** 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.
* Ann. of Math. (2) * ** 171 ** (2010) 673-730

[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*) *Amer. J. Math. * ** 132 ** (2010) 711-730

[29] * Constructing a relativistic heat flow by transport time steps , * with Marjolaine Puel. * Ann. Inst. H. Poincare Anal. Non Lineaire * ** 26 ** (2009) 2539-2580.

[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) 375-395.

[32] * Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness*, with Pierre-Andre Chiappori and Lars Nesheim. *Econom. Theory* **42** (2010) 317-354

[33] * Continuity, curvature, and the general covariance of optimal transportation , * with Young-Heon Kim. * J. Eur. Math. Soc. (JEMS) *
** 12** (2010) 1009-1040

[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. *J. Reine Angew. Math.* 664 (2012) 1-27.

[37] * A family of nonlinear fourth order equations of gradient flow type*, with Daniel
Matthes and Giuseppe Savare.
* Comm. Partial Differential Equations. * ** 34 ** (2009) 1352-1397.

[38] * Optimal transportation, topology and uniqueness* (formerly titled *Extremal doubly stochastic measures and
optimal transportation)*, with Najma
Ahmad and Hwa Kil Kim.
* Bull. Math. Sci.* ** 1** (2011) 13-32

[39] * Pseudo-Riemannian geometry
calibrates optimal transportation*, with Young-Heon
Kim and Micah Warren . *Math. Res. Lett.* **17** (2010) 1183-1197.

[40] * The Ma-Trudinger-Wang curvature for natural mechanical actions*, with Paul W.Y.
Lee .
*Calc. Var. Partial Differential Equations.* **41** (2011) 285-299

[41] * When is multidimensional screening a convex program?*, with Alessio
Figalli
and Young-Heon
Kim .
*J. Econom. Theory* **146** (2011) 454-478.

[42] * Regularity of optimal transport maps on multiple products of spheres*,
with Alessio
Figalli
and Young-Heon
Kim . * J. Eur. Math. Soc. (JEMS)* 15 (2013) 1131-1166.

[43] * Hoelder continuity for optimal multivalued mappings, * with Maria Sosio. *
SIAM J. Math. Anal.* ** 43 ** (2011) 1855-1871

[44] * Rectifiability of optimal transportation plans*, with Brendan
Pass and Micah
Warren .
*Canad. J. Math.* ** 64 ** (2012) 924-934

[45] * Unique equilibria and substitution effects in a stochastic model of the marriage market*, with Colin Decker,
Elliott H. Lieb, and Benjamin K.
Stephens .
*J. Econom. Theory* 148 (2013) 778-792

[46] * Five lectures on optimal transportation: geometry, regularity and applications, * with Nestor Guillen.
In * Analysis and Geometry of Metric Measure Spaces: Lecture Notes
of the Seminaire de Mathematiques Superieure (SMS) Montreal 2011. *
G. Dafni et al, eds. Providence: Amer. Math. Soc. (2013) 145-180.

[47] * Hoelder continuity and injectivity of
optimal maps*, with Alessio
Figalli
and Young-Heon
Kim .
* Arch. Rational Mech. Anal.* ** 209 ** (2013) 747-795

[48] * On supporting hyperplanes to convex bodies*, with Alessio
Figalli
and Young-Heon
Kim .
* Methods Appl. Anal.* ** 20** (2013) 261-272.

[49] * Optimal transportation with capacity constraints*, with Jonathan
Korman
* Trans. Amer. Math. Soc.* ** 367** (2015) 1501-1521.

