Seminar on optimal transport

October 17, 2007

Talks

29.10. Kantorovich duality, chapter 1 of [26]. Michael Munzert

05.11. Geometry of optimal transportation, chapter 2 of [26]. Tobias Marxen

12.11. Brenier’s polar factorization theorem, chapter 3 of [26]. Carla Cederbaum (?) or Tobias Lamm

19.11. The Monge-Ampère equation, chapter 4 of [26]. Anda Degeratu

References

[1] L. Ambrosio, L. Caffarelli, Y. Brenier, G. Buttazzo and C. Villani. Optimal transportation and applications. Lecture Notes in Mathematics, 1813. Springer Verlag, Berlin, 2003.

[2] L. Ambrosio, N. Gigli and G. Savaré. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics, ETH Zürich. Birkhäuser Verlag, Basel, 2005.

[3] D. Cordero-Erausquin, R. McCann and M. Schmuckenschläger. A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math., 146:219–257, 2001.

[4] D. Cordero-Erausquin, R. McCann and M. Schmuckenschläger. Prékopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields and optimal transport. Ann. Fac. Sci. Toulouse Math., 15:613–635, 2006.

[5] L. Evans. Partial differential equations and Mong-Kantorovich mass transfer. Lecture notes, available at http://math.berkeley.edu/ evans/

[6] Y.-H. Kim. Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds. Preprint, available at http://www.math.toronto.edu/yhkim/research.html

[7] Y.-H. Kim and R. McCann. Continuity, curvature, and the general covariance of optimal transportation. Preprint, available at http://www.math.toronto.edu/yhkim/research.html [8] G. Loeper. Continuity of maps solutions of optimal trans-

portation problems. Preprint, available at http://math.univlyon1.fr/ loeper/publications.htm.

[9] J. Lott. Optimal transport and Ricci curvature for metric-measure spaces. Preprint, to appear in Surveys of Differential Geometry X, available at http://www.math.lsa.umich.edu/ lott/papers.html.

[10] J. Lott. Some geometric calculations on Wasserstein space. Preprint, to appear in Comm. Math. Phys., available at http://www.math.lsa.umich.edu/ lott/papers.html.

[11] J. Lott and C. Villani. Weak curvature conditions and functional inequalities. J. Funct. Anal., 245:311–333, 2007.

[12] J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Preprint, to appear in Annals of Math., available at http://www.math.lsa.umich.edu/ lott/papers.html.

[13] X.-N. Ma, N. Trudinger and X.-J. Wang. Regularity of potential functions of the optimal transportation problem. Arch. Rational Mech. Anal., 177:151–183, 2005.

[14] R. McCann. Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal., 11:589–608, 2001.

[15] R. McCann and P. Topping. Ricci flow, entropy and optimal transportation. Preprint, 2007, available at http://www.warwick.ac.uk/ maseq/

[16] F. Otto and C. Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal., 173:361–400, 2000.

[17] F. Otto. The geometry of dissipative evolution equations: the porous medium equation. Comm. PDE, 26:101–174, 2001.

[18] S. Rachev and L. Rüschendorf. Mass transportation problems I. Probability and its applications. Springer Verlag, New York, 1998.

[19] S. Rachev and L. Rüschendorf. Mass transportation problems II. Probability and its applications. Springer Verlag, New York, 1998.

[20] K.-T. Sturm. On the geometry of metric measure spaces. Acta Math., 196:65–131, 2006.

[21] K.-T. Sturm. On the geometry of metric measure spaces II. Acta Math., 196:133–177, 2006.

[22] K.-T. Sturm and M.-K. von Renesse. Transport inequalities, gradient estimates, entropy and Ricci curvature. Comm. Pure Appl. Math., 58:923–940, 2005.

[23] P. Topping. Lectures on the Ricci flow. London Mathematical Society Lecture Notes Series, 325. Cambridge University Press, Cambridge, 2006.

[24] P. Topping. L-optimal transportation for Ricci flow. Preprint, 2007, available at http://www.warwick.ac.uk/ maseq/

[25] N. Trudinger. Recent developments in elliptic partial differential equations of Monge-Ampére type. ICM Lecture 2006, available at http://www.icm2006.org/proceedings/vol3.html

[26] C. Villani. Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003.

[27] C. Villani. Optimal transport, old and new. Book in preparation, available at http://www.umpa.ens-lyon.fr/ cvillani/surveys.html.