Robert J. McCann.

Professor, Department of Mathematics, University of Toronto

Courses Offered 2005--06:
SCI 199 H1S-LEC 0291 Aha! Mathematical discovery and creative problem solving

Research Interests:
- mathematical physics and mathematical economics
- convex analysis, geometry and optimization
- partial differential equations

DEPARTMENTAL SEMINAR LISTINGS
Analysis / Applied Math / PDE Seminar
Fields Analysis Working Group

Publications:
[1] Uniform density theorem for the Hubbard model, with Elliott H. Lieb and Michael Loss. J. Math. Phys 34, 891-898 (1993)
[2] A Convexity Theory for Interacting Gases and Equilibrium Crystals. PhD Thesis, Princeton University (1994)
[3] Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80, 309-323, (1995)
[4] Optimal maps in Monge's mass transport problem, with Wilfrid Gangbo. C.R. Acad. Sci. Paris. Ser. I. Math. 325, 1653-1658 (1995)
[5] The geometry of optimal transportation, with Wilfrid Gangbo. Acta Math. 177, 113-161 (1996)
[6] A convexity principle for interacting gases. Adv. Math. 128, 153-179 (1997)
[7] Equilibrium shapes for planar crystals in an external field. Comm. Math. Phys. 195, 699-723 (1998)
[8] Exact solutions to the transportation problem on the line. Proc. Royal Soc. London Ser. A 455, 1341-1380 (1999)
[9] Shape recognition via Wasserstein distance, with Wilfrid Gangbo. Quart. Appl. Math. 58, 705-737 (2000)
[10] Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589-608
[11] A Riemannian interpolation inequality a la Borell, Brascamp and Lieb, with Dario Cordero-Erausquin and Michael Schmuckenschlaeger. Invent. Math. 146 (2001) 219-257
[12] Constructing optimal maps in Monge's transport problem as a limit of strictly convex costs, with Luis A. Caffarelli and Mikhail Feldman. J. Amer. Math. Soc. 15 (2002) 1-26
[13] Uniqueness and transport density in Monge's mass transportation problem , with Mikhail Feldman. Calc. Var. Partial Differential Equations. 15 (2002) 81-113
[14] Monge's transport problem on a Riemannian manifold, with Mikhail Feldman. Trans. Amer. Math. Soc. 354 (2002) 1667-1697
[15] Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, with Jose A. Carrillo and Cedric Villani. Revista Mat. Iberoamericana 19 (2003) 971-1018
[16] Stable rotating binary stars and fluid in a tube. Houston J. Math. 32 (2006) 603-632
[17] Phase transitions and symmetry breaking in singular diffusion, with Jochen Denzler. Proc. Natl. Acad. Sci. USA 100 (2003) 6922-6925.
[18] Exact semi-geostrophic flows in an elliptical ocean basin, with Adam Oberman. Appendix by Maxim Trokhimtchouk. Nonlinearity 17 (2004) 1891-1922
[19] A least action principle for steepest descent in a non-convex landscape , with Nassif Ghoussoub. Contemp. Math. 362 (2004) 177-187.
[20] Fast diffusion to self-similarity: complete spectrum, long time asymptotics, and numerology, with Jochen Denzler. Arch. Rational Mech. Anal. 175 (2005) 301-342
[21] Contractions in the 2-Wasserstein length space and thermalization of granular media , with Jose A. Carrillo and Cedric Villani. Arch. Rational Mech. Anal. 179 (2006) 217-263
[22] Prekopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport , with Dario Cordero-Erausquin and Michael Schmuckenschlaeger. Ann. Fac. Sci. Toulouse Math. (6) 15 (2006) 613-635.
[23] Sharp decay rates for the fastest conservative diffusions, with Yong-Jung Kim. C. R. Acad. Sci. Paris Ser. I Math. 341 (2005) 157-162
[24] Potential theory and optimal convergence rates in fast nonlinear diffusion, with Yong-Jung Kim. J. Math. Pures Appl. 86 (2006) 42-67
[25] Second-order asymptotics for the fast-diffusion equation , with Dejan Slepcev. Int. Math. Res. Not. 24947 (2006) 1-22
[26] Free boundaries in optimal transport and Monge-Ampere obstacle problems, with Luis A. Caffarelli. To appear in Ann. of Math. (2)
[27] Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, & nonradial focusing geometries , with Jochen Denzler. Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) 865-888
[28] Ricci flow, entropy, and optimal transportation , with Peter Topping. (formely titled "Diffusion is a 2-Wasserstein contraction on any manifold evolving by reverse Ricci flow") To appear in Amer. J. Math.
[29] Constructing a relativistic heat flow by transport time steps , with Marjolaine Puel. To appear in Ann. Inst. H. Poincare Anal. Non Lineaire .
[30] Chaotic response of the 2D semi-geostrophic and 3D quasi-geostrophic equations to gentle periodic forcing , with Dorian Goldman. Nonlinearity 21 (2008) 1455-1470 doi 10.1088/0951-7715/21/7/005.
[31] Optimal partition of a large labor force into working pairs, with Maxim Trokhimtchouk. Econom. Theory 42 (2010) page 375.
[32] Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness, with Pierre-Andre Chiappori and Lars Nesheim. To appear in Econom. Theory
[33] Continuity, curvature, and the general covariance of optimal transportation , with Young-Heon Kim. To appear in J. Eur. Math. Soc. (JEMS)
[34] Curvature and the continuity of optimal transport (joint work with Young-Heon Kim). Oberwolfach Rep. 4 (2007) 2060-2062
[35] Explicit Yamabe flow of an asymmetric cigar , with Almut Burchard and Aaron Smith. Methods Appl. Anal. 15 (2008) 65-80
[36] Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular) , with Young-Heon Kim. To appear in J. Reine Angew. Math.
[37] A family of nonlinear fourth order equations of gradient flow type with Daniel Mattes and Giuseppe Savare. To appear in Comm. Partial Differential Equations.
[38] Extremal doubly stochastic measures and optimal transportation with Najma Ahmad and Hwa Kil Kim.
[39] Calibrating optimal transportation: a new pseudo-Riemannian geometry with Young-Heon Kim and Micah Warren.
[40] The Ma-Trudinger-Wang curvature for natural mechanical actions with Paul W.Y. Lee .
[41] When do systematic gains uniquely determine the number of marriages between different types in the Choo-Siow matching model? Sufficient conditions for a unique equilibrium with Colin Decker and Benjamin K. Stephens .
[42] Continuity and injectivity for optimal maps with non-negatively cross-curved costs with Alessio Figalli and Young-Heon Kim .
[43] When is multidimensional screening a convex program? with Alessio Figalli and Young-Heon Kim .

