Karen Uhlenbeck

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Karen Uhlenbeck (1982)

Karen Keskulla Uhlenbeck (born August 24, 1942 in Cleveland , Ohio ) is an American mathematician who works on partial differential equations , calculus of variations , geometric analysis, and differential geometry . In 2019 she was the first woman to be awarded the Abel Prize, one of the most prestigious awards for achievements in the field of mathematics.

Career

Education and career

Karen Uhlenbeck was born in Cleveland, Ohio, the oldest of four children. Her grandfather was Estonian and her grandmother was German. Her father Arnold Keskulla was an engineer and her mother Carolyn Windeler Keskulla was a teacher and artist. The family later moved to New Jersey , where Karen attended school and during that time developed an interest in books and science in general. After graduating from high school, she first studied physics at the University of Michigan , but then switched to mathematics and received her bachelor's degree in 1964 . She moved to the Courant Institute of Mathematical Sciences of New York University , but temporarily interrupted her studies because of her marriage. In 1966 she completed her master’s degree from Brandeis University , where she received her doctorate in 1968 from Richard Palais (The Calculus of Variations and Global Analysis). In 1968 she went to the Massachusetts Institute of Technology (MIT) as a post-doctoral student as an instructor and from 1969 to 1971 as a lecturer at the University of California, Berkeley . Uhlenbeck was an Assistant Professor at the University of Illinois at Urbana-Champaign from 1971 to 1976 . In 1976 she became an Associate Professor at the University of Illinois at Chicago and in 1983 Professor at the University of Chicago . In 1988 Uhlenbeck was appointed to the University of Texas at Austin , where she holds the Sid W. Richardson Foundation Regents Chair in Mathematics. Together with Dan Freed , she founded the IAS / Park City Mathematics Institute, where summer courses (supervised by the Institute for Advanced Study in Princeton) are held. Uhlenbeck lives in retirement in Texas.

In 1979 and 1980 she conducted research at the Institute for Advanced Study . She became a visiting professor at Harvard University in 1983 and at the Max Planck Institute for Mathematics in Bonn in 1985 , was a visiting scientist at MSRI in 1982 , Chancellors Distinguished Visiting Professor at the University of Berkeley in 1979 and at the University of California, San Diego in 1986 . From 1983 to 1986 she was on the Council of the Institute of Mathematics and its Applications.

Scientific activity

Uhlenbeck began her scientific career with research on the calculus of variations with her doctoral supervisor Richard Palais and later became known primarily for her work on nonlinear partial differential equations in various geometric and physical problems.

Some of her most important work concerned harmonic mappings between Riemannian manifolds, that is, those that minimize the Dirichlet energy , which is a problem of variation. The following analogue can be clearly formulated: One is looking for a mapping of a surface M onto a surface N, for example, let M be a rubber skin and N a stone and the mapping consists in putting the rubber skin over the stone. Then the Dirichlet energy would correspond to the elastic energy in the rubber skin that is generated when it is put on. In the case of topologically more complicated manifolds (measured for example via the number of holes, the topological gender ), the solution sought is often ambiguous and it is difficult to prove convergence when approximating harmonic mappings. In 1964, Palais and Stephen Smale came up with the idea of ​​using more general energy measures than the Dirichlet energy that meet the so-called Palais-Smale condition. Initially, this only worked in the one-dimensional case; in higher-dimensional cases, the Dirichlet energy often did not meet the condition. This was investigated in more detail by Karen Uhlenbeck in the mid-1970s and she and her post-doctoral student Jonathan Sacks tested various energy functionals on two-dimensional surfaces that all meet the Palais-Smale condition (which ensured that the mapping minimized an energy) and against that Dirichlet energies converge. The question was whether the images also became harmonious when the energy measures approached the Dirichlet energy. They found that this for almost all been points of the surface of the case up to a finite number with bubbles singularity (bubble singularity). These can only arise at topological holes of the target manifold assumed to be compact, and the question of the existence of a harmonic mapping thus allowed statements about the topology of the target manifold. The work of Sacks and Uhlenbeck is considered to be one of the fundamental works in the field of geometric analysis, which is partly founded on it. Similar phenomena with bubble singularities of images were found later in many other contexts.

