UHLENBECK, KAREN

Karen Uhlenbeck

Professor
Department of Mathematics

Sid W. Richardson Foundation Regents Chair in Mathematics #3

Non-linear Analysis, Gauge Theory and Integrable Systems
uhlen@math.utexas.edu

Phone: 512-471-7711

Office Location
RLM 8.100

Postal Address
The University of Texas at Austin
Department of Mathematics, College of Natural Sciences
1 University Station C1200
Austin, TX 78712

Ph.D., Brandeis University (1968)

Research Interests

Non-linear Analysis, Gauge Theory and Integrable Systems

Geometric Partial Differential Equations, The Calculus of Variations, (especially scale invariant problems) Gauge Theory (now mostly for wave and non-linear Schoedinger systems) Integrable Systems, Virasoro actions, Non-linear waves and non-linear Schroedinger Equations, (especially scale invariant problems).

I am a member of the geometry group in the mathematics department. The geometry group maintains a website with general and current information. http://www.ma.utexas.edu/users/plowrey/dev/

My recent research has been mainly on two subjects, with two groups of coauthors. I have written four papers of with Chuu-Lian Terng on the geometric aspects of integrable systems. With Andrea Nahmod and Atanas Stefanov I authored two papers on wave maps and Schoedinger maps. These are all posted on the LANL preprint server (in geometry or anaylsis).

Articles

Regularity Theorems for solutions of elliptic polynomial equations, Amer. Math. Soc., Symposium in Pure Math. XVI (1970), 225-231.

Morse theory on Banach manifolds, Bull. Amer. Math. Soc., 76(1970), 105-107.

Integrals with nondegenerate critical points, Bull. Amer. Math. Soc., 76(1970), 125-128.

Harmonic maps; a direct method in the calculus of variations, Bull. Amer. Math. Soc., 76(1970), 1082-1087.

Morse theory on Banach manifolds, J. Funct. Anal., 10(1972), 430-445.

Bounded sets and Finsler structures of manifolds of maps, J. Diff. Geom., 7(1972), 589-599.

Eigenfunctions of Laplace operators, Bull. Amer. Math. Soc., 78(1972), 1073-76.
Lorentz geometry, global analysis and its applications, Vol. 3, International Center for Theoretical Physics, Trieste (1972), 235-42.
The Morse index theorem in Hilbert space, J. Diff. Geom., 8(1973), 555-564.

A new proof of a regularity theorem for elliptic systems, Proc. Amer. Math. Soc., 37(1973), 315-316.

Morse theory for geodesics on a Lorentz manifold, Topology, 14(1975), 69-90.

Generic properties of Eigenfunctions, Amer. J. Math., 98(1976), 1059-1078.

Regularity for a class of non-linear elliptic equations, Acta Math., 48(1977), 217-238.

(with J. Sacks) The existence of minimal immersions of two-spheres, Bull. Amer. Math. Soc., 83(1977), 1033-1036.

Removeable singularities in Yang-Mills fields, Bull. Amer. Math. Soc., 1(New Series, 1979), 112-114.

(with J. Sacks) The existence of minimal immersions of 2-spheres, Ann. of Math., 113(1981), 1-24.

Morse theory by perturbation methods with applications to harmonic maps, Trans. Amer. Math. Soc., 267(1981), 569-583.

Variational problems for gauge fields, Seminar on Differential Geometry, S. T. Yau (ed.), Princeton University Press (1982), 455-464.

 

Removeable singularities in Yang-Mills fields, Comm. Math. Phys., 83(1982), 11-29.

Connections with tex2html_wrap_inline212 bounds on curvature, Comm. Math. Phys., 83(1982), 31-42.

Equivariant harmonic maps into spheres, Proceedings of the Tulane Conference on Harmonic Maps, Knill, Kalka and

Sealey (eds.), Lecture Notes in Mathematics 949, Springer-Verlag (1982), 146-158.

(with J. Sacks) Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc., 271(1982), 639-652.

(with R. Schoen) A regularity theory for harmonic maps, J. Diff. Geom., 17(1982), 307-335.

Minimal spheres and other conformal variational problems, Seminar on Minimal Submanifolds, E. Bombieri (ed.),

Princeton University Press (1983), 169-176.

Closed minimal surfaces in hyperbolic 3-manifolds, Seminar on Minimal Submanifolds, E. Bombieri (ed.), Princeton

University Press (1983), 147-168.

Boundary regularity and Dirichlet Problem for harmonic maps, J. Diff. Geom., 18(1983), 253-268.

Conservation laws and their application in Global Differential Geometry, in Srinivasan and Sally (eds.), Emmy Noether in

Bryn Mawr (1983), Springer-Verlag, 103-115.

Regularity of minimizing harmonic maps into spheres, Invent. Math., 78(1984).

Variational problems for Gauge fields, Proceedings of the International Congress of Mathematics, August 16-24, 1983,

Warsaw (1984), 613-619.

The Chern classes of Sobolev connections, Comm. Math. Phys., 101(1985), 449-457.

(with S.-T. Yau) On the existence of Hermitian Yang-Mills connections in stable bundles, Comm. Pure Appl. Math., 39(1986), S257-S297.

