Time: 3:00pm EST

Zoom Meeting ID:  975 4547 4357

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

Moment inequalities for Gaussian vectors

The Gaussian product inequality is a long-standing conjecture relating the moments of an arbitrary centred normal random vector to the moments of a standard one. In connection to this problem we present some new extensions of moment inequalities for Gaussian vectors.

Time: 3:00 pm (New York)

Zoom Meeting ID:  676 808 9500

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Correlation bounds for polynomials and all that

"Correlation bounds" in this context mean estimates of the correlation of a bounded-degree polynomial over F_2^n with some other function on F_2^n with respect to some probability measure. (Note these polynomials are evaluated mod 2, so for example Parity = x_1+...+x_n). Correlation bounds are fundamental to complexity theory and present formidable mathematical challenges. Emanuele will present some work from the recently-updated survey found here: https://www.khoury.northeastern.edu/home/viola/papers/corr-survey.pdf.

Time: 3:00pm EST

Zoom Meeting ID:  676 808 9500

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

Sparse "Juntas"

Boolean function analysis concerns functions f: {0,1}ⁿ → {0,1}. We often consider the behavior of f when its input is chosen uniformly at random from {0,1}ⁿ. Sometimes it is more natural to consider a biased input — the classical example being the random graph model G(n,p). Analyzing functions with respect to a biased measure often reveals phenomena that do not occur under an unbiased measure. We describe one such example: the structure of functions which are almost of low degree.

Joint work with Irit Dinur (Weizmann institute) and Prahladh Harsha (TIFR): https://arxiv.org/pdf/1711.09428.pdf

Time: 3:00pm EST

Zoom Meeting ID:  743 668 7250

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Approximately Hadamard matrices and random frames

We will discuss a problem concerning random frames which

arises in signal processing. A frame is an overcomplete set of vectors

in the n-dimensional linear space which allows a robust decomposition

of any vector in this space as a linear combination of these vectors.

Random frames are used in signal processing as a means of encoding

since the loss of a fraction of coordinates does not prevent the

recovery. We will discuss a question when a random frame contains a

copy of a nice (almost orthogonal) basis.

Despite the probabilistic nature of this problem it reduces to a

completely deterministic question of existence of approximately

Hadamard matrices.  An n by n matrix with plus-minus 1 entries is

called Hadamard if it acts on the space as a scaled isometry. Such

matrices exist in some, but not in all dimensions. Nevertheless, we

will construct plus-minus 1 matrices of every size which act as

approximate scaled isometries. This construction will bring us back to

probability as we will have to combine number-theoretic and

probabilistic methods.

Joint work with Xiaoyu Dong.

Time: 3:00pm EST

Zoom Meeting ID: 946 5127 6631

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Title: Sharp stability for the Brunn-Minkowski inequality for arbitrary sets

Abstract: The Brunn-Minkowski inequality states that for (open) sets A and B in R^d, we have |A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}. Equality holds if and only if A and B are convex and homothetic sets in R^d. In this talk, we present a sharp stability result for the Brunn-Minkowski inequality, concluding a long line of research on this problem. We show that if we are close to equality in the Brunn-Minkowski inequality, then A and B are close to being homothetic and convex, establishing the exact dependency between the three notions of closeness. This is joint work with Alessio Figalli and Peter van Hintum.

Time: 3:00pm EST

Zoom Meeting ID: 989 4826 3906

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Title:  Universality and matrix concentration inequalities 

Abstract: Random matrices are ubiquitous across many fields — physics, computer science, applied and pure mathematics. Oftentimes the random matrix of interest will have non-trivial structure — entries that are dependent and have potentially different means and variances (e.g. sparse Wigner matrices, matrices corresponding to adjacencies of random graphs, sample covariance matrices). However, current understanding of such complex random matrices remains lacking. In this talk, I will discuss recent results concerning the spectrum of sums of independent random matrices with a.s. bounded operator norms. In particular, I will demonstrate that under some fairly general conditions, such sums will exhibit the following universality phenomenon — their spectrum will lie close to that of a Gaussian random matrix with the same mean and covariance. (joint with Ramon van Handel)

Time: 3:00pm EST

Zoom Meeting ID:  948 6994 8619

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Random Geometric Graphs, and how to find their underlying metrics

Abstract: Consider the following scenario: We have a smooth compact manifold M embedded in a Euclidean space. We sample n points X_1, ..., X_n in M according to some probability measure μ on M. Now, we construct a graph G with the vertex set {1, 2, ..., n} and connect the edge {i, j}, independently, with a probability that depends on the Euclidean distance between X_i and X_j, such that the greater the distance, the smaller the probability.

