• December 07, 2022 03:30 PM
Speaker: Lior Alon
Title: Fourier quasicrystals and stable polynomials
Venue: Harvard Science Center

Probability Seminar Note location change: Science Center Room 300H Speaker: Lior Alon (MIT) Title: Fourier quasicrystals and stable polynomials Abstract: The Poisson summation formula says that the countable sum of exp(int), over all integers n, vanishes as long as t is not an integer multiple of 2 pi. Can we find a non-periodic discrete set A, such that the sum of exp(iat), over a in A, vanishes for all t outside of a discrete set? The surprising answer is yes. Yves Meyer called the atomic measure supported on such a set a crystalline measure. Crystalline measures provide another surprising connection between physics (quasicrystals) and number theory (the zeros of the Zeta and L functions under GRH). A recent work of…

  • November 30, 2022 03:00 PM
Speaker: Dan Mikulincer
Title: Lipschitz properties of transport maps under a log-Lipschitz condition
Venue: Harvard Science Center

Probability Seminar Location: Room 109, Harvard Science Center, 1 Oxford Street, Cambridge MA 02138 Speaker: Dan Mikulincer (MIT) Title: Lipschitz properties of transport maps under a log-Lipschitz condition Abstract: Consider the problem of realizing a target probability measure as a push forward, by a transport map, of a given source measure. Typically one thinks about the target measure as being ‘complicated’ while the source is simpler and often more structured. In such a setting, for applications, it is desirable to find Lipschitz transport maps which afford the transfer of analytic properties from the source to the target. The talk will focus on Lipschitz regularity when the target measure satisfies a log-Lipschitz condition. I will present a construction of…

  • November 16, 2022 03:30 PM
Speaker: Ainesh Bakshi
Title: Outlier-Robust Algorithms for Clustering Non-Spherical Mixtures
Venue: CMSA, 20 Garden St, G10

Probability Seminar Speaker: Ainesh Bakshi (MIT) Title: Outlier-Robust Algorithms for Clustering Non-Spherical Mixtures Abstract: In this talk, we describe the first polynomial time algorithm for robustly clustering a mixture of statistically-separated, high-dimensional Gaussians. Prior to our work this question was open even in the special case of 2 components in the mixture. Our main conceptual contribution is distilling analytic properties of distributions, namely hyper-contractivity of degree-two polynomials and anti-concentration of linear projections, which are necessary and sufficient for clustering.

  • November 09, 2022 03:30 PM
Speaker: Hugo Falconet
Title: Liouville quantum gravity from random matrix dynamics
Venue: CMSA Room G10

Probability Seminar Speaker: Hugo Falconet (Courant Institute, NYU) Title: Liouville quantum gravity from random matrix dynamics Abstract: The Liouville quantum gravity measure is a properly renormalized exponential of the 2d GFF. In this talk, I will explain how it appears as a limit of natural random matrix dynamics: if (U_t) is a Brownian motion on the unitary group at equilibrium, then the measures $|det(U_t – e^{i theta}|^gamma dt dtheta$ converge to the 2d LQG measure with parameter $gamma$, in the limit of large dimension. This extends results from Webb, Nikula and Saksman for fixed time. The proof relies on a new method for Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols, which extends to multi-time settings. I will explain this…