Research

Research Interests: Optimal Transport, Partial Differential Equations, Calculus of Variations, and Geometry:

I am a member of The Kantorovich Initiative, which is a interdisciplinary research group focusing on optimal transport.
There will be a summer school in June 2022 in Seattle. We also accept postdoc applications.

Collaborators (in alphabetical order):

Xiaohui Chen Sunhi Choi Raymond Chu Gonzalo Davila Nabarun Deb Samer Dweik Alessio Figalli Nassif Ghoussoub Seung-Yeal Ha Matthieu Heitz Hyunseok Kim Inwon C. Kim Seick Kim Jun Kitagawa Forest Kobayashi Hugo Lavenant Hanjin Lee Tongseok Lim Robert McCann Emanuel Milman Javier Morales Mathav Murugan Kyeongsik Nam Aaron Zeff Palmer Jinyeong Park Soumik Pal Brendan Pass Yuan Long Ruan Geoff Schiebinger Dave Schneider Jeff Streets Micah Warren Jane Wang Roomina Zand Stephen Zhang

Publications & Preprints:

Existence for the Supercooled Stefan Problem in General Dimensions, with Sunhi Choi and Inwon Kim. 27 pages. (Mar. 2024).
The nonlocal Stefan problem via a martingale transport, with Raymond Chu, Inwon Kim, Young-Heon Kim, Kyeongsik Nam. 42 pages. (Nov. 2023).
Wasserstein mirror gradient flows as the limit of the Sinkhorn Algorithm, with Nabarun Deb, Soumik Pal, and Geoffrey Schiebinger. 49 pages. Oct. 2023.
Backward and Forward Wasserstein Projections in Stochastic Order, with Yuan Long Ruan. Journal of Functional Analysis. Volume 286, Issue 2, 15 January 2024, 110201 (arXiv:2110.04822, Oct. 2021)
The Stefan problem and free targets of optimal Brownian martingale transport, with Inwon C. Kim To appear in Annals of Applied Probability. (arXiv:2110.03831. Oct. 2021.)
Towards a mathematical theory of trajectory inference, with Hugo Lavenant, Geoffrey Schiebinger, and Stephen Zhang. the Annals of Applied Probability, 34(1A): 428-500 (February 2024) (arXiv:2102.09204. Feb. 2021.)
Hidden convexity in a problem of nonlinear elasticity, with Nassif Ghoussoub, Hugo Lavenant and Aaron Zeff Palmer. SIAM J. Math. Anal., 53(1), 1070–1087. (arXiv:2004.10287. April 2020.)
Optimal stopping of stochastic transport minimizing submartingale costs. with Nassif Ghoussoub and Aaron Zeff Palmer. Trans. Amer. Math. Soc. 374 (2021), 6963-6989 (arXiv:2003.06465. March. 2020.)
Optimal transport and barycenters for dendritic measures with Brendan Pass and Dave Schneider (Oct. 2019.) Pure Appl. Anal. 2(3): 581-601 (2020). DOI: 10.2140/paa.2020.2.581 (MSP journals)
Stochastic optimal transport with free end time with Samer Dweik, Nassif Ghoussoub and Aaron Zeff Palmer. (Sept. 2019.) Ann. Inst. H. Poincaré Probab. Statist. 57(2): 700-725 (May 2021).
A solution to the Monge transport problem for Brownian martingales , with Nassif Ghoussoub and Aaron Zeff Palmer. (March 2019.) the Annals of Probability 2021, Vol. 49, No. 2, 877-907.
PDE Methods For Optimal Skorokhod Embeddings, with Nassif Ghoussoub and Aaron Zeff Palmer. July 2018. Calc. Var. (2019) 58: 113. https://doi.org/10.1007/s00526-019-1563-7
Optimal Transport with Controlled Dynamics and Free End Times, with Nassif Ghoussoub and Aaron Zeff Palmer. March 2018. SIAM J. Control Optim., 56(5), 3239–3259. (21 pages)
Optimal Brownian Stopping for radially symmetric marginals in general dimensions, with Nassif Ghoussoub and Tongseok Lim. (March 2018.) SIAM J. Control Optim., 58(5), 2765–2789.
Canonical barycenters on Riemannian manifolds via Wasserstein regularization, with Brendan Pass. SIAM J. Math. Anal., 50(2), 1817–1828.
Emergence of phase concentration for the Kuramoto-Sakaguchi equation, with Seung-Yeal Ha, Javier Morales, and Jinyeong Park. (Oct. 2016.) Physica D: Nonlinear Phenomena Volume 401, January 2020, 132154. 24pages
Dynamics of Optimal Partial Transport, with Gonzalo Davila. Aug. 2015. Calculus of Variations and Partial Differential Equations (CVPDE). First Online:22 September 2016. DOI: 10.1007/s00526-016-1052-1
Structure of optimal martingale transport in general dimensions, with Nassif Ghoussoub and Tongseok Lim. Aug. 2015. Ann. Probab. Volume 47, Number 1 (2019), 109-164.
Nonpositive curvature, the variance functional, and the Wasserstein barycenter, with Brendan Pass. Submitted. March 22, 2015. Proceedings of the American Mathematical Society.148 (2020), 1745-1756.
