**Probability and Analysis Webinar**

### Fall 2022

Time: 3PM (New York)

Zoom Meeting ID: 962 5899 4879

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

**Quantum KKL, Talagrand and Friedgut's theorems**

As a fundamental theorem in Boolean analysis, the KKL theorem says that any Boolean function has an influential variable. Two related results are Talagrand's inequality which implies the KKL theorem, and Friedgut's theorem on juntas. Montanaro and Osborne proposed a quantum extension of Boolean functions, namely self-adjoint unitary matrices of certain sizes. In this context, many classical results have been extended to quantum Boolean functions, such as Talagrand's inequality. However, since the L1 influences and L2 influences do not agree for general quantum Boolean functions, this Talagrand's inequality does not help to derive a quantum KKL theorem. So a quantum version of the KKL theorem seems to be missing, as conjectured by Montanaro and Osborne. In this talk, I will present an alternative answer to this question. We prove that every balanced quantum Boolean function has a geometrically influential variable. This is based on a quantum analog of L1 Talagrand's inequality which we prove using recently studied hypercontractivity and gradient estimates. We also prove Friedgut's Junta Theorem in the quantum setting that has applications in the learnability of quantum observables. This is based on joint work with Cambyse Rouzé (TUM) and Melchior Wirth (IST Austria)

**Optimal transport with respect to non-traditional costs**

Given a cost function and two probability measures, the optimal transport problem is that of finding a transport map (or a plan) which minimises total cost. The case of finite-valued costs is well-understood and, under mild assumptions, the optimal plan has a special geometric structure. In particular, there exists a function, which we call a potential, whose c-subgradient contains the support of the optimal transport plan (for the quadratic cost |x-y\|^2 the gradient of the potential is famously known as the Brenier map). However, if a cost function attains infinite values, which corresponds to prohibiting certain pairs of points to be mapped to one another, only special families of costs were studied. We present a unified approach to transportation with respect to infinite-valued costs: we reduce the question of existence of a potential to solvability of a system of linear inequalities and give necessary and sufficient conditions for existence of a solution. The talk is based on joint work with S. Artstein-Avidan and S. Sadovsky.

**Mean inequalities for symmetrizations of convex sets**

The arithmetic-harmonic mean inequality can be generalized for convex sets, considering the intersection, the harmonic and the arithmetic mean, as well as the convex hull of two convex sets. We study those relations of symmetrization of convex sets, i.e., dealing with the means of some convex set C and -C. We determine the dilatation factors, depending on the asymmetry of C, to reverse the containments between any of those symmetrizations, and tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.

Time: 3:30pm (New York time zone)

Zoom Meeting ID: 983 3049 1122

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

**The Gaussian Brunn-Minkowski Theory for convex bodies**

The Brunn-Minkowski Theory concerns the behaviour of convex bodies in R^n (compact, convex sets with non-empty interior), by studying their properties e.g. volume, surface area, projections, and Minkowski sum. We shall discuss a generalization of this theory to the measure theoretic setting (replacing volume with some Borel measure with density). Throughout, there is a focus on the Gaussian measure (i.e. the normal distribution on R^n). In particular, we show analogues of Minkowski’s First Inequality and Second Inequality, and other inequalities.

Joint work with M. Fradelizi, M. Madiman and A. Zvavitch

Time: 3:00pm (New York time zone)

Zoom Meeting ID: 676 808 9500

Password: last **6** digits of the zoom meeting ID** in reverse order**

**Convex influences and a quantitative Gaussian correlation inequality**

The Gaussian correlation inequality (GCI), proved by Royen in 2014, states that any two centrally symmetric convex sets (say K and L) in Gaussian space are positively correlated. We establish a new quantitative version of the GCI which gives a lower bound on this correlation based on the "common influential directions" of K and L. This can be seen as a Gaussian space analogue of Talagrand's well known correlation inequality for monotone Boolean functions.

To obtain this inequality, we propose a new approach, based on analysis of Littlewood type polynomials, which gives a recipe for transferring qualitative correlation inequalities into quantitative correlation inequalities. En route, we also give a new notion of influences for symmetric convex symmetric sets over Gaussian space which has many of the properties of influences of Boolean functions over the discrete cube. Much remains to be explored about this new notion of influences for convex sets.

Based on joint works with Anindya De and Shivam Nadimpalli.