[50] * Becker meets Ricardo: Multisector matching
with communication and cognitive skills*, with Xianwen
Shi , Aloyius
Siow and Ronald
Wolthoff.
Originally titled
* Becker meets Ricardo: multisector matching
with social and cognitive skills*. * Journal of Law, Economics
and Organization * 31 (4) (2015) 690-720 doi: 10.1093/jleo/ewv002

[51] * Higher order time asymptotics of fast
diffusion in Euclidean space (via dynamical systems methods)
, * with Jochen
Denzler and
Herbert Koch.
* Mem. Amer. Math. Soc.* **234** (2015) 1-94

[52] * A glimpse into the differential
topology and geometry of optimal transport . *
* Discrete Contin. Dyn. Syst.* **34** (2014) 1605-1621.

[53] * Insights into capacity
constrained optimal transport, * with Jonathan Korman.
* Proc. Natl. Acad. Sci. USA,* **110** (2013) 10064-10067.

[54] * Dual potentials for capacity constrained
optimal transport, * with Jonathan Korman and Christian Seis. * Calc. Var. Partial Differential Equations* ** 54 ** (2015) 573-584.

[55] * An elementary approach to linear programming duality with application to capacity constrained transport, * with Jonathan Korman and Christian Seis.
*J. Convex Anal.* ** 22** (2015) 797-808.

[56] * The spectrum of a family
of fourth-order nonlinear diffusions near the global attractor, * with
Christian Seis.
* Comm. Partial Differential Equations.* ** 40** (2015) 191-218.

[57] * Long-time asymptotic expansions
for nonlinear diffusions in Euclidean space
, * with Jochen
Denzler and
Herbert Koch.
* Contemp. Math. * ** 656 ** (2016) 85-94.

[58] *
Academic wages, singularities, phase transitions and pyramid schemes
, *
Proceedings of the International Congress of Mathematicians (Seoul 2014)
S.Y. Jang et al, eds., vol III Invited Lectures, Seoul, Kyung Moon SA, 2014, pp 835-849.

[59] * Academic wages and pyramid schemes: a mathematical model*, with Alice Erlinger, Xianwen
Shi , Aloyius
Siow and Ronald
Wolthoff.
* J. Functional Analysis * ** 269 ** (2015) 2709-2746.

[60] * The intrinsic dynamics of optimal transport*, with Ludovic
Rifford
* J. Ecole Polytechnique - Math. * ** 3** (2016) 67-98.

[62] * Multi- to one-dimensional optimal transport*, with Pierre-Andre Chiappori and Brendan Pass.
* Comm. Pure Appl. Math. * ** 70 ** (2017) 2405-2444.

[63] * Multidimensional matching: theory and empirics*, with Pierre-Andre Chiappori and Brendan Pass.

[64] * On concavity of the monopolist's
problem facing consumers with nonlinear price preferences*, with Kelvin Shuangjian
Zhang.
* Comm. Pure and Applied Math.* **72**(7) (2019) 1386-1423

[65] *Free discontinuties in
optimal transport*, with Jun
Kitagawa.
* Arch. Rational Mech. Anal.* ** 232** (2019) 1505-1541.
DOI 10.1007/s00205-018-01348-3

[66] * Transition to nestedness in multi- to one-dimensional optimal transport*, with Pierre-Andre Chiappori and Brendan Pass. Euro. J. Appl. Math. ** 30 ** (2019) 1220-1228 doi.org/10.1017/S0956792518000578

[67] * Optimal transportation between unequal dimensions*, with Brendan Pass. arXiv:1805.11187v2. *Arch. Rational Mech. Analysis* ** 238 ** (2020) 1475-1520 doi.org/10.1007/s00205-020-01569-5

[68] * Displacement convexity of Boltzmann's entropy characterizes the strong energy condition from general relativity*, arXiv1808.1536v2. *Camb. J. Math.* **8**:3 (2020) 609-681.

[69] *Dimensionality Reduction has Quantifiable Imperfections: Two Geometric Bounds*, with Yik Chau (Kry) Lui, Gavin Weiguang Ding, and Ruitong Huang.
In *Advances in Neural Information Processing Systems 31 (NeurIPS)* S. Bengio et al, eds. Curran Associates Inc.
2018, pp 8453-8463.