Wasserstein Barycenters over Riemannian manfolds, with Brendan Pass. Submitted. Dec. 25, 2014. Advances in Mathematics, Volume 307, 5 February 2017, Pages 640–683.
Prohibiting isolated singularities in optimal transport, with Jun Kitagawa. Submitted. March. 2014. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. XVI (2016), 277-290.
On weak solutions of linear elliptic equations with singular drift terms, with Hyunseok Kim. Submitted. March 2014. SIAM J. Math. Anal. 47 (2015), no. 2, 1271–1290.
A general condition for Monge solutions in the multi-marginal optimal transport problem, with Brendan Pass. Submitted. July 24, 2013. SIAM J. Math. Anal. 46 (2014) no. 2, 1538-1550.
Multi-marginal optimal transport on Riemannian manifolds, with Brendan Pass. Submitted. March 2013. Amer. J. Math. 137 (2015), no. 4, 1045–1060
On the degeneracy of optimal transportation, with Jun Kitagawa. Submitted. Jan. 2013. Comm. Partial Differential Equations, 39(7), pp. 1329-1363, 2014.
Lectures on Ma-Trudinger-Wang curvature and regularity of optimal transport maps. Analysis and Geometry of Metric Measure Spaces: Lecture Notes of the 50th Seminaire De Mathematiques Superieures (SMS), Montreal, 2011, CRM Proceedings and Lecture Notes, 56 (2013), 119--143.
On supporting hyperplanes to convex bodies, with Alessio Figalli and Robert J. McCann. Methods Appl. Anal. Volume 20 (2013) no. 3, 261-272.
Hoelder continuity and injectivity of optimal maps,with Alessio Figalli and Robert J. McCann. Arch. Rational Mech. Anal. 209 (2013) 747-795. This paper supercedes the previous paper "Continuity and injectivity for optimal maps with non-negatively cross-curved costs" below.
Efficient minimax estimation of high-dimensional sparse precision matrices, with Xiaohui Chen and Z. Jane Wang. IEEE Transactions on Signal Processing, Vol. 60, No. 6, June 2012: 2899-- 2912.
Parabolic optimal transport equations on manifolds, with Jeff Streets and Micah Warren. Int Math Res Notices (2012) Vol. 2012 4325-4350 first published online October 3, 2011 doi:10.1093/imrn/rnr188
Regularity of optimal transport maps on multiple products of spheres, with Alessio Figalli and Robert J. McCann. J. Eur. Math. Soc. (JEMS) 15 (2013) 1131-1166.
A generalization of Caffarelli's contraction theorem via (reverse) heat flow, with Emanuel Milman. Math. Ann. 354 (3), 827-862, 2012
Partial regularity of Brenier solutions of the Monge-Ampère equation, with Alessio Figalli. Discrete Contin. Dyn. Syst. Series A, 28 (2010), no. 2, 559-565.
When is multidimensional screening a convex program? with Alessio Figalli and Robert J. McCann. J. Econom. Theory 146 (2011) 454-478.
Continuity and injectivity for optimal maps with non-negatively cross-curved costs with Alessio Figalli and Robert J. McCann. Submitted. Nov. 2009 (superseded by a more recent paper Hoelder continuity and injectivity of optimal maps, listed above).
Pseudo-Riemannian geometry calibrates optimal transportation, with Robert J. McCann and Micah Warren. Math. Res. Lett. 17 (2010) 1183-1197
Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular), with Robert J. McCann . J. Reine Angew. Math. (Crelle's journal), (2012), vol 664, 1--27
Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds. International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn120, 15 pages, doi:10.1093/imrn/rnn120.
Continuity, curvature, and the general covariance of optimal transportation, with Robert J. McCann. J. Eur. Math. Soc. (JEMS) 12 (2010) 1009-1040.
Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality. Duke Mathematical Journal, vol 144, no. 1 (2008) 73--107.
Holomorphic Extensions of Laplacians and Their Determinants. Advances in Mathematics 211(2007) 517--545. (This is the paper version of my thesis for my PhD degree for Northwestern University in 2005. My thesis advisor was Ezra Getzler.)
Determinants of Laplacians, quasifuchsian spaces, and holomorphic extensions of Laplacians. In the Tradition of Ahlfors-Bers, IV, Contemporary Mathematics 432 (2007) 143--150. (A summary of my thesis.)
Harmonic maps into open manifolds with nonnegative curvature, with Jin-Whan Yim. Commun. Korean Math. Soc. 11 (1996) no.3, 789--796.

