastic processes. The question easily translates to a harmonic analysis question: for what matrix weights the Hilbert transform is bounded with respect to this weight? We solved this problem with Sergei Treil in 1996 introducing the matrix A_2 condition. But what is the sharp estimate of the Hilbert transform in terms of matrix A_2 norm is still unknown in a striking difference with scalar case. Convex body valued operators helped to get the estimate via norm in the power 3/2. But shouldn't it be power 1? We construct an example of a rather natural operator for which the estimate in scalar and vector case is indeed different. But it is neither the Hilbert transform, nor a sparse Lerner transform.

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In this talk, we will discuss order reversing quasi involutions, which are dualities on their image, and their properties. We prove that any order reversing quasi involution can be represented in a special (and useful) form. We will provide with many examples and ways to construct new order reversing quasi involutions from given ones. In particular, we will analyse the example of ‘dual polarity’. This talk is based on joint work with Shiri Artstein-Avidan and Shay Sadovsky.

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In this talk we will discuss the interplay between the sharp range of parameters for each one has sparse domination for certain spherical maximal functions and the sharp Lp improving boundedness region of corresponding localized spherical maximal operators, which first appeared in a work of Lacey. I will then talk about a joint work with B. Foster, Y. Ou, J. Pipher, and Z. Zhou, in which we proved sparse domination results for a bilinear generalization of the spherical maximal function in any dimension d ≥ 2, and in dimension 1 for its lacunary version. Such sparse domination results allows one to recover the known sharp Lp × Lq → Lr bounds for the bilinear spherical maximal operator and to deduce new quantitative weighted norm inequalities with respect to bilinear Muckenhoupt weights.

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We estimate the Banach-Mazur distance of $\ell_{p}^{n^{2}}$ and $\ell_{q}^{n}\otimes_{\epsilon}\ell_{r}^{n}$.

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In this talk we will discuss old and not so old inequalities on the volume of the orthogonal projections (sometimes called local Loomis-Whitney type estimates). We will explore connections of those inequalities to inequalities for Mixed Volumes. We will also show the links between those inequalities and a number of interesting inequalities in Convex Geometry which are inspired by sumsets estimates in additive combinatorics and classical facts from the information theory. This is a joint work with Matthieu Fradelizi, Mokshay Madiman and Mathieu Meyer.

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In this talk we discuss an extension of a result of Rudelson concerning an inequality of the Rogers-Shephard type for sections. The original motivation of Rudelson's result was to find a bound for the MM*-estimate for non-symmetric convex bodies. Here we present extensions of his result to the case of measures and log-concave functions. We will also pose some open questions.

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The classical isoperimetric theorem states that among all borel sets in $\mathbb{R}^n$ with a given volume, the euclidean unit ball has the minimal surface. Many other isoperimetric theorems are known, such as the gaussian isoperimetric theorem and the vertex isoperimetric theorem on the hypercube. In this talk we report on a paper that establishes a vertex isoperimetric theorem for the cartesian product of complete graphs. Analogous to the euclidean ball, the theorem says that the hamming ball minimizes the vertex neighborhood over all vertex subsets of a fixed size.

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In this talk, the framework of Lp operations for functions including the extension of convolution sum and the Asplund sum for functions will be presented. Based on the properties of these summations for functions, we establish the Lp Borell-Brascamp-Lieb inequalities and discover the integral formula for Lp mixed quermassintegral for functions. This talk is based on a joint work with Michael Roysdon.

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We look at two weight inequalities for CZO operators on spaces of homogeneous type including their weighted generalizations to multilinear CZO operators.

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We investigate existence of isometric copies of “many” dilates of products of given non-degenerate simplices, in subsets of positive Lebesgue measure of the unit cube. We obtain a quantitative lower bound on the largeness of the family of dilates whose isometric copies are detected in the set. We approach the problem via harmonic analysis, passing through certain cancellation estimates for multilinear singular integrals associated with hypergraphs. This is joint work with Mario Stipčić.