Could we recover (M, μ) by observing the graph G?

Indeed, under some reasonable assumptions about the triple (M, μ, p), when n is large, with high probability, we can recover both the Euclidean distance and the Geodesic distance of every pair X_i and X_j with an error proportional to n^{-c/d}, where d is the dimension of M. From this, we could also derive a result of approximating (M, μ) by (G', ν) in the Gromov-Hausdorff distance, where G' is a weighted graph constructed from G equipped with the path distance, and ν is a probability measure on G'.

This is a joint work with Pakawut Jiradilok and Elchanan Mossel. 

Time: 3:00pm EST

Zoom Meeting ID:  823 7967 6983

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Global well-posedness and scattering in nonlinear wave equations

Abstract:  In this talk, we will consider the wave equation in odd space dimensions with energy-supercritical nonlinearity, in the radial setting. We will review the concentration compactness and rigidity arguments starting from the earlier work by Kenig-Merle and Duyckaerts-Kenig-Merle in the energy-critical and energy-supercritical cases and outline the key ideas behind the proof of global well-posedness and scattering results in 3d and higher odd dimensions. 

Time: 3:00pm EST

Zoom Meeting ID:  980 2030 3035

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An entropy power inequality for dependent random variables

Abstract: We present recent work with James Melbourne (CIMAT) and Cyril Roberto (Paris Nanterre) on a new entropy power inequality for dependent random vectors. The inequality takes a particularly pleasant form, reminiscent of work of Johnson involving conditional entropies, under conditions involving log-supermodularity of the joint density.

Time: 3:00pm EST

Zoom Meeting ID:  930 5953 4357

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Random approximation of convex bodies in Hausdorff metric

Abstract: While there is extensive literature on approximation, deterministic as well as random, of general convex bodies K in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance when the approximant K_n is given by the convex hull of n independent random points chosen uniformly on the boundary or in the interior of K. When K is a polygon and the points are chosen on its boundary, we determine the exact limiting behavior of the expected Hausdorff distance between a polygon as n -> infinity. From this we derive the behavior of the asymptotic constant for a regular polygon in the number of vertices.

Based on joint work with J. Prochno, C. Schütt and M. Sonnleitner. 

Time: 3:00pm EST

Zoom Meeting ID:  940 4914 8205

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Vector-Valued Concentration on the Symmetric Group

Concentration inequalities for real-valued functions are well understood in many settings and are classical probabilistic tools in theory and applications -- however, much less is known about concentration phenomena for vector-valued functions. We present a novel vector-valued concentration inequality for the uniform measure on the symmetric group. Furthermore, we discuss the implications of this result regarding the distortion of embeddings of the symmetric group into Banach spaces, a question which is of interest in metric geometry and algorithmic applications. We build on prior work of Ivanisvili, van Handel, and Volberg (2020) who proved a vector-valued inequality on the discrete hypercube, resolving a question of Enflo in the metric theory of Banach spaces. This talk is based on joint work with Ramon van Handel.

Time: 3:00pm EST

Zoom Meeting ID: 924 4808 5460

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A degree one Carleson operator along the paraboloid

Carleson proved in 1966 that the Fourier series of any square integrable function converges pointwise to the function, by establishing boundedness

of the maximally modulated Hilbert transform from L^2 into weak L^2. This talk is about a generalization of his result, where the Hilbert transform is replaced by a singular integral operator along a paraboloid. I will review some previous extensions of Carleson's theorem, and then discuss the two main ingredients needed to deduce our result: sparse bounds for singular integrals along the paraboloid, and a square function argument relying on the geometry of the paraboloid.