Commentary on Analysis in the Large, in A Century of Mathematics in America, Part II, Peter Duren (ed.), American Mathematical Society (1988), 357-359.

Harmonic Maps into Lie Groups (classical solutions of the chiral model), J. Diff. Geom., 30 (1989), 1-50.

(with L. M. Sibner, R. J. Sibner) Solutions to Yang-Mills Equations which are not self-dual, Proc. Nat. Acad. Sci., 89(1989), 8610-8613.

Applications of non-linear analysis in topology, Proc. of the ICM-90, Mathematical Society of Japan, 1991, 261-279.

Instantons and their relatives, Mathematics into the 21st Century, Browder (ed.), American Mathematical Society (1992), 467-479.

On the connection between harmonic maps and the self-dual Yang-Mills and the Sine-Gordon equations, J. Geom. Phys., 8(1992), 283-316.

(with M. Atiyah, et al.) Responses to ``Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics'', by A. Jaffe and F. Quinn, Bull. Amer. Math. Soc. (new series), 30(1994), 201-202.

(with G. Daskalopoulos) An application of traversality to the topology of the moduli space of stable bundles, Topology, 34(1995), 203-215.

(with G. Daskalopoulos and R. Wentworth) Moduli of extensions of holomorphic bundles on Kähler manifolds, Comm. Anal. Geom, 3(1995), 479-522.

Moduli spaces and adiabatic limits, Notices of the AMS, 42(1995), 41-42.

(with R. Mazzeo and D. Pollack) Connected sum structions for constant scalar curvature metrics, Topological Methods in Nonlinear Analysis, 6(1995), 207-233.

(with R. Mazzeo and D. Pollack) Moduli spaces of singular Yamabe metrics, J. Amer. Math. Soc., 9(1996), 303-343.

(with C.-L. Terng) Poisson Actions and Scattering Theory for Integrable Systems, Survey in Surveys in Differentiable Geometry, vol 4, International Press (l998).

(with C.L. Terng) Introduction to the volume "Survey's in Differential Geometry," Integrable Systems, Volume 4 (2000), 1-5.

(with C.L. Terng) The Geometry of Solitons (cover article) Notices of the AMS 47 (2000),17-25.

(with C L Terng) Backlund Transformations and Loop Group Actions, Communications in Pure and Applied Mathematics 53 (2000, 1-75.

(with N.H. Chang and J. Shatah) Schroedinger Maps, Comm. Pure Appl. Math 53 (2000),590-602.

(with J. Viaklovsky) Regularity of weak solutions to critical exponent variational problems, Mathematical Research Letters, I P, vol 7 no 5-6 (2000), 652-656.

(with A. Nahmod and A. Stefanov) On the well-posedness of the wave map problem in high dimensions, Comm. Analysis and Geometry, vol 11, no 1 (2003) 29-83.

(with A. Nahmod and A Stephanov) On Schroedinger Maps, Comm. Pure Appl. Math 56 (2003) no 1, 114-151.

(with C.L. Terng) Wave maps into symmetric spaces, Comm. Anal. Geom. 12 (2004), no 1-2, 345-388.

(with C. L. Terng) Schroedinger flows on Grassmannians, Integrable systems, geometry and topology, 234-256, AMS/IP Stud. Adv.Math, AMS, Providence (2006), 235-256.

(with M. Vajiac) Virasoro Actions on harmonic maps, submitted to JDG.

(with C. L. Terng) On Space-time Monopoles.

(with A. Goncalves) On constant mean curvature embeddings of surfaces in space forms, To appear in CAG.

Books

(with D. Freed) Instantons and 4-manifold Topology, Mathematical Sciences Research Institute Lecture Notes, 1, Springer Verlag, 1984.

(edited) Global Analysis in Modern Mathematics: a Symposium in Honor of Richard Palais' sixtieth birthday. Publish or Perish, 1993.

(edited with D. Freed) Geometry and Quantum Field Theory, American Mathematical Society, 1995.

  • 2007 Steele Prize, Amercian Mathematics Society
  • 2004 Honorary Degree: Doctor of Science; University of Michigan
  • 2001 National Medal of Science
  • 2001 Honaray Degree: Doctor Of Science: University of Ohio
  • 2000 National Medal of Science
  • 2000 Honorary Degree: Doctor of Science, University of Illinois, Champaign
  • 1995 CommonWealth Award for Science and Technology
  • 1988 Alumni Achievement Award, Brandeis University
  • 1988 Honorary Degree: Doctor of Science, Knox College, Illinois
  • 1986 National Academy of Sciences, Fellow
  • 1985 American Academy of Arts and Sciences, Fellow
  • 1984 Alumna of the Year, University of Michigan
  • 1983-88 MacArthur Prize Fellowship
  • 1979-80 Albert Einstein Fellow, Institute for Advanced Study, Princeton
  • 1974-76 Sloan Foundation Fellowship
  • 1964-68 National Science Foundation Graduate Fellowship
  • 1964-65 Woodrow Wilson Fellow
  • 1964 Phi Beta Kappa