Research Interests: Partial Differential Equations and Geometric Analysis:

Analysis and Geometry of Optimal Transportation / Monge-Ampere Equations (some references)
Evolution equations and applications.
Variational calculus.
Riemannian Geometry
Spectral Geometry: Analytic Torsion (Determinants of Laplacians), Isospectrality 
Analysis of Riemann surfaces (Teichmueller Theory)
Inverse Problems
Mathematical economics 

Collaborators (in alphabetical order):

Xiaohui Chen    Alessio Figalli    Hyunseok Kim    Seick Kim    Jun Kitagawa    Robert McCann    Emanuel Milman   Hanjin Lee   Brendan Pass    Jeff Streets    Micah Warren    Jane Wang         

• 
  
• Multi-marginal optimal transport on Riemannian manifolds, with Brendan Pass. Submitted. March 2013.
  
• On the degeneracy of optimal transportation, with  Jun Kitagawa. Submitted. Jan. 2013.   
• On supporting hyperplanes to convex bodies, with Alessio Figalli and Robert J. McCann. Submitted. July 2011.
  
• Hoelder continuity and injectivity of optimal maps,with Alessio Figalli and Robert J. McCann.  Submitted. July 2011. To appear in  Archive for Rational Mechanics and Analysis (accepted on March 2013).  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. To appear in J. Eur. Math. Soc. (JEMS). 
• 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 Nov. 2009 (superseded by a more recent paper  Hoelder continuity and injectivity of optimal maps, listed above).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.
• Curvature and continuity of optimal transport, with  Robert J. McCann. An extended abstract. Oberwolfach Report 35/2007:26--29.
• 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.

• 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.
  

Presentation Slides

• Curvature and the continuity of optimal transportation maps (joint work with Robert J. McCann) June, 2007.

ETC

• One-Page Summary of Research (OUTDATED: written in Oct. 2007)

Conference/workshop organizing: 

• 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)   

Invited Talks and Lectures

•  
• On a multi-marginal optimal transport problem. March 9, 2013. Southern California Analysis & Partial Differential Equations (SCAPDE) meeting, Mar 9 -10, 2013, University of California at Los Angeles, Los Angeles, CA.
  
• Hoelder continuity of optimal transport maps. August 6, 2012 Workshop “Recent trends in geometric and nonlinear analysis”, August 5-10, 2012. Banff International Research Station, Banff, AB.  
• Hoelder continuity of optimal transport maps. July 23, 2012. Workshop “Calculus of Variations”, July 22-28, 2012. Oberwolfach, Germany.
• Calibration, pseudo-Riemannian geometry, and optimal transportation. June 26, 2012. Mini workshop on geometry, June 25 –27, 2012. University of Montpellier, France.
• Hoelder continuity of optimal transport maps. June 21, 2012.  Workshop “Optimal transport to Osray” June 18--22. University of Paris-Orsay. France
• Dynamics to optimal transport. June 4, 2012. Conference ``Matching Problems: Economics Meets Mathematics” June 4—6, University of Chicago, Chicago, IL, USA.
  
• Analysis and geometry in regularity of optimal maps.  April 27, 2012. Centre de Recherches Mathematiques, University of Montreal, Montreal. (Andre-Aisenstadt Prize lecture)
  
• Parabolic optimal transport equations. Analysis seminar. August 30, 2011. Yonsei University, Korea.
  
• Continuity of optimal transport maps and the curvature of Ma, Trudinger and Wang. Three hour minicourse. August 29, 2011. Seoul National University, Korea.
  
• Regularity of optimal transport maps.  Workshop “PRIMA-PARC-PIMS meeting in PDE”. August 25 –27, 2011. Seoul National University, Korea
  
• Parabolic optimal transport equations. Analysis seminar. August 24, 2011. KAIST, Korea.
  
