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京东 11.11 红包
Justin Salez:Entropy and curvature of Markov chains on metric spaces
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https://www.youtube.com/watch?v=8KJC55FZlYM Hausdorff Center for Mathematics 2024 Justin Salez: Entropy and curvature of Markov chains on metric spaces This talk is devoted to the celebrated Peres-Tetali conjecture, which asserts that any Markov chain exhibiting contraction in the Wasserstein distance should also exhibit contraction in relative entropy, by the same amount. I will give a brief historical overview of this question, describe our main result, and illustrate it with several applications. This is based on joint works with Pietro Caputo and Florentin Münch. Pietro Caputo, Florentin Münch, Justin Salez——”Entropy and curvature: beyond the Peres-Tetali conjecture“:https://doi.org/10.48550/arXiv.2401.17148 We study Markov chains with non-negative sectional curvature on finite metric spaces. Neither reversibility, nor the restriction to a particular combinatorial distance are imposed. In this level of generality, we prove that a 1-step contraction in the Wasserstein distance implies a 1-step contraction in relative entropy, by the same amount. Our result substantially strengthens a recent breakthrough of the second author, and has the advantage of being applicable to arbitrary scales. This leads to a time-varying refinement of the standard Modified Log-Sobolev Inequality (MLSI), which allows us to leverage the well-acknowledged fact that curvature improves at large scales. We illustrate this principle with several applications, including birth and death chains, colored exclusion processes, permutation walks, Gibbs samplers for high-temperature spin systems, and attractive zero-range dynamics. In particular, we prove a MLSI with constant equal to the minimal rate increment for the mean-field zero-range process, thereby answering a long-standing question.
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【系列课程】Wangjian Jian--Ricci Flow课程38
微分几何速通!流形和张量场
Max Hallgren:Tensor Maximum Principle——1
Max Hallgren:Tensor maximum principle——2
Ben Andrews:Ricci flow on surfaces
Jian Song:Geometric analysis on singular complex spaces
微分几何速通!协变导数算符
Carlos Simpson:Moduli Spaces of Sheaves on Surfaces——Lectures 1-2
微分几何速通!测地线和里奇张量
Kyle Broder:Kähler-Ricci flow and the Wu-Yau theorem——2
Boáz Klartag:Isoperimetric inequalities in high dimensional convex sets——3——2
Jack Thompson:Ricci solitons
Vishnu Mangalath:Evolution of curvature for the Ricci Flow
Joseph Lehec:Isoperimetric inequalities in high dimensional convex sets——4——3
Gunhee Cho:Calabi conjecture的Kähler-Ricci flow方法和Kähler–Einstein metric的存在
Joseph Lehec:Isoperimetric inequalities in high dimensional convex sets——1——4
Ben Andrews:Harmonic functions of polynomial growth——2
Aaron Naber:Nonlinear Harmonic Maps and the Energy Identity
Vishnu Mangalath:Estimates for Mean Curvature Flow in R^3
James Stanfield:Background on differential geometry——2
Boáz Klartag:Isoperimetric inequalities in high dimensional convex sets——1——1
Boáz Klartag:Isoperimetric inequalities in high dimensional convex sets——3——1
Otis Chodosh:Generalizations of the Bernstein Problem
Alexandros Eskenazis:Functional inequalities in Metric Geometry——II
Joseph Lehec:Isoperimetric inequalities in high dimensional convex sets——3——4
Tobias Colding:Geometry of PDEs
Timothy Buttsworth:Preserved curvature conditions for Ricci Flow——1
Noam Lifshitz:Inverse results for isoperimetric inequalities——II
Karl-Theodor Sturm:Metric measure spaces and synthetic Ricci bounds(度量测度空间)
Glen Wheeler:Ancient solutions of the heat equation with polynomial growth——2
Joseph Lehec:Isoperimetric inequalities in high dimensional convex sets——3——3
【CRM】Simon Brendle:Singularity models in 3D Ricci flow
Vishnu Mangalath:The Haslhofer-Kleiner Gradient Estimate
Ben Andrews:Harmonic functions with polynomial growth——1
Boáz Klartag:Isoperimetric inequalities in high dimensional convex sets——2——1
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Vishnu Mangalath:Gradient Estimates in Mean Curvature Flow——1
Daniele Semola:The metric measure boundary of non collapsed RCD spaces
Andrea Mondino:Time-like Ricci curvature bounds via optimal transport
Kyle Broder:Kähler-Ricci flow and the Wu-Yau theorem——1