# On the Fourier-Entropy Conjecture

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Characterizing Boolean functions with small total influence is one of the most fundamental questions in analysis of Boolean functions.

The seminal results of Kahn-Kalai-Linial and of Friedgut address this question for total influence $K = o(\log n)$, and show that

a function with total influence $K$ (essentially) depends on $2^{O(K)}$ variables.

The Fourier-Entropy Conjecture of Friedgut and Kalai is an outstanding conjecture that strengthens these results, and remains

meaningful for $K \geq \log n$. Informally, the conjecture states that the Fourier transform of a function with total influence $K$,

is concentrated on at most $2^{O(K)}$ distinct characters.

In this talk, we will discuss recent progress towards this conjecture. We show that functions with total influence $K$ are concentrated

on at most $2^{O(K\log K)}$ distinct Fourier coefficients. We also mention some applications to learning theory and sharp thresholds.

Based on a joint work with Esty Kelman, Guy Kindler, Noam Lifshitz and Muli Safra.

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# Kakeya maximal estimates via real algebraic geometry

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The Kakeya (maximal) conjecture concerns how collections of long, thin tubes which point in different directions can overlap. Such geometric problems underpin the behaviour of various important oscillatory integral operators and, consequently, understanding the Kakeya conjecture is a vital step towards many central problems in harmonic analysis. In this talk I will discuss work with K. Rogers and R. Zhang which apply tools from the theory of semialgebraic sets to yield new partial results on the Kakeya conjecture. Also, more recent work with J. Zahl has used these methods to improve the range of estimates on the Fourier restriction conjecture.

# L^p improving bounds for spherical maximal operators

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Consider families of spherical means where the radii are restricted to a given subset of a compact interval. One is interested in the L^p improving estimates for the associated maximal operators and related objects. Results depend on several notions of fractal dimension of the dilation set, or subsets of it. There are some unexpected statements on the shape of the possible type sets. Joint works with J. Roos, and with T. Anderson, K. Hughes and J. Roos.

# Matrix-valued logarithmic Sobolev inequalities

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Logarithmic Sobolev inequalities (LSI) first were introduced by Gross in 1970s as an equivalent formulation of hypercontractivity. LSI have been well studied in the past few decades and found applications to information theory, optimal transport, and graphs theory. Recently matrix-valued LSI have been an active area of research. Matrix-valued LSI of Lindblad operators are closely related to decoherence of open quantum systems. In this talk, I will present recent results on matrix-valued LSI, in particular a geometric approach to matrix-valued LSI of Lindblad operators. This talk is based on joint work with Li Gao, Marius Junge, and Nicholas LaRacuente.

# On a reversal of Lyapunov's inequality for log-concave sequences

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Log-concave sequences appear naturally in a variety of

fields. For example in convex geometry the Alexandrov-Fenchel

inequalities demonstrate the intrinsic volumes of a convex body to be

log-concave, while in combinatorics the resolution of the Mason

conjecture shows that the number of independent sets of size n in a

matroid form a log-concave sequence as well. By Lyapunov's

inequality we refer to the log-convexity of the (p-th power) of the

L^p norm of a function with respect to an arbitrary measure, an

immediate consequence of Holder's inequality. In the continuous

setting measure spaces satisfying concavity conditions are known to

satisfy a sort of concavity reversal of both Lyapunov's inequality,

due to Borell, while the Prekopa-Leindler inequality gives a reversal

of Holder. These inequalities are foundational in convex geometry,

give Renyi entropy comparisons in information theory, the Gaussian

log-Sobolev inequality, and more generally the HWI inequality in

optimal transport among other applications. An analogous theory has

been developing in the discrete setting. In this talk we establish a

reversal of Lyapunov's inequality for monotone log-concave sequences,

settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. A

strengthened version of the same conjecture is disproved through

counterexamples.