# TBA

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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.

# Norms of structured random matrices

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We consider the structured Gaussian matrix G_A=(a_{ij}g_{ij}), where g_{ij}'s are independent standard Gaussian variables. The exact behavior of the spectral norm of the structured Gaussian matrix is known due to the result of Latala, van Handel, and Youssef from 2018. We are interested in two-sided bounds for the expected value of the norm of G_A treated as an operator from l_p^n to l_q^m. We conjecture the sharp estimates expressed only in the terms of the coefficients a_{ij}'s. We confirm the conjectured lower bound up to the constant depending only on p and q, and the upper bound up to the multiplicative constant depending linearly on a certain (small) power of ln(mn). This is joint work with Radoslaw Adamczak, Joscha Prochno, and Michal Strzelecki.

# TBA

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# Large deviations for triangle densities

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Take a uniformly random graph with a fixed edge density e. Its triangle density will typically be about e^3, and we are interested in the large deviations behavior: what's the probability that the triangle density is about e^3 - delta? The general theory for this sort of problem was studied by Chatterjee-Varadhan and Dembo-Lubetzky, who showed that the solution can be written in terms of an optimization over certain integral kernels. This optimization is difficult to solve explicitly, but Kenyon, Radin, Ren and Sadun used numerics to come up with a fascinating and intricate set of conjectures regarding both the probabilities and the structures of the conditioned random graphs. We prove these conjectures in a small region of the parameter space.

Joint work with Charles Radin and Lorenzo Sadun