• Regularity of optimal transport maps. Applied Analysis and Applied PDEs. July 12 -- July 15, 2011. Victoria, Canada.
• Ma-Trudinger-Wang curvature and regularity of optimal transport. Invited Lecturer (4 series lectures): 50th SMS 2011. Metric Measure Spaces: Geometric and Analytic Aspects. June 27 -- July 8, 2011. Centre de Recherches Mathematiques, Montreal, Quebec.
  
• Regularity of optimal transport maps on multiple products of spheres.. The second renunion conference for IPAM’s Optimal Transport Program. Dec. 12--17, 2010. Lake Arrowhead, CA.
• Regularity of optimal transport maps on multiple products of spheres.. Workshop on Geometric Probability and Optimal Transportation. Nov. 1--5, 2010 (tentative dates) Fields Institute, Toronto. Canada.
• Regularity of optimal transport maps on multiple products of spheres..  Optimal Transportation and Applications. Oct. 12 --16, 2010. SNS, Pisa, Italy.   
  
• Regularity of optimal transport maps on multiple products of spheres. Sept. 3, 2010.  SAGE workshop: Optimal Transport and Riemannian Geometry (Bakry-Emery curvature, Ma-Trudinger-Wang curvature, and applications to Ricci flow). Sept. 1--3, 2010. U. Texas, Austin, TX.
  
• Regularity of optimal transport maps on multiple products of spheres. Aug. 16, 2010. Colloquium. University of Iowa. Iowa City, IA.
  
• Regularity of optimal transport maps on multiple products of spheres. July 22, 2010. Calculus of Variations, July 18--24, 2010. Oberwolfach, Germany.
• Regularity of optimal transport maps on multiple products of spheres. July 14, 2010. FAWG seminar. U. Toronto, Toronto, Canada.
  
• Non-negatively cross-curved transportation costs, March, 22, 2010.  International Conference on Nonlinear PDE and Related Analysis/Applications (in conunction with the 64th Midwest PDE seminar), March 20--24, 2010. Northwestern University, Evanston, IL, USA.         
• Non-negatively cross-curved transportation costs. Dec  22, 2009. PDE Seminar. Sungkyunkwan University, Suwon, Korea.
• c-Monge-Ampere equations. Dec 21, 2009. PDE Seminar. Seoul National University, Seoul, Korea.              
  
• Non-negatively cross-curved transportation costs, Dec 19, 2009. Joint meeting of the Korean Mathematical Society and the American Mathematical Society: special session --- Nonlinear Partial Differential Equations and Viscosity Solutions. (Dec. 16--20, 2009)
• Non-negatively cross-curved transportation costs, Dec 8, 2009. The first reunion conference for IPAM’s Optimal Transport Program. Dec. 6--11, 2009. Lake Arrowhead, CA.
  
• Non-negatively cross-curved transportation costs, Nov 6, 2009. University of Washington, Seattle, WA.
• Non-negatively cross-curved transportation costs, Oct 2, 2009.  Praire Analysis Seminar Oct 2--3, 2009 (a contribution talk). Kansas State University, KS.
• Geometry and regularity of optimal transportation. June 1, 2009. Geometry/Physics seminar.Northwestern Univ. Evanston. IL.
• Determinants of Laplacians as functions on spaces of Riemannian metrics. March 9, 2009. Univ. of Maryland. College Park, MD.
• Determinants of Laplacians as functions on spaces of Riemannian metrics. Aug. 25, 2008. Seminar. Kyung Hee Univ. Seoul, Korea.
• On continuity of optimal transportation. Aug. 22, 2008. Seminar. Pohang Univ. Sci. & Tech. Pohang, Korea.                 
  
• Determinants of Laplacians as functions on spaces of Riemannian metrics. Aug. 22, 2008. Seminar. Pohang Univ. Sci. & Tech. Pohang, Korea.
• On continuity of optimal transportation. Aug. 8, 2008. Seminar. Korea Adv. Inst. Sci. & Tech. Daejun, Korea 
• Analysis of surfaces. Aug. 6, 2008. Geometry seminar. Seoul National Univ. Seoul, Korea.
• Continuity of optimal transportation. Aug. 5, 2008. PDE seminar. Seoul National Univ. Seoul, Korea.Curvature and continuity of optimal transportation. July 2009.
  
