# Endpoint sparse domination for oscillatory Fourier multipliers

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Sparse domination was first introduced in the context of Calderón--Zygmund theory. Shortly after, the concept was successfully extended to many other operators in Harmonic Analysis, although many endpoint situations have remained unknown. In this talk, we will present new endpoint sparse bounds for oscillatory and Miyachi-type Fourier multipliers using Littlewood—Paley theory. Furthermore, the results can be extended to more general dilation-invariant classes of multiplier transformations via Hardy space techniques, yielding results, for instance, for multi-scale sums of radial bump multipliers.

This is joint work with Joris Roos and Andreas Seeger

# An integer parallelotope with small surface area

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We will prove that there exists an n-dimensional convex body whose surface area is at most n^{1/2} times a lower order factor, yet its translates by the integer lattice tile space.

This is joint work with Oded Regev.

# Szemeredi regularity, matrix decompositions, and covariance loss

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We will discuss a new kind of weak Szemeredi regularity lemma. It allows one to decompose a positive semidefinite matrix into a small number of "flat" matrices, up to a small error in the Frobenius norm. The proof utilizes randomized rounding based on Grothendieck’s identity. The regularity lemma can be interpreted as a probabilistic statement about "covariance loss" – the amount of covariance that is lost by taking conditional expectation of a random vector. This talk is based on a joint work with March Boedihardjo and Thomas Strohmer.

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# Moments, concentration, and entropy of log-concave distributions

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In this talk I will present a simple mechanism, combining log-concavity and majorization in the convex order to derive moments, concentration, and entropy inequalities for random variables that are log-concave with respect to a reference measure