Problems
Barycenters in metric spaces
Clustering in metric spaces
Sequential prediction/optimization in metric spaces
Chen-Chvatal conjecture
Applications of discrete curvature to your favourite graph pbl
Books
Warmup in differential geometry
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M. P. do Carmo. Differential geometry of curves and surfaces. 2nd Edition. Dover. 2016
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J. McCleary. Geometry from a differentiable viewpoint. 2nd Edition. Cambridge Univesrity Press. 2013.
Riemannian geometry
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P. Petersen. Riemannian Geometry. 2nd Edition. Springer. 2006
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M. P. do Carmo. Riemannian Geometry. Birkhauser. 1992
Metric geometry
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D. Burago, Y. Burago and S. Ivanov. A course in metric geometry. Graduate Studies in Mathematics. American Mathematical Society. 2001
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S. Alexander, V. Kapovitch and A. Petrunin. Alexandrov geometry: preliminary version no. 1. (book in preparation )
Optimal transport
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C. Villani. Topics in Optimal Transportation. Graduate Studies in Mathematics. American Mathematical Society. 2003
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C. Villani. Optimal transport: Old and new. Springer. 2008
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L. Ambrosio, N. Gigli and G. Savare. Gradient flows in metric spaces and in the space of probability measures. Springer. 2008
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L. Ambrosio and N. Gigli. A user's guide to optimal transport problems and applications. Lecture notes of the C.I.M.E. summer school. 2009.
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Y. Ollivier, H. Pajot and C. Villani (Editors). Optimal Transport: Theory and Applications. London Mathematical Society Lecture Note Series. 2014.
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F. Santambrogio. Optimal transport for applied mathematicians. Birkhauser. 2015.
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M. Cuturi and G. Peyre. Computational optimal transport. Foundations and trends in Machine Learning. vol 11(5-6). pp 355-607. 2019.
Graph theory
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J. A. Bondy and and U. S. R. Murty. Graph theory. Springer. 2008
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R. Diestel. Graph theory. Springer. 2017.
Discrete Geometry
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L. Najman and P. Romon (Editors). Modern Approaches to Discrete Curvature. Lecture Notes in mathematics. Springer. 2017.
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N. Peyerimhoff. Curvature notions on graphs. Lecture notes from the Leeds Summer School. from July 18-19, 2019.
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Discrete differential geometry (Graduate course at Carnegie Mellon University):
Articles
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Y Ollivier. A survey of Ricci curvature for metric spaces and Markov chains. in Probabilistic approach to geometry, Adv. Stud. Pure Math. 57, Math. Soc. Japan, pp: 343–381. 2010.
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Y. Ollivier. Discrete Ricci curvature: Open problems. Manuscript (2008)(
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F. Memoli. Gromov–Wasserstein Distances and the Metric Approach to Object Matching. Found Comput Math. vol 11, pp:417-487 (2011)
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M. Bacak. Computing Medians and Means in Hadamard Spaces. SIAM J. Optim., vol. 24(3), pp: 1542–1566 (2014)
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S-I. Ohta and M. Palfia. Discrete-time gradient flows and law of large numbers in Alexandrov spaces, Calc. Var. Partial Differential Equations. vol. 54, pp: 1591-1610 (2015)
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S-I. Ohta and M. Palfia. Gradient flows and a Trotter-Kato formula of semi-convex functions on CAT(1)-spaces. Amer. J. Math. vol. 139. pp: 937-965 (2017)
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V. Chepoi et al. Packing and Covering with Balls on Busemann Surfaces. Discrete Comput. Geom. vol 57. pp:985-1011 (2017)
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A. Samal et al. Comparative analysis of two discretizations of Ricci curvature for complex networks. Sci. Rep. vol. 8, #8650 (2018)
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C. Ni et al. Community Detection on Networks with Ricci Flow. Sci. Rep. vol. 9, #9984 (2019)
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H. Farooq et al. Network curvature as a hallmark of brain structural connectivity. Nat Commun. vol. 10, #4937 (2019)
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V. Chepoi et al. Fast approximation of eccentricities and distances in hyperbolic graphs. Journal of Graph Algorithms and Applications. vol. 23(2), pp.393–433 (2019)
Videos
Short talks
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Gradient descent algorithms for Bures-Wasserstein barycenters. T. Maunu. (link)
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Estimation of the Wasserstein distance in the spiked transport model. J. Niles-Weed. (link)
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Recent Progress and New Challenges in Learning Undirected Graphical Models. G. Bresler. (link)
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An introduction to optimal transport. N. Gigli. (link)
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Of triangles, gases, prices and men. C. Villani. (link)