• Curvature and continuity of optimal transportation. Workshop on Transport, Optimization, Equilibrium in Economics. July 14 -- July 20, 2008.  PIMS, University of British Columbia, Vancouver, BC.
• Continuity of optimal transport maps under a degenerate MTW condition. July 10, 2008. Calculus of varations. Oberwolfach workshop, Germany.
• Some geometric aspects of regularity of optimal transportation. June 9, 2008. Optimal Transport --Culminating workshop at Lake Arrowhead. IPAM. CA.
• Some geometric aspects of regulairty of optimal transportation. May 2, 2008. Analysis and PDE seminar. UCLA, Los Angeles, CA.
• Curvature and continuity of optimal transport. Mar. 16, 2008. AMS 2008 Spring Eastern Meeting, Mar. 15-16, 2008 --Special Session on  Nonlinear Elliptic Equations and Geometric Inequalities. New York, NY.
  
• Curvature and continuity of optimal transport. Feb. 20, 2008. Colloquium. University of Oregon, Eugene, OR.
  
• Curvature and continuity of optimal transport. Feb. 18, 2008. Colloquium. Simon Fraser University, Vancouver, BC.
  
• Curvature and continuity of optimal transport. Jan 25 2008. Colloquium. University of Arizona, Tucson, AZ.
• Determinants of Laplacians as functions on spaces of metrics. Jan. 17, 2008. Diff. Geom.-Math. Phys.-PDE Seminar. University of British Columbia, Vancouver, BC.
• Curvature and continuity of optimal transport,  Jan 16. 2008, Colloquium. University of British Columbia, Vancouver, BC.
• Curvature and continuity of optimal transport. Jan. 15, 2008. Colloquium. Lehman College CUNY, New York, NY.
  
• Determinants of Laplacians as functions on spacecs of metrics. Jan 12. 2008. Southern California Analysis/PDE Conference at UCSD. San Diego, CA.  (Sat-Sun) January 12-13, 2008.
• Curvature and continuity of optimal transport, Jan. 11, 2008. Partial Differential Equations Seminar.  UCSD, San Diego, CA.
• Curvature and continuity of optimal transport. Jan. 8, 2008. University of Edinburgh. Edinburgh, UK.
• Curvature and continuity of optimal transport. Jan. 7, 2008. Hausdorff Center for Mathematics. Bonn, Germany.
• Curvature and continuity of optimal transport. Dec. 2007. Canadian Mathematical Society Winter 2007 Meeting, Dec. 8 -10, 2007--Special session: Calculus of Variations in Physics, Geometry and Economics. London, Ontario, Canada.
  
• Curvature and continuity of optimal transport. Nov. 2007. Prairie Analysis Seminar 2007 Nov 2-3, 2007. (A contribution talk) Kansas State University, Manhattan, Kansas.
• Curvature and the continuity of optimal transportation maps, July 24, 2007, MSRI(Berkeley)Summer Microprogram on Nonlinear Partial Differential Equations (July 23, 2007 to August 10, 2007 ).
• Curvature and the continuity of optimal transportation maps, July 19, 2007, Conference Optimal Transportation, and Applications to Geophysics and Geometry (Jul 16, 2007 -- Jul 20, 2007) , ICMS, University of Edinburgh.
• Determinants of Laplacians and isospectrality, July 10, 11, & 12, 2007,  KIAS (Seoul) Summer school on Dirac and Laplace Operators  (Jul 09, 2007 ~ Jul 13, 2007).
  
• Determinants of Laplacians and their holomorphic extensions, July 3, 2007, KIAS (Seoul) Seminar.
• Curvature and the continuity of optimal transportation maps, June 20, 2007, Banff workshop Geometric Inequalities (Sun. June 17, 2007 -- Fri. June 22, 2007).
• Curvature and the continuity of optimal transportation maps, April 13, 2007, Fields Institute, Young Mathematicians' Conference in PDE and Dynamical Systems IV (April 13-14, 2007).
  