Time: 3PM (New York)

Zoom Meeting ID: 962 5899 4879

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

**Uniform in N entropy production bounds for some quantum Kac models**

We prove a logarithmic Sobolev inequality for a family of quantum Kac models with constants that are uniform in the number of particles. A key role is played by recent progress in understanding the rate of return to equilibrium of a classical random transposition process on the multislice. This is joint work with Michael Loss.

Time: 3PM (New York)

Zoom Meeting ID: 951 5914 7034

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

**Fourier Restriction to the sphere is extremizable more often than not**

Inequalities play a central role in harmonic analysis. However, in many cases the fundamental question "Is equality possible?" is left unanswered. Resolving this question is a first step toward proving stronger versions of the inequality. In this talk, we'll consider this question in the context of Fourier restriction to the sphere. In particular, we'll show that if valid, the $L^p-L^q$ Fourier extension inequality passesses extremizers whenever $p<q <(d+2)p'/d$. Joint work with Betsy Stovall.

Time: 3PM (New York)

Zoom Meeting ID: 954 5747 4477

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

**The Korányi Spherical Maximal Function on Heisenberg groups**

In this talk, we discuss the problem of obtaining sharp Lp→Lq estimates for the local maximal operator associated with averaging over dilates of the Korányi sphere on Heisenberg groups. This is a codimension one surface compatible with the non-isotropic Heisenberg dilation structure. I will describe the main features of the problem, some of which are helpful while others are obstructive. These include the non-Euclidean group structure (the extra “twist” due to the Heisenberg group law), the geometry of the Korányi sphere (in particular, the flatness at the poles) and an “imbalanced” scaling argument encapsulated by a new type of Knapp example, which we shall describe in detail.

Time: 3PM (New York)

Zoom Meeting ID: 954 5747 4477

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

**Time-frequency localization operators, their eigenvalues and relationship to elliptic PDE**

In the classical realm of time-frequency analysis, a classical object of interest is the short-time Fourier transform of a function. This object is a modified Fourier transform of a signal f(x), modified by a certain 'window function', in order to make joint time-frequency analysis of functions more feasible.

Since the pioneering work of Daubechies, time-frequency localisation operators have been of extreme importance in that analysis. These are defined through V^∗1_Ω V f=P_Ω f, where V denotes the short-time Fourier transform with some fixed window. These operators seek to measure how much a function concentrates in the time-frequency plane, and thus the study of their eigenvalues and eigenfunctions is intimately connected to the previous questions.

In this talk, we will explore the case of a Gaussian window function φ(x)=e−πx^2, and the operators thus obtained. We will discuss some classical and recent results on domains of maximal time-frequency concentration, their eigenvalues, and inverse problems associated with such properties. During this investigation, we shall see that many of these problems possess some rather unexpected connections with overdetermined elliptic boundary value problems and free boundary problems in general. This is based on recent joint work with Paolo Tilli.

Time: 3PM (New York)

Zoom Meeting ID: 938 4189 8163

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

** Affine Subspace Concentration Conditions for Centered Polytopes**

The subspace concentration condition is a property of the volume distribution inside a convex body (in our case, a polytope). Given an n-polytope P, which contains the origin as an interior point, one associates to each facet F a cone C = conv(0,F). We say that P satisfies the subspace concentration condition, if for any linear d-subspace L, the volume of all cones corresponding to facets with normal vector in L is not more than (d/n)*vol(P).

The subspace concentration condition plays a key role in the study of the log-Minkowski problem, one of the major open problems in convex geometry these days. Moreover, Hering, Nill and Süß recently found an interpretation of the subspace concentration condition within toric geometry, which lead Wu to prove an elegant affine variant for special lattice polytopes.

In the talk, we study the convex geometry behind Wu's affine subspace concentration condition and generalize it to arbitrary centered polytopes, exploiting a curious phenomenon in high dimensions. Also, we discuss open question around the equality case. This is joint work with Martin Henk and Christian Kipp.

Time: 3PM (New York)

Zoom Meeting ID: 930 0738 6410

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

**Fourier Analysis in Quantum Circuit Complexity**

Fourier analysis over the hypercube has been instrumental for understanding the computational power of constant-depth Boolean circuits. Famous among its applications is a 1993 result of Linial, Mansour, and Nisan: functions implemented by constant-depth circuits have nearly all their Fourier mass concentrated on the lowest levels of the Fourier spectrum.

It is natural to ask whether functions implemented by shallow quantum circuits (with classical postprocessing) exhibit the same concentration properties. As a first step towards this question, we consider whether this model of computation fails to approximate the Parity function. We find *unitary* constant-depth quantum computation indeed cannot approximate Parity noticeably better than random guessing, and in the non-unitary case we connect to the question to the approximate degree of AC0, an important open problem in concrete complexity theory.

