# Dimension reduction for incompressible subsets of $\ell_p$

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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.

# On a j-Santaló conjecture

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Let $k\geq 2$ be an integer. In the spirit of Kolesnikov-Werner, for each $j\in\{2,\ldots,k\}$, we conjecture a sharp Santaló type inequality (we call it $j$-Santal\'{o} conjecture) for many sets (or more generally for many functions), which we are able to confirm in some cases, including the case $j=k$ and the unconditional case. Interestingly, the extremals of this family of inequalities are tuples of the $l_j^n$-ball.

Our results also strengthen one of the main results of Kolesnikov-Werner, which corresponds to the case $j=2$. All members of the family of our conjectured inequalities can be interpreted as generalizations of the classical Blaschke-Santaló inequality.

Related, we discuss an analogue of a conjecture due to K. Ball in the multi-entry setting and establish a connection to the $j$-Santaló conjecture.

# Noncommutative Bohnenblust--Hille inequalities and applications to quantum learning

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Bohnenblust--Hille (BH) inequalities are an extension of Littlewood's 4/3 inequality and have found many applications to harmonic analysis. A variant of BH inequalities for Boolean cubes has been proven with constants that are dimension-free and subexponential in degree. Such inequalities have found great applications in learning low-degree Boolean functions. Motivated by learning quantum observables, a quantum analog of BH inequality for Boolean cubes was recently conjectured and resolved unaware of the conjecture. In this talk, we give a simpler proof with better constants. As applications, we study learning problems for quantum observables of low degrees. Joint work with Alexander Volberg.

# Continuous analytic capacity, rectifiability and holomorphic motions

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In this talk, I will present several geometric and analytic characterizations of purely unrectifiable quasicircles. These necessary and sufficient conditions are expressed in terms of various notions such as Dirichlet algebras, harmonic measure, analytic capacity and continuous analytic capacity. As an application, I will explain how to construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion. This answers a question raised by Paul Gauthier.

# A stationary set method for estimating oscillatory integrals

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Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a ``stationary set'' method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

# The characteristic polynomial of sums of random permutations

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Let $A_n$ be the sum of $d$ permutations matrices of size $n×n$, each drawn uniformly at random and independently. We prove that $\det( I_n−zA_n/\sqrt{d})$ converges when $n\to\infty$ towards a random analytic function on the unit disk. As an application, we obtain an elementary proof of the spectral gap of random regular digraphs with a sharp constant. Our results are valid both in the regime where $d$ is fixed and for $d$ slowly growing with $n$. Joint work with Simon Coste and Gaultier Lambert.

# Sharp multiplicative inequality on BMO

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We find the sharp constant C in the inequality $\|\phi\|_{L^r} \leq C \|\phi\|_{L^p}^{p/r} \|\phi\|_{BMO}^{1-p/r}$, where $1\leq p\leq r<+\infty$. We use the Bellman function machinery to solve this problem. The Bellman function of three variables corresponding to this problem has a rather complicated structure, however, we managed to provide the explicit formulas for this function. Based on joint works with D. Stolyarov, V. Vasyunin, and I. Zlotnikov.

# Measures with given marginals and information economics

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There is a connection between the problem of Bayesian persuasion (an informed agent aims to induce desirable behavior of uninformed ones by tailoring the information available to them) and feasibility questions for measures with given marginals. We will discuss the two problems, their connection, and related open questions.

The talk is based on two papers:

"Feasible Joint Posterior Beliefs" with Itai Arieli, Yakov Babichenko, and Omer Tamuz

and "Private Private information" with Kevin He and Omer Tamuz.

# Decidability and periodicity of translational tilings

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Translational tiling is a covering of a space using translated copies of some building blocks, called the “tiles”, without any positive measure overlaps. Which are the possible ways that a space can be tiled? In the talk, we will discuss the study of this question as well as its applications, and report on recent progress, joint with Terence Tao.