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In this talk, I will report some recent work on the stability for a class of Caffarelli-Kohn-Nirenberg (CKN) inequalities in Euclidean spaces. By establishing a parameter family of Caffarelli-Kohn-Nirenberg identities, we prove sharp stability for a class of CKN inequalities (including the Heisenberg uncertainty principle) with optimal constants. Moreover, we also show that there exist extremal functions for these sharp stable CKN inequalities. This is joint work with C. Cazacu, J. Flynn and N. Lam.

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Almgren-type monotonicity formulas In this talk we will explore the celebrated Almgren’s monotonicity formula. This beautiful result with far reaching consequences states that if u is harmonic in the unit ball, then a certain frequency function N(r) is non-decreasing. Moreover, N(r)=k for all 0<r<1 if, and only if, u is homogeneous of degree k. We will then discuss some of the many applications of this formula, and recent developments connected to it.

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Projections detect information about the size, geometric arrangement, and dimension of sets. In recent years, there has been significant interests in determining the rates of decay of the classical Favard length (or average orthogonal projection length) for various fractal sets. For orthogonal projections, quantitative estimates rely on a separation condition: most points are well-differentiated by most projections. It turns out that this idea also applies to a broad class of nonlinear projection-type operators satisfying a transversality condition. This begs the question of obtaining quantitative upper & lower bounds for decay rates for nonlinear variants of Favard length, including Favard curve length (as well as a new generalization to higher dimensions, called Favard surface length) and visibility measurements associated to radial projections. As one application, we consider the decay rate of the Favard curve length of generations of the four corner Cantor set, first established by Cladek, Davey, and Taylor. Our upper bound utilizes the seminal work of Nazarov, Peres, and Volberg, while energy techniques play a role in achieving a lower bound.

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The extremizers for the endpoint Stein-Tomas Fourier restriction estimate to the sphere in three dimensions are known. I will present an old proof with a new twist. The extremizers for endpoint Stein Tomas Fourier restriction to the circle are not known. I will present some numerical evidence and some ideas on this problem.

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Extending the Sobolev theory of quasiconformal and quasiregular maps in the complex plane to subdomains motivates our investigation of Sobolev boundedness of truncated Calderón-Zygmund operators. We introduce certain Carleson measures and give a complete weighted Sobolev theory in some special situations. In particular, we have weighted Sobolev estimates for the truncated Beurling transform which imply self-improving Sobolev regularity for certain quasiregular distributions.

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In this talk, we will present an open question by Saroglou, Soprunov and Zvavitch, which states the $n$-simplex is the only minimizer of a certain set of inequalities (Bezout inequalities). At the other extreme, one may ask which convex bodies maximize this set of inequalities. We show the cube is a maximizer, and state open questions for extremizers of some related inequalities. Regarding the original Bezout inequalities, uniqueness is still an open question, for both extremes.

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The inequality of Berwald is a reverse-Hölder like inequality for the p-th average of a concave function over a convex body in R^n . We prove Berwald’s inequality for averages of concave functions with respect to measures that have some concavity conditions, e.g. s-concave measures, s ∈ [−∞, 1/n]. As applications, we apply shown results to generalizations of the concepts of radial means bodies and the projection body of a convex body.

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The Busemann intersection inequality is a fundamental isoperimetric inequality for intersection bodies. I will discuss a new stochastic approach based on a random approximation inspired by a construction of Anttila, Ball, and Perissinaki. In particular, we show how symmetrization methods apply to empirical approximations of average volume of central sections. Based on a joint work with P. Pivovarov.

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Consider a vector space and a finite group acting unitary on that space. We study the general problem of constructing a stable embedding. The domain of this embedding is the quotient of the vector space modulo the group action and the target space is an Euclidean space.

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We calculate the average decay rate of the Favard length of certain disk-like planar Cantor set. Our result is the same as the case of the random 4-corners Cantor set studied by Peres and Solomyak.

