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Speaker: Vishesh JainTitle: Thresholds for edge coloringsVenue: virtualProbability Seminar Speaker: Vishesh Jain (University of Illinois Chicago) Title: Thresholds for edge colorings Abstract: We show that if each edge of the complete bipartite graph K_{n,n} is given a random list of C(\log n) colors from [n], then with high probability, there is a proper edge coloring where the color of each edge comes from the corresponding list. We also prove analogous results for Latin squares and Steiner triple systems. This resolves several related conjectures of Johansson, Luria-Simkin, Casselgren-Häggkvist, Simkin, and Kang-Kelly-Kühn-Methuku-Osthus. I will discuss some of the main ingredients which go into the proof: the Kahn-Kalai conjecture, absorption, and the Lovasz Local Lemma distribution. Based on joint work with Huy Tuan Pham. |
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Speaker: Zhigang YaoTitle: Manifold Fitting: An Invitation to StatisticsVenue: CMSA Room G10Probability Seminar Speaker: Zhigang Yao (Harvard CMSA/National University of Singapore) Title: Manifold Fitting: An Invitation to Statistics Abstract: This manifold fitting problem can go back to H. Whitney’s work in the early 1930s (Whitney (1992)), and finally has been answered in recent years by C. Fefferman’s works (Fefferman, 2006, 2005). The solution to the Whitney extension problem leads to new insights for data interpolation and inspires the formulation of the Geometric Whitney Problems (Fefferman et al. (2020, 2021a)): Assume that we are given a set $Y \subset \mathbb{R}^D$. When can we construct a smooth $d$-dimensional submanifold $\widehat{M} \subset \mathbb{R}^D$ to approximate $Y$, and how well can $\widehat{M}$ estimate $Y$ in terms of distance and smoothness? To address these problems, various… |
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Speaker: Roland BauerschmidtTitle: Bakry-Emery theory and renormalisationVenue: HybridProbability Seminar Speaker: Roland Bauerschmidt (Cambridge) Title: Bakry-Emery theory and renormalisation Abstract: I will discuss an approach to log-Sobolev inequalities that combines the Bakry-Emery theory with renormalisation and present several applications. These include log-Sobolev inequalities with polynomial dependence for critical Ising models on Z^d when d>4 and singular SPDEs with uniform dependence of the log-Sobolev constant on both the regularisation and the volume. The talk is based on joint works with Thierry Bodineau and Benoit Dagallier. |
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Speaker: Lior AlonTitle: Fourier quasicrystals and stable polynomialsVenue: Harvard Science CenterProbability 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… |
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Speaker: Dan MikulincerTitle: Lipschitz properties of transport maps under a log-Lipschitz conditionVenue: Harvard Science CenterProbability 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… |
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Speaker: Ainesh BakshiTitle: Outlier-Robust Algorithms for Clustering Non-Spherical MixturesVenue: CMSA, 20 Garden St, G10Probability 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. |
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Speaker: Hugo FalconetTitle: Liouville quantum gravity from random matrix dynamicsVenue: CMSA Room G10Probability 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… |