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):

Alessio Figalli     Robert McCann     Emanuel Milman    Raanan Schul   Jeff Streets   Ben Stephens    Micah Warren

• 
  
• 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. (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. Submitted. June 2011.
  
• Parabolic optimal transport equations on manifolds, with Jeff Streets and Micah Warren.  To appear in International Mathematics Research Notices. 
• 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. To appear in Math. Ann.
  
• 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 Nov. 2009 (superseded by a more recent paper  Hoelder continuity and injectivity of optimal maps, listed above).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 . To appear in J. Reine Angew. Math. (Crelle's journal).
• 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: 

• 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

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• 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

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• 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)

• 
  
• 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: Aug. 2010