Notes/extended abstracts, etc:

Hoelder continuity of optimal transport maps, with Alessio Figalli and Robert McCann. Oberwolfach Report 36/2012: 2216-2218.
Regularity of optimal transport maps on multiple products of spheres, with Alessio Figalli and Robert McCann. Oberwolfach Report 31/2010: 1859— 1861.
Continuity of optimal transport maps under a degenerate MTW condition, with Alessio Figalli and Robert McCann. Oberwolfach Report 31/2008: Vol 5, issue 3, 1751—1753.
Curvature and continuity of optimal transport, with Robert J. McCann. An extended abstract. Oberwolfach Report 35/2007:26--29.

Some unpublished manuscripts:

An old version (with Appendix on regularity) "Continuity, curvature, and the general covariance of optimal transportation". with Robert J. McCann. Dec. 2007.
On the cost-subdifferentials of cost-convex functions, with Robert J. McCann. Preprint, June 2007.

Conference/workshop organizing:

CMC Thematic Program on Many-Body Interacting Systems and Optimal Mass Transport. April - August, 2017. Seoul National University and Korea Institute for Advanced Study (KIAS). (with Seung-Yeal Ha).
Geometric Measure Theory and Optimal Transport (minischool). August 7 - 10, 2017. Korea Institute for Advanced Study (KIAS), Seoul, Korea.
Optimal transport and related topics. July 31 - August 4, 2017. Centre for Mathematical Challenge (CMC) conference: Korea Institute for Advanced Study (KIAS), Seoul, Korea.
Optimal Transport and Machine Learning (minischool). July 3 - 7, 2017. Korea Institute for Advanced Study (KIAS), Seoul, Korea.
Monge-Ampere type equations in geometry, optics and optimal transportation. June 27 - July 1, 2016. A special session in 7th Pacific Rim Conference on Mathematics, Seoul National University, Korea. (with Neil Trudinger).
KIAS CMC conference on Analysis, Geometry, and Optimal Transport. June 20 -24, 2016. Korea Institute for Advanced Study (KIAS), Seoul, Korea.
KIAS CMC minischool on Analysis, Geometry, and Optimal Transport. Aug. 24-28, 2015. Korea Institute for Advanced Study (KIAS), Seoul, Korea.
Thematic Program on Variational Problems in Physics, Economics and Geometry. July-December 2014. Fields Institute, Toronto, ON. With Robert McCann and Wilfrid Gangbo.
CONFERENCE OF CALCULUS OF VARIATIONS: Geometry, Inequalities, and Design. November 10-14, Fields Institute, Toronto, ON. With Almut Burchard (Toronto), Panagiota Daskalopoulos (New York), Ludovic Rifford (Nice), Neil Trudinger (Canberra)
Minischool on Variational Problems in Geometry. November 6-7, 2014, Fields Institute, Toronto, ON. With Almut Burchard (Toronto), Panagiota Daskalopoulos (New York), Ludovic Rifford (Nice), Neil Trudinger (Canberra)
Workshop on Nonlinear Analysis and Nonlinear Elliptic PDEs. August 9 – 12, 2014. Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea. (with Jaeyoung Byeon)
Analysis and PDE July 7--12, 2013. "Conference in the honor of the 60th birthday of Nassif Ghoussoub". University of British Columbia, Vancouver, Canada. With Stephen Gustafson (UBC), Dong Li (UBC), Abbas Moameni (Lethbridge), Tai-Peng Tsai (UBC), Juncheng Wei (UBC), Meijun Zhu (Oklahoma).
Optimal Transportation and Differential Geometry May 20-- 25, 2012. Banff International Research Station (BIRS), Banff, AB, Canada. (with Alessio Figalli)
PRIMA-PARC-PIMS meeting in PDE. Dec. 7--8, 2010. University of British Columbia, Vancouver, Canada. (with Nassif Ghoussoub, Stephen Gustafson and Tai-Peng Tsai)
Analysis and Geometry of Nonlinear Partial Differential Equations. Special session in Canadian Mathematical Society 2010 Winter Metting. Dec 4--6. 2010, Vancouver, Canada. (with Nassif Ghoussoub and Robert J. McCann.)
Optimal transportation and applications. April 18--23, 2010. Banff International Research Station (BIRS), Banff, AB, Canada. (with Alessio Figalli, Yuxin Ge, Robert J. McCann and Neil Trudinger. )
PIMS Mini-symposium in PDE, Nov. 12--13, 2009. University of British Columbia, Vancouver, Canada. (with Stephen Gustafson)

Updated: 2021