• Determinants of Laplacians as functions on spaces of metrics,  Nov. 17, 2006, Princeton University, Geometric Analysis Seminar.
  
• Determinants of Laplacians and isospectrality of compact surfaces with boundary,  Oct. 30, 2006, University of Waterloo,
  Geometry/Topology Seminar.
  
• Determinants of Laplacians and their holomorphic extensions,Apr. 18, 2006, Ohio State University, Analysis/Geometry Seminar.
  
• Determinants of Laplacians and Quasifuchsian Spaces, Feb. 3, 2006, Northwestern University, Geometry/Physics Seminar.
  
• Determinants of Laplacians and their holomorphic extensions , Feb. 2, 2006, University of Chicago, Geometry/Topology Seminar.
  
• Determinants of Laplacians and their holomorphic extensions,  May 19,  2005, Workshop on The Weil-Petersson Geometry of Teichmuller Space,  2005 Ahlfors Bers Colloquium, University of Michigan, Ann Arbor, Michigan.
  
• Holomorphic extensions of determinants of Laplacians, April 9, 2005   2005 Bloomington Geometry Workshop, Indiana University, Bloomington, Indiana.
  

Local Seminar Talks

• 
  
• Nonnegatively cross-curved transportation costs. Nov. 24, 2009. DG-MP-PDE seminar. UBC, Vancouver, BC.
  
• Continuity of optimal transport maps under a degenerate MTW condition. Dec. 2, 2008. Member seminar. Inst. for Advanced Study. Princeton, NJ 
• Curvature and the continuity of optimal transport maps, March 8,  March 22, 2007  Fields Analysis Working Group Seminar 
• Determinants of Laplacians as functions on spaces of metrics,  Nov. 20, 2006, University of Toronto, Geometry and Topology Seminar.
• Regularity results for optimal transport problems with general costs ,  Oct. 26,  Nov. 2, 2006. Fields Analysis Working Group Seminar.
• Determinants of Laplacians and isospectrality of compact surfaces  with boundary. Oct. 13,  2006, University of Toronto, PDE/Applied Math/Analysis Seminar.
• Determinants of Laplacians and their holomorphic extensions, Nov. 17, 2005, Fields Insititute, Mini course.
  

Conferences and Programs (which I have participated, will attend, or am interested in since 2007)

• 
  
• Southern California Analysis & Partial Differential Equations (SCAPDE) meeting, Mar 9 -10, 2013, University of California at Los Angeles, Los Angeles, CA.
• Workshop “Recent trends in geometric and nonlinear analysis”, August 5-10, 2012. Banff International Research Station, Banff, AB. 
  
• Workshop “Calculus of Variations”, July 22-28, 2012. Oberwolfach, Germany.
  
• Mini workshop on geometry, June 25 –27, 2012. University of Montpellier, France.
  
• Workshop “Optimal transport to Osray” June 18--22. University of Paris-Orsay. France
  
• Conference ``Matching Problems: Economics Meets Mathematics” June 4—6, University of Chicago, Chicago, IL, USA.
  
• Workshop “PRIMA-PARC-PIMS meeting in PDE”. August 25 –27, 2011. Seoul National University, Korea.
• Applied Analysis and Applied PDEs. July 12 -- July 15, 2011. Victoria, Canada.
• SMS 2011. Metric Measure Spaces: Geometric and Analytic Aspects. June 27 -- July 8, 2011. Montreal, Canada.
• The second renunion conference for IPAM’s Optimal Transport Program. Dec. 12--17, 2010. Lake Arrowhead, CA.
• PRIMA-PARC-PIMS meeting in PDE. Dec. 7 --8, 2010. UBC, Vancouver, Canada.
  
