Norms of structured random matrices

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.

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

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 counter-examples.

Matrix-valued logarithmic Sobolev inequalities

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.

3PM NYT

Zoom Meeting ID: 928 3597 1183

Password: last 4 digits of the zoom meeting in reverse order

L^p improving bounds for spherical maximal operators

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.

3PM NYT

Zoom Meeting ID: 954 5747 4477

Password: last 4 digits of the zoom meeting in reverse order

Kakeya maximal estimates via real algebraic geometry

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.

3PM NYT

Zoom Meeting ID: 928 3597 1183

Password: last 4 digits of the zoom meeting in reverse order

Bilinear embedding in Orlicz spaces for divergence-form operators with complex coefficients

Abstract: We will discuss a bi(sub)linear embedding for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form and for certain mutually dual pairs of Orlicz-space norms. This generalizes a result by Carbonaro and Dragičević from power functions to more general Young functions that still behave like powers. To achieve this, we generalize a classic Bellman function constructed by Nazarov and Treil. The talk is based on joint work with Vjekoslav Kovač.

3PM NYT

Zoom Meeting ID: 962 5899 4879

Password: last 4 digits of the zoom meeting in reverse order

George Boole meets Harald Bohr

Abstract: view abstract

Limit Profiles of Reversible Markov chains

It all began with card shuffling. Diaconis and Shahshahani studied the random transpositions shuffle; pick two cards uniformly at random and swap them. They introduced a Fourier analysis technique to prove that it takes $1/2 n \log n$ steps to shuffle a deck of $n$ cards this way. Recently, Teyssier extended this technique to study the exact shape of the total variation distance of the transition matrix at the cutoff time from the stationary measure, giving rise to the notion of a limit profile. In this talk, I am planning to discuss a joint work with Olesker-Taylor, which extends the above technique from conjugacy invariant random walks to general, reversible Markov chains. I will also present a new technique that allows to study the limit profile of star transpositions, which turns out to have the same limit profile as random transpositions.

Sharp restriction theory: rigidity, stability, and symmetry breaking

We report on recent progress concerning two distinct problems in sharp restriction theory to the unit sphere.

Firstly, the classical estimate of Agmon-Hörmander for the adjoint restriction operator to the sphere is in general not saturated by constants. We describe the surprising intermittent behaviour exhibited by the optimal constant and the space of maximizers, both for the inequality itself and for a stable form thereof. Secondly, the Stein-Tomas inequality on the sphere is rigid in the following rather strong sense: constants continue to maximize the weighted inequality as long as the perturbation is sufficiently small and regular, in a precise sense to be discussed. We present several examples highlighting why such assumptions are natural, and describe some consequences to the (mostly unexplored) higher dimensional setting.

This talk is based on joint work with E. Carneiro and G. Negro.

On the Fourier-Entropy Conjecture

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.

Toward a fundamental gap of convex sets in hyperbolic space

The difference between the first two eigenvalues of the Dirichlet Laplacian on convex sets of R^n and, respectively S^n, satisfies the same strictly positive lower bound depending on the square of the diameter of the domain. In work with collaborators, we have found that the gap of the hyperbolic space on convex sets behaves strikingly different even if a stronger notion of convexity is employed. This is very interesting as many other features of first two eigenvalues behave in the same way on all three spaces of constant sectional curvature. We will discuss the possibility of a different lower bound on the fundamental gap in the hyperbolic space.

Time: 3PM (New York time)

Zoom Meeting ID: 962 5899 4879

Password: last 4 digits of the meeting ID in reverse order

Generalized Collatz maps

The Collatz map is defined as follows: for any natural number, we divide by two when possible, and otherwise we multiply by 3 and add 1. The well known Collatz Conjecture states that every orbit of this map eventually hits one. In this talk we will discuss the behavior of a general class of Collatz maps.

Time: 3PM (New York time)
Zoom Meeting ID: 962 5899 4879

Password: last 4 digits of the meeting ID in reverse order

Phase space projections

A partition into tiles of the area covered by a convex tree in the Walsh phase plane gives an orthonormal basis for a subspace of L2. There exists a related projection operator, which has been an important tool for dyadic models of the bilinear Hilbert transform. Extending such an approach to the Fourier model is strictly speaking not possible, but satisfactory substitutes can be constructed. This approach was pursued by Muscalu, Tao and Thiele (2002) for proving uniform bounds for multilinear singular integrals with modulation symmetry in dimension one. I discuss a multidimensional variant of the problem. This is based on joint work with Marco Fraccaroli and Christoph Thiele.

Lower bounds for the L^p norms of some Fourier multipliers

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.