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

  • M. P. do Carmo. Differential geometry of curves and surfaces2nd Edition. Dover. 2016

  • J. McCleary. Geometry from a differentiable viewpoint. 2nd Edition. Cambridge Univesrity Press. 2013.

Riemannian geometry​

  • P. Petersen. Riemannian Geometry. 2nd Edition. Springer. 2006

  • M. P. do Carmo. Riemannian Geometry. Birkhauser. 1992

Metric geometry

  • D. Burago, Y. Burago and S. Ivanov. A course in metric geometryGraduate Studies in Mathematics. American Mathematical Society. 2001

  • S. Alexander, V. Kapovitch and A. Petrunin. Alexandrov geometry: preliminary version no. 1. (book in preparation )

Optimal transport​

  • C. Villani. Topics in Optimal Transportation. Graduate Studies in Mathematics. American Mathematical Society. 2003

  • C. Villani. Optimal transport: Old and new. Springer. 2008

  • L. Ambrosio, N. Gigli and G. Savare. Gradient flows in metric spaces and in the space of probability measures. Springer. 2008

  • 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.

  • Y. Ollivier, H. Pajot and C. Villani (Editors)Optimal Transport: Theory and Applications. London Mathematical Society Lecture Note Series. 2014.

  • F. Santambrogio. Optimal transport for applied mathematicians. Birkhauser. 2015.

  • M. Cuturi and G. Peyre. Computational optimal transport. Foundations and trends in Machine Learning. vol 11(5-6). pp 355-607. 2019.

Graph theory​

  • J. A. Bondy and and U. S. R. Murty. Graph theory. Springer. 2008

  • R. Diestel. Graph theory. Springer. 2017.

Discrete Geometry

Articles

  • 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.

  • Y. Ollivier. Discrete Ricci curvature: Open problems. Manuscript (2008)(http://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf)

  • F. Memoli. Gromov–Wasserstein Distances and the Metric Approach to Object MatchingFound Comput Math. vol 11, pp:417-487 (2011) 

  • M. Bacak. Computing Medians and Means in Hadamard SpacesSIAM J. Optim., vol. 24(3), pp: 1542–1566 (2014)

  • 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)

  • 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)

  • V. Chepoi et al. Packing and Covering with Balls on Busemann SurfacesDiscrete Comput. Geom. vol 57. pp:985-1011 (2017)

  • A. Samal et al. Comparative analysis of two discretizations of Ricci curvature for complex networksSci. Rep. vol. 8, #8650 (2018)

  • C. Ni et al. Community Detection on Networks with Ricci FlowSci. Rep. vol. 9, #9984 (2019)

  • H. Farooq et al. Network curvature as a hallmark of brain structural connectivityNat Commun. vol. 10, #4937 (2019) 

  • V. Chepoi et al. Fast approximation of eccentricities and distances in hyperbolic graphsJournal of Graph Algorithms and Applications. vol. 23(2), pp.393–433 (2019)

Videos

​Short talks

  • Gradient descent algorithms for Bures-Wasserstein barycenters. T. Maunu. (link)

  • Estimation of the Wasserstein distance in the spiked transport model. J. Niles-Weed. (link)

  • Recent Progress and New Challenges in Learning Undirected Graphical Models. G. Bresler. (link)

  • An introduction to optimal transport. N. Gigli. (link)

  • Of triangles, gases, prices and men. C. Villani. (link)

Longer video series​

  • Short course on gradient flows by F. Otto (part 1, part 2, part 3part 4, part 5

  • MIT course on Geometric Data Analysis (link)

  • Great lectures on Riemannian Geometry by Pr Frederic P. Schuller (link)