• CMS winter meeting. Dec 4 --6. 2010. UBC, Vancouver, Canada.
• Workshop on Geometric Probability and Optimal Transportation. Nov. 1--5, 2010 (tentative dates) Fields Institute, Toronto. Canada.
  
• Optimal Transportation and Applications. Oct. 12 --16, 2010. SNS, Pisa, Italy.
• SAGE workshop: Optimal Transport and Riemannian Geometry (Bakry-Emery curvature, Ma-Trudinger-Wang curvature, and applications to Ricci flow). Sept. 1--3, 2010. U. Texas, Austin, TX. 
• PRIMA conference on Geometric Analysis. July 20 --30, 2010.  PIMS, UBC, Vancouver, Canada.
  
• Calculus of Variations, July 18--24, 2010. Oberwolfach, Germany.
  
• CNA summer school. June 7--12, 2010. Carnegie-Melon University. Pittsburg, PA  (Not attended)
  
• Optimal transportation and applications. April 18--23, 2010. Banff International Research Station (BIRS), Banff, AB, Canada.
• International Conference on Nonlinear PDE and Related Analysis/Applications (in conunction with the 64th Midwest PDE seminar), March 20--24, 2010. Northwestern University, Evanston, IL, USA.         
• KMS AMS Joint meeting, Dec. 16--20, 2009. Seoul, Korea
• The first renunion conference for IPAM’s Optimal Transport Program. Dec. 6--11, 2009. Lake Arrowhead, CA.
• Pacific Northwest Geometry Seminar (Nov 7--8, Portland State U., Oregon)
• Fall school on weak KAM. Oct. 19--23, 2009. CIRM. Luminy, France.
• Prairie Analysis Seminar. Oct. 2--3., 2009. Kansas State Univ. Manhattan, Kansas.
• PIMS thematic program on PDE. June, July, Aug, 2009. UBC, Vancouver, BC.
• Geometric Partial Differential Equations. Sep 2008 -- April 2009. IAS, Princeton, NJ.
• Workshop on Transport, Optimization, Equilibrium in Economics. July 14 -- July 20, 2008. University of British Columbia, Vancouver, BC.
• Calculus of Variations, July 6th - 12th, 2008. Oberwolfach, Germany.
• 2008 CNA Summer School  May 29 – June 7, 2008,  Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA (Not attended)
• Optimal Transport, March 10 - June 13 2008, IPAM(UCLA).
• AMS 2008 Spring Eastern Meeting, Mar. 15-16, 2008 (Sat-Sun) --Special Session on  Nonlinear Elliptic Equations and Geometric Inequalities. New York, NY.
• Southern California Analysis/PDE Conference at UCSD. San Diego, CA.  (Sat-Sun) January 12-13, 2008.
• Canadian Mathematical Society Winter 2007 Meeting, Dec. 8 -10, 2007. Hilton Hotel London (Ontario, Canada) --special session: Calculus of Variations in Physics, Geometry and Economics.
• Prairie Analysis Seminar 2007 (Fri-Sat. Nov 2-3, 2007). Kansas State University, Manhattan, Kansas.
• MSRI(Berkeley) Summer Microprogram on Nonlinear Partial Differential Equations (July 23, 2007 to August 10, 2007 ). Berkeley, CA.
• ICMS (Edinburgh): Optimal Transportation, and Applications to Geophysics and Geometry (Jul 16, 2007 -- Jul 20, 2007). Edinburgh, UK.
• KIAS (Seoul) Summer school on Dirac and Laplace Operators  (Jul 09, 2007 ~ Jul 13, 2007). Korea.
• Banff Geometric Inequalities  (Sun. June 17, 2007 - Fri. June 22, 2007).  Banff, Alberta, Canada.
• MPIM (Bonn, Germany) (May 4, 2007 - May 31, 2007).
• IPAM(UCLA): Random Curves, Surfaces, and Transport. (April 16 - 20, 2007).
• Fields Institute, Toronto:Young Mathematicians' Conference in PDE and Dynamical Systems IV (April 13-14, 2007).
  

  
  

Updated: March 2013