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In this talk, the Musielak-Orlicz-Gauss image measure for a convex body is proposed. Such a measure can be produced by a variational formula of the general dual volume of a convex body under the perturbation of the Musielak-Orlicz addition. The Musielak-Orlicz-Gauss image problem contains many intensively studied Minkowski type problems and the recent Gauss image problem as its special cases. Under the condition that the Musielak-Orlicz function is decreasing, the existence of solutions to this problem is established. This talk is based on a joint work with Dr. Qingzhong Huang, Deping Ye and Baocheng Zhu.

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In this talk, I will report on some results joint with Almut Burchard and Galia Dafni regarding boundedness and continuity of the decreasing and symmetric decreasing rearrangements on function spaces defined by mean oscillation.

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The arithmetic-harmonic mean inequality can be generalized for convex sets, considering the intersection, the harmonic and the arithmetic mean, as well as the convex hull of two convex sets. We study those relations of symmetrization of convex sets, i.e., dealing with the means of some convex set C and -C. We determine the dilatation factors, depending on the asymmetry of C, to reverse the containments between any of those symmetrizations, and tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.

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We introduce a generalization of the classical ellipticity (or accretivity) condition for complex matrices, explain its provenance and argue that it may be useful for the Lp theory of elliptic PDE with complex coefficients. To this purpose we present a couple of examples which were obtained in the last several years. The talk is based on collaboration with Andrea Carbonaro.

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This talk gives an easy review of sparse domination and extends it to a setting with a continuous index. In addition, a process 'with infinite memory' is dominated via a novel (non-)stopping procedure. This process targets a model by X.D. Li for the Riesz vector by Bakry. As an application, we discuss the dimensionless L^p estimates for said Riesz vector. This is a novel proof (almost) free of any Bellman function and in some cases it extends to a dimensionless bound in the weighted setting (with appropriate classes of weights).

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We present a dyadic model Hilbert transform of Haar-shift type that allows new norm estimates.

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One of complexity theory’s “greatest hits” is Håstad‘s Fourier-analytic proof that constant-depth Boolean circuits cannot approximate the Parity function. We extend this argument to the case of constant-depth quantum circuits. Connections to other open questions in Analysis of Boolean Functions, such as the approximate degree of AC0, are highlighted.

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Inequalities for maximal operators play a foundational role in mathematics, and the question about optimal (or at least tight) constants involved is of significant importance for applications. The purpose of the talk will be to survey several recent results in this direction, both in the classical and the non-commutative setting. The main emphasis will be put on the weighted context.

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We study weighted norm inequalities for families of operators depending on a parameter, $\varepsilon=\{\varepsilon_Q\}_{Q\in\mathcal{D}}$, representing a sequence of real numbers indexed by a dyadic system $\mathcal{D}$ in $\mathbb{R}^n$. We give necessary and sufficient conditions describing the weights for which such operators satisfy the corresponding weighted strong-type and weak-type bounds. Our results use a more general modification of the classical Muckenhoupt $A_p$ condition involving the parameter $\varepsilon$.

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The basic question we are going to ask is, how large does the Hausdorff dimension of a subset of Euclidean space need to be to ensure that it contains a similar copy of a given finite point configuration? In vector spaces over finite fields, a similar question can be asked in terms of the size of a subset of the vector space. We are also going to explore these and similar problems from the standpoint of the Vapnik-Chervonenkis dimension which, in some sense, points to the most complicated configurations one can hope to construct under a given set of constraints. Connections with the previous work by Falconer, Bourgain, Furstenberh-Katznelson-Weiss, and others will be described.

