Young-Heon's Home

Young-Heon Kim's Research

Research Interests:
Differential Geometry, Geometric Analysis, and Partial Differential Equations:

Analysis and Geometry of Optimal Transportation / Monge-Ampere Equations
Riemannian Geometry
Spectral Geometry: Analytic Torsion (Determinants of Laplacians), Isospectrality
Analysis of Riemann surfaces (Teichmueller Theory)
Inverse Problems

Publications & Preprints: One-Page Summary of Research

Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds. Preprint, Sept 2007. Submitted.
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. Preprint, Sept 2007. Submitted.
On the cost-subdifferentials of cost-convex functions, with Robert J. McCann. Preprint, June 2007.
Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality. To appear in Duke Mathematical Journal.
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.

Papers in preparation

Riemannian submersions, products, and curvature of optimal transport, with Robert J. McCann.In preparation.
On the Hoelder continuity of optimal partial transport maps, with Robert J. McCann. In preparation.

Presentation Slides

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

Invited Talks and Lectures

TBA.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. 2008. 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). 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, 20052005 Bloomington Geometry Workshop, Indiana University, Bloomington, Indiana.

Local Seminar Talks

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 in, will attend, or am interested in)

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.
Evolution equations. June 23 - July 18, 2008. Clay Institute 2008 Summer School. ETH, Zurich, Switzerland.
2008 CNA Summer School May 29 – June 7, 2008, Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA
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.
BanffGeometric 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).

Back to Young-Heon's main page.