No knowledge of quantum computation or circuit complexity is assumed.

**Curvature of graphs and a discrete notion of log-concavity**

The utility of log-concavity in asymptotic geometric analysis is well-known. One very fruitful perspective on this condition is provided by the formalism of Γ-calculus due to Bakry and Émery, according to which log-concave measures are simply measures with "nonnegative curvature." In this talk, we will explain this formalism and propose a new method for extending it to the setting of graphs, which yields a replacement for the notion of log-concavity on graphs. As an application, we show that the Poincaré constant of a log-concave sequence decreases along the heat flow, which is a discrete variant of a previous result of Klartag and the speaker.

Time: 3:00pm (New York time zone)

Zoom Meeting ID: 676 808 9500

Password: last **6** digits of the zoom meeting ID** in reverse order**

**Lower bounds for the directional discrepancy**

The discrepancy of a point set in the unit cube provides a measure of how well-distributed the point set is. However, precise behavior of the discrepancy strongly depends on the properties of the underlying collection of geometric test sets. In two dimensions, the discrepancy with respect to axis-parallel rectangles and rectangles rotated in arbitrary directions is well-understood. The increased complexity of the latter collection leads to discrepancy bounds of polynomial order, in contrast to logarithmic order in the axis-parallel case. In this talk we will examine the "in between" and study what happens when we restrict the allowed set of rotations to a smaller interval. We will discuss similarities between the directional discrepancy and the discrepancy of certain classes of convex sets with particular smoothness properties. We will also touch on some bounds for even sparser classes of allowed rotations such as infinite sequences.

Time: 3PM (New York)

Zoom Meeting ID: 942 8880 0425

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

**Multi-Bubble Isoperimetric Problems - Old and New**

The classical isoperimetric inequality in Euclidean space R^n states that among all sets ("bubbles") of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the n-sphere S^n and on n-dimensional Gaussian space G^n (i.e. R^n endowed with the standard Gaussian measure). Furthermore, one may consider the "multi-bubble" isoperimetric problem, in which one prescribes the volume of p ≥ 2 bubbles (possibly disconnected) and minimizes their total surface area -- as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to p=1; the case p=2 is called the double-bubble problem, and so on.

In 2000, Hutchings, Morgan, Ritoré and Ros resolved the double-bubble conjecture in Euclidean space R^3 (and this was subsequently resolved in R^n as well) -- the boundary of a minimizing double-bubble is given by three spherical caps meeting at 120-degree angles. A more general conjecture of J. Sullivan from the 1990's asserts that when p ≤ n+1, the optimal multi-bubble in R^n (as well as in S^n) is obtained by taking the Voronoi cells of p+1 equidistant points in S^n and applying appropriate stereographic projections to R^n (and backwards).

In 2018, together with Joe Neeman, we resolved the analogous multi-bubble conjecture for p ≤ n bubbles in Gaussian space G^n -- the unique partition which minimizes the total Gaussian surface area is given by the Voronoi cells of (appropriately translated) p+1 equidistant points. In the talk, I will describe our approach in that work, as well as recent progress we have made on the multi-bubble problem on R^n and S^n. In particular, we show that minimizing bubbles in R^n and S^n are always spherical when p ≤ n, and we resolve the latter conjectures when in addition p ≤ 5 (e.g. the triple-bubble conjectures when n ≥ 3 and the quadruple-bubble conjectures when n ≥ 4).

Time: 3PM (New York)

Zoom Meeting ID: 962 5899 4879

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

**Universality in nonasymptotic random matrix theory**

Nonasymptotic random matrix theory aims to estimate spectral statistics (such as the extreme eigenvalues) of rather general random matrix models in a quantitative fashion. Such results are often first established under Gaussian or sub-Gaussian assumptions, and much work is then devoted to extending such bounds to more general situations. In this talk I will discuss a very different perspective on such problems: under remarkably weak structural assumptions, one can show in a precise nonasymptotic manner that the behavior of random matrices is accurately captured by that of an associated Gaussian model, regardless of the behavior of the Gaussian model itself. When combined with recent developments in the understanding of Gaussian random matrices, this nonasymptotic universality principle yields a powerful "black box" tool for understanding the behavior of extremely general nonhomogeneous and non-Gaussian random matrix models. If time permits, I will discuss applications to random graphs, spiked models, sample covariance matrices, and/or free probability theory. (Based on joint work with Tatiana Brailovskaya.)