In the case of the Yang-Mills equations , which are important in physics (here one also deals with mappings between manifolds, and the solutions of the Yang-Mills equation minimize a functional similar to the Dirichlet energy in harmonic mappings), proved in four dimensions they 1982 a theorem on the removability of singularities ( removable singularities theorem ). It showed that there cannot be bubble singularities around isolated points. Solutions of the Yang-Mills equation with finite energy, which behave in a nonsingular manner in the vicinity of a point, are also nonsingular in that point. Uhlenbeck proved the existence of Coulomb calibrations in Yang-Mills equations and derived analytical properties of their solutions from the fact that these become elliptical in a calibration. Her estimates of (self-dual) instanton solutions of Yang-Mills equations in particular were important analytical preparatory work for Simon Donaldson's classification of differentiable structures on four-dimensional manifolds, for which he received the Fields Medal . Uhlenbeck also researches non-linear wave equations and integrable systems with an infinite number of conserved quantities ( solitons ).

Karen Uhlenbeck has become a leading figure for women in mathematics through her scientific successes since the early 1980s. She was the first woman since 1932 to give a plenary lecture at an international mathematicians' congress in 1990 . In 1994 she and Chuu-Lian Terng founded a mentoring program for women in mathematics (Women and Mathematics, WAM) at the Institute for Advanced Study.

Memberships and honors

Private life

Until 1976 she was married to the biophysicist Olke Cornelis Uhlenbeck (* 1942), the son of George Uhlenbeck . She later married the mathematician Robert F. Williams.

Fonts

  • With J. Sacks: The existence of minimal immersions of 2-spheres. In: Annals of Mathematics. Volume 113, 1981, pp. 1-24.
  • Morse theory by perturbation methods with applications to harmonic maps. In: Trans. Amer. Math. Soc. Vol. 267, 1981, pp. 569-583, online.
  • With J. Sacks: Minimal immersions of closed Riemann surfaces. In: Trans. AMS. Volume 271, 1982, pp. 639-652.
  • Removable Singularities in Yang Mills Fields. In: Communications in Mathematical Physics. Vol. 83, 1982, No. 1, ISSN  0010-3616 , pp. 11-29, Project Euclid.
  • Connections with bounds on curvature. In: Communications in Mathematical Physics. Vol. 83, 1982, No. 1, ISSN  0010-3616 , pp. 31-42, Project Euclid.
  • With R. Schoen : A regularity theory for harmonic maps. In: J. Diff. Geom. Vol. 17, 1982, pp. 307-335, Project Euclid.
  • With DS Freed: Instantons and Four-Manifolds. In: Mathematical Sciences Research Institute publications 1. Springer-Verlag, New York et al. 1984, ISBN 0-387-96036-8 .
  • With C.-L. Terng: Geometry of Solitons. (PDF; 212 kB), Notices AMS, 2000.

literature

  • Simon Donaldson : Karen Uhlenbeck and the Calculus of Variations, Notices AMS, March 2019

Web links

Individual evidence

  1. ↑ Life and career data from: Pamela Kalte u. a .: American Men & Women of Science. Thomson Gale, 2004.
  2. ^ Allyn Jackson: Interview with Karen Uhlenbeck. In: celebratio.org. 2018, accessed on May 22, 2019 .
  3. ↑ Jim Al-Khalili: A biography of Karen Uhlenbeck. (PDF; 380 kB) In: abelprize.no. Retrieved March 19, 2019 .
  4. ↑ Karen Uhlenbeck in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used Template: MathGenealogyProject / Maintenance / name used
  5. ^ R. Palais, S. Smale: A generalized Morse theory. In: Bull. AMS. Volume 70, 1964, Issue 1, pp. 165-172.
  6. ↑ K. Uhlenbeck, J. Sacks: The existence of minimal immersions of 2-spheres. In: Annals of Mathematics. Volume 113, 1981, pp. 1-24.
  7. ↑ Erica Klarreich: Karen Uhlenbeck, Uniter of Geometry and Analysis, Wins Abel Prize. In: Quanta Magazine. 19th March 2019.
  8. ^ Women and Mathematics Program Celebrates Twenty-Five Years. In: IAS. June 2018.
  9. ↑ 2020 Steele Prize for Lifetime Achievement
personal data
SURNAME Uhlenbeck, Karen
ALTERNATIVE NAMES Uhlenbeck, Karen Keskulla
BRIEF DESCRIPTION American mathematician
DATE OF BIRTH August 24, 1942
PLACE OF BIRTH Cleveland , Ohio, USA