September

October

November

December

Krzysztof Oleszkiewicz (University Warsaw): On the asymptotics of the optimal constants in the Khinchine-Kahane inequality

Let us consider a sequence of indepenedent symmetric +/-1 random variables, often called the Rademacher system. A linear combination of these random variables is a real random variable called a (weighted) Rademacher sum. There are also vector-valued Rademacher sums, in which the real coefficients in the linear combination are replaced by vectors from some normed linear space. Rademacher sums, both real and vector-valued, have been studied for more than 100 years now. In the talk, classical moment inequalities for Rademacher sums will be described, going back to Khinchine (1923) and Kahane (1964), as well as some more recent results.

Devraj Duggal (University of Minnesota): On Spherical Covariance Representations

We first motivate the study of covariance representations by surveying preceding results in the Gaussian space. Their spherical counterparts are then derived thereby allowing applications to the spherical concentration phenomenon. The applications include second order concentration inequalities. The talk is based on joint work with Sergey Bobkov.

(No recording available)

Rafał Latała (University of Warsaw): On the spectral norm of Rademacher matrices

We will discuss two-sided non-asymptotic bounds for the mean spectral norm of nonhomogenous weighted Rademacher matrices. We will present a lower bound and show that it may be reversed up to log log log n factor for arbitrary n×n Rademacher matrices. Moreover, the triple logarithm may be eliminated for matrices with 0,1-coefficients.

(No recording available)

Miquel Saucedo (Universitat Autònoma de Barcelona): Weighted inequalities for the Fourier transform

In this talk we discuss inequalities for the Fourier transform between weighted Lebesgue spaces and their connection with an interpolation technique due to Calderón.

(No recording available)

Guy Kindler (The Hebrew University of Jerusalem): Bounded functions with small tails are Juntas

There seems to be some recent interest in structural results concerning functions whose Fourier transform is mostly supported on `low-degrees'  while their range is restricted to a specific set. This is at least partly motivated by applications to Theoretical Computer Science. In this talk I will go over a not-so-recent result with this theme, which shows that a function f over the discrete hypercube whose Fourier representation is `mostly' of low degree and which obtains values in the (continuous) segment [-1,1] must be close to a junta, namely it can be approximated by only looking at a constant number of input-coordinates. The proof goes by showing a large-deviation lower bound for low degree functions that uses some tricks that may be of interest. If time permits I may also talk about some improvements to our result made by O’Donnell and Zhao. Joint work with Irit Dinur, Ehud Friedgut, and Ryan O'Donnell.

(No recording available)

 Marco Fraccaroli (Basque Center for Applied Mathematics): The L^p theory for outer measure spaces

The theory of L^p spaces for outer measures, or outer L^p spaces, was introduced by Do and Thiele, as tool in the proof of estimate for multilinear forms arising in the context of harmonic analysis (Calderón-Zygmund theory, time-frequency analysis). To this end, they developed the theory in the direction of the interpolation properties of the spaces, such as Hölder’s inequality and Marcinkiewicz interpolation. However, the outer L^p spaces can be defined in a broader generality of settings, for example extending the classical notion of mixed L^p spaces on the Cartesian product of measure spaces. In this talk we expose further developments in the theory of the outer L^p spaces, focusing on their Banach space properties, such as Fubini’s theorem, Köthe duality, and Minkowski’s inequality.

(No recording available)

Diogo Oliveira e Silva (Instituto Superior Técnico Lisboa): Global maximizers for spherical restriction

We prove that constant functions are the unique real-valued maximizers for all L²-L²ⁿ adjoint Fourier restriction inequalities on the unit sphere S^d-1⊆ R^d, d ∈ {3,4,5,6,7}, where n≥ 3 is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character e^iξ·ω, for some ξ, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions d≥ 2 and general even exponents.

(No recording available)

Jaume de Dios Pont (ETH Zürich): Query lower bounds for log-concave sampling

A central step in the implementation of probabilistic algorithms is that of sampling from known, complicated probability distributions: Given the density of a random variable (for example, as a black-box function that one can query) generate samples from a random variable that has a distribution "similar enough" to the given one. Significant effort has been devoted to designing more and more efficient algorithms, ranging from relatively simple algorithms, such as rejection sampling, to increasingly sophisticated such as Langevin or diffusion based models. In this talk we will focus on the converse question: Finding universal complexity lower bounds that no algorithm can beat.  We will do so in the case when the log-density is a strictly concave smooth function. In this case we will be able to construct tight bounds in low dimension using a modification of Perron's sprouting construction for Kakeya sets. Based on joint work with Sinho Chewi, Jerry Li, Chen Lu and Shyam Narayanan.

(No recording available)

Alexander Borichev (Aix-Marseille Université): The zero distribution for Taylor series with random and pseudo-random coefficients

We study the local distribution of zeros of Taylor series for different classes of coefficients: random ones (independent, stationary, arithmetic random) and pseudo-random ones (exponential-polynomial, Rudin-Shapiro, Thue-Morse).