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Inspired by the work of (Charles) Fefferman and Stein concerning real Hardy spaces, (Robert) Fefferman, Kenig and Pipher (FKP) produced a characterization of Muckenhoupt weights in terms a Carleson measure condition for the heat extension of a doubling measure. Fefferman, Kenig and Pipher used this characterization to show their main theorem was sharp, by producing a counterexample (an elliptic measure that did not satisfy their Carleson condition). Though this was not their main theorem, this Carleson characterization of weights has inspired other very interesting works. Dyadic versions of this characterization are the genesis of the Bellman function technique. Recently, with Toro and Zhao, I directly connected the FKP Carleson condition to certain elliptic measures. Because of this connection, I reinvestigated the FKP Carleson condition with Egert and Saari and we proved what appear to be sharp `small constant’ bounds for the inequality we are most interested in. However, there are still adjacent questions that remain; in particular, those that have been resolved in the dyadic setting using the Bellman function technique.

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We talk about some specific discrete optimization problems, like best-packing or best-covering, on sets of low smoothness; for example, fractal sets. We will, in particular, study the behavior of the optimizers depending on how bad the fractal set is.

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We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group and then by Lieb and Solovej for SU(2) and by Kulikov for SU(1,1) and the affine group. In this paper, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber--Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1,1) cases.

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In this talk I will present some recent results from joint work with David Beltran and Andreas Seeger concerning bilinear sparse domination bounds for large classes of Fourier multipliers. Applications include endpoint sparse domination results for classical oscillatory multipliers, Miyachi classes and certain multiscale radial bumps.

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I will speak about vector valued Kahn--Kalai--Linial (KKL) inequality for functions with values in a Banach spaces having Rademacher type 2. This is joint work with Yonathan Stone.

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Quite often we wonder about the sharpness of estimates for certain singular integral operators. In theory, their sharpness can be confirmed by constructing extremizers or approximate extremizers, but, in practice, such extremizers might not be obvious, or they might be impossibly complicated to work with. In this talk we will discuss a reasonably general way of proving lower bounds for the exact L^p norms of unimodular homogeneous Fourier multipliers. We will then apply it to solve three open problems: one by Iwaniec and Martin (from 1996) on the powers of the complex Riesz transform, one by Maz'ya (traced back to the 1970s) on multipliers with smooth phases, and one by Dragičević, Petermichl, and Volberg (from 2006) on the two-dimensional Riesz group. This is joint work with Aleksandar Bulj, Andrea Carbonaro, and Oliver Dragičević.

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We discuss the characterization of compactness for the sparse operator (associated with symbol in weighted VMO space) in the two-weight setting on the spaces of homogeneous type. As a direct application we obtain the compactness characterization for the maximal commutators with respect to the weighted VMO functions and the commutator of Calderon–Zygmund operators on the homogeneous spaces. We will look at the applications of this approach to multilinear Bloom setting.

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In this talk, I will describe an elliptic PDE that models electric conduction, and the electric field concentration phenomenon between closely spaced inclusions of high contrast. In the first part, I will present some results on the insulated conductivity problem (jointly with Hongjie Dong and Yanyan Li). We obtained an optimal gradient estimate in terms of the distance between inclusions. This solved one of the major open problems in this area. In the second part, I will discuss a recent work regarding optimal estimates for higher derivatives of the conductivity problem with circular inclusions in 2D, when the relative conductivities of inclusions have different signs (Jointly with Hongjie Dong). This improves a recent result of Ji and Kang.

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An integrable function and its Fourier transform cannot both have support of finite measure. Amrein-Berthier theorem is an inequality from the late 70s refining the latter statement. We will present this classical theorem, give ideas of the proof, state one or two corollaries, mention recent counter-examples to some strengthening of the theorem, as well as an open question almost 30 years old.

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This work devises a new smooth representation formula for the compression of CZ operators on domains. As a first order consequence of this representation, we obtain a weighted, sharply quantified T(1)-type theorem on Sobolev spaces. Previous results of Prats and Prats-Tolsa are limited to unweighted bounds for convolution-type operators. Our weighted Sobolev inequalities are subsequently applied to obtain quantitative regularity results for solutions to the Beltrami equation with symbol in the critical class W^{k,2}(Omega). Alll past results, due to Prats among others, based on the Iwaniec scheme are of qualitative nature. Talk is based on current and ongoing joint work with Walton Green and Brett Wick (WUSTL)

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