[70] *Isodiametry, variance, and regular simplices from particle interactions*, with Tongseok Lim
arXiv:1907.13593.
*Archive Rational Mech. Analysis* ** 241 ** (2021) 553-576.

[71] *Geometrical bounds for the variance and recentered moments*, with Tongseok Lim arXiv:2001:11851. *Math. Oper. Res.* ** 47** (2022) 286-296.

[72] *Inscribed radius bounds for lower Ricci bounded metric measure spaces with mean convex boundary*, with Annegret
Burtscher,
Christian
Ketterer, and Eric
Woolgar,
arXiv:2005.07435. *SIGMA Symmetry Integrability Geom. Methods Appl.* **16** (2020),
131, 29 pages. (Special issue in honor of Gromov's 75th birthday)

[73] *Independence of synthetic Curvature Dimension conditions on transport distance exponent*, with Afiny
Akdemir,
Fabio
Cavalletti, Andrew
Colinet, and Flavia
Santarcangelo,
* Trans. Amer. Math. Soc.* ** 374** (2021) 5877-5923. arXiv:2005.07435

[74] *On Fejes Toth's conjectured maximizer for the sum of angles between lines*, with Tongseok Lim, *Appl. Math. Optim.* **84**(3) (2021) 3217-3227. DOI 10.1007/s00245-020-09745-5 arXiv:2007.08698

[75] *Maximizing expected powers of the angle between pairs of points in projective space*, with Tongseok Lim, and an appendix by Dmitriy Bilyk, Alexey Glazyrin, Ryan Matzke, Josiah Park, and Oleksandr Vlasiuk. *Probab. Theory Related Fields* **184 ** (2022) 1197-1214. arXiv:2007.13052

[76] *On the cardinality of sets in Rd obeying a slightly obtuse angle bound*, with Tongseok Lim, *SIAM J. Discrete Math,* ** 36** (2022) 1093-1101, arXiv:2007.13871

[77] *Classifying minimum energy states for interacting particles: spherical shells*, with Cameron Davies and Tongseok Lim, *SIAM J. Appl. Math.* **82** (2022) 1520-1536. arXiv:2107.11718

[78] *Classifying minimum energy states for interacting particles: regular simplices*, with Cameron Davies and Tongseok Lim, *Comm. Math. Phys.* ** 399 ** (2023) 577-598 (2022). arXiv:2109.07091

[79] *A Lorentzian analog for Hausdorff dimension and measure*, with Clemens Saemann, *Pure and Applied Analysis* (2022) 367-400. arXiv:2110.04386

[80] *Asymptotics near extinction for nonlinear fast diffusion on a bounded domain*, with Beomjun Choi and Christian Seis, *Archive Rational Mech. Analysis* ** 247 ** (2023) 2:16 (48 pp) arXiv:2202.02769

[81] * A duality and free boundary approach to adverse selection*, with Kelvin Shuangjian
Zhang. * arXiv 2301.07660*

[82] * C{1,1} Regularity for principal-agent problems *, with Cale Rankin and Kelvin Shuangjian
Zhang. * arXiv 2303.04937*

[83] * A synthetic null energy condition*,
to appear in * Comm. Math. Phys.* arXiv 2304.14341

[84] * Comment on "Ironing, sweeping, and multidimensional screening"*, with Kelvin Shuangjian
Zhang. * arXiv 2311.13012*

[85] * A geometric approach to apriori estimates for optimal maps*, with Simon Brendle, Flavien Leger and Cale Rankin * arXiv 2311.10208*

[86] * Causal convergence conditions through variable timelike Ricci curvature bounds*, with Mathias
Braun. *
arXiv 2311.13012*

BOOK REVIEWS

[61] * Villani's `Birth of a Theorem'*,
SIAM News ** 48** #10 (2015) p9.

Professor Robert J. McCann, FRSC

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 -at- math -dot- toronto -dot- edu

Office location: Room BA 6124

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Monday 19 February 2024

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