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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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