Graduate Students:
Najma Ahmad, PhD 2004, The Geometry of Shape Recognition Via the Monge-Kantorovich Optimal Transport Problem
Nathan Killoran, MSc 2007, Supports of Extremal Doubly and Triply Stochastic Measures
Dorian Goldman, MSc 2008, Weak Lagrangian Solutions to a One-dimensional Model of the Moist Semi-geostrophic Equations
Jiayong Li, MSc 2009, Smooth Optimal Transportation on Hyperbolic Space

Conferences Organized:
CMS Winter Meeting Session December 8-10, 2007 at London, Ontario and December 11 Minisymposium in Toronto
Geometric Inequalities, June 17- 22, 2007 at Banff, Canada.
Calculus of Variations, July 9-15, 2006 at Oberwolfach, Germany.
Nonlinear diffusions: entropies, asymptotic behaviour, and applications, April 15-20, 2006 at Banff, Canada.
CMS Winter Meeting Session. December 10-12, 2005 at Victoria, Canada.
CMS Summer Meeting Session. June 4-6, 2005 at Waterloo, Canada.
AMS Sectional Meeting Session. March 18-19, 2005 at Bowling Green, KY.
Optimal Transport Theory and Applications. October 9-13, 2003 in Pisa, Italy.
Calculus of Variations: Geometric Problems, Superconductivity, and Material Microstructures. August 25-29, 2003 in Toronto, Canada.
Optimal Transportation and Nonlinear Dynamics. August 11-15, 2003 in Vancouver, Canada.
Calcul des Variations: Transport optimal... June 10-13, 2003 in Savoie, France.
Measure Transportation and Geometric Inequalities: July 8-12, 2002 at the Pacific Institute for the Mathematical Sciences, Vancouver BC.
UofT Nonlinear and Geometric Analysis Day: 7 December 2001. (Satellite to the December 8-10 CMS meeting in Toronto, Canada).
Problems and Perspectives on the Calculus of Variations: Physics Economics and Geometry. August 20-25, 2001 in Toronto, Canada.

Slide presentation:
Spreading versus shape in a gradient flow. July 11, 2002. From the PIMS conference above.

Professor Robert J. McCann
Department of Mathematics,
University of Toronto, Bahen Centre
40 St George St Room 6290
Toronto, Ontario M5S 2E4
Office: (416) 978-4658
FAX: 978-4107
E-mail: mccann@@math.toronto.edu
Office location: Room BA 6124

Last modified on Monday May 14, 2001
Comments and questions to mccann@@math.toronto.edu @