(No recording available)

 Joseph Slote (Caltech): Testing monotonicity from quantum data

This talk is about testing properties of Boolean functions from data, where it turns out quantum algorithms can have dramatic speedups. We will focus on monotonicity testing. Here, a classical algorithm given access only to uniformly random samples (x,f(x)) requires at least 2^Ω(√(n)) samples to test if f is monotone. On the other hand, we will describe a quantum algorithm for monotonicity testing that requires only poly(n) quantum data, in the form of so-called function states: Σ_x (x,f(x)). We will also prove an n^3/2 lower bound for such quantum algorithms via a careful analysis of certain matrix ensembles. This is one of the first works to consider such lower bound arguments, and we welcome discussion and improvements to our techniques.

Based on joint work with Matthias Caro and Preksha Naik.

(No recording available)

Friedrich Götze (Universität Bielefeld): Counting Lattice Points in Ellipsoids and the Central Limit Theorem for Quadratic Forms

In this talk we review classical results on lattice point counting problems for ellipsoids and describe in dimensions five and larger some older and recent results on explicit error bounds. We outline their relation to corresponding errors estimates in the multivariate central limit theorem in Probability and the importance of gap principles for bounding Fourier integrals.

Gautam Aishwarya (Michigan State University): Stability in the Banach isometric conjecture for planar sections

Banach asked whether a normed space all whose k-dimensional linear subspaces are isometric to each other, for some fixed 2 ≤ k < dim(V), must necessarily be Euclidean. At present, an affirmative answer is known for k=2 (Auerbach-Mazur-Ulam, 1935), all even k (Gromov, 1967), all k=1 mod 4 but k=133 (Bor-Hernandez Lamoneda-Jimenez Desantiago-Montejano Peimbert, 2021), and k=3 (Ivanov-Mamaev-Nordskova, 2023). These developments, except perhaps the recent resolution of the k=3 case, can be considered spiritual successors to the original argument of Auerbach-Mazur-Ulam for k=2 which is based on a topological obstruction. In this talk, I will present a stable version of their result: if all 2-dimensional linear subspaces are approximately isometric to each other, then the normed space is approximately Euclidean.

This talk is based on joint work with Dmitry Faifman.

Felipe Gonçalves (IMPA and University of Texas at Austin): Sharp Strichartz Estimates via Hermite Polynomials and Hypercontractivity

We will present an approach to prove sharp inequalities for free-range Schrödinger propagator using a pseudo-conformal transformation (the Lens transform) that reformulates the whole problem as a sort of average hypercontractivity statement in Gauss space We will indicate how to solve this in the even exponent case. We will also explain an old idea from my phd thesis on how to solve the general case via 3-symbol Hamming cube approximations.

Gennady Uraltsev (University of Arkansas): Banach-space valued multilinear singular integrals and time-frequency analysis

Since the 1960s, Calderón-Zygmund theory has provided a framework for studying singular integral operators that resemble the Hilbert transform, a ubiquitous operator in complex analysis, PDEs, and many other areas of mathematics. In the 1980s, Bourgain and Burkholder characterized the Banach spaces where linear Calderón-Zygmund theory is available through the UMD property—a purely probabilistic condition.

More recently, there has been significant interest in multilinear singular integral operators, which arise e.g. in fractional Leibniz rules and multilinear ergodic theory. However, a complete Banach-space valued theory remains elusive, especially for operators that exhibit stronger singularities than Calderón-Zygmund operators, such as the bilinear Hilbert transform—a prototypical operator of time-frequency type.

We will review the techniques involved in these problems, with emphasis on time-frequency analysis methods, presenting both our current understanding and open directions for extending the theory to the multilinear setting.

(The recording has not been edited, yet)

Lars Becker (Universität Bonn): Discrete Brunn-Minkowski inequality for subsets of the cube

The classical Brunn-Minkowski inequality implies that for any two subsets A and B of Euclidean space, the volume of the Minkowski sum A + B is bounded from below by (|A||B|)^1/2. This inequality continues to hold in the discrete setting, on the integer lattice with counting measure. The topic of this talk is an optimal sharpening of this inequality when the sets A, B are contained in the cube {0, 1, 2}^d. I will discuss some applications and a useful method for proving inequalities in one variable.

This is based on joint work with Paata Ivanisvili, Dmitry Krachun and José Madrid.

Xinyuan Xie (University of California, Irvine): Jackson's inequality on the hypercube

This talk explores how well real-valued functions on the hypercube can be uniformly approximated by degree-d polynomials. I will present techniques for obtaining upper and lower bounds. As a first application, we show that the reverse Bernstein inequality fails in the tail space L¹_≥0.499n, improving previous counterexamples in L¹_{≥Clog log(n)}. As a second application, we show that no sensitivity theorem holds for bounded-valued functions, even when degree is relaxed to approximate degree.

This is a joint work with Paata Ivanisvili and Roman Vershynin.

(No recording available)

Benjamin Jaye (Georgia Tech): Uncertainty Principles Associated to Sets Satisfying the Geometric Control Condition

In this talk we shall introduce and prove some forms of the uncertainty principle suggested by problems in control theory.

Joint work with Walton Green and Mishko Mitkovski.

(No recording available)