Andreas Floer

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Andreas Floer 1986 in Bochum

Andreas Floer [fløːɐ] (born August 23, 1956 in Duisburg ; † May 15, 1991 in Bochum ) was a German mathematician who made important contributions to (symplectic) topology , differential geometry and mathematical physics. He developed what is now known as Floer's homology , which has proven to be an important mathematical instrument.

life and work

Floer studied mathematics at the Ruhr University Bochum and received his diploma in 1982. He then went to the University of Berkeley in California, where he worked on monopolies (in Yang-Mills theories ) on three-dimensional manifolds with Alan Weinstein and Clifford Taubes . His doctorate was interrupted by doing alternative military service, but he received his doctorate in 1984 in Bochum with Eduard Zehnder .

In his dissertation in Bochum, Floer proved a special case (for mappings close to identity) of Arnold's conjecture about the fixed points of symplectic mappings (symplectomorphisms) of a symplectic manifold . With the partial proof of Arnold's conjecture and with his development of the Floer homology from 1985 in seminars in Berkeley, he attracted great attention and gave one of the plenary speeches at the International Congress of Mathematicians in Kyoto in 1990 (Elliptic methods in variational problems). The topology of low-dimensional manifolds is notoriously difficult - as the case of the Poincaré conjecture shows, which was proven in the higher-dimensional cases by Stephen Smale as early as 1960 , in the four-dimensional case by Michael Freedman around 1984 and in the three-dimensional case by Grigori Perelman in 2002 . The Floer homologies (there are several) are now a common tool in topology and differential geometry, especially of low dimensions.

In 1986 Floer was at Stony Brook University in New York, then at the Courant Institute . In 1988 he became an assistant professor of mathematics at Berkeley. In 1990 the assistant professorship was converted to a full professorship. In the same year he also became a mathematics professor in Bochum. 1991 took it surprisingly life.

His theory also has applications in quantum field theory (e.g. Seiberg-Witten theory ), and vice versa from there, in particular in the work of Edward Witten , new methods flowed into differential geometry, especially in the classification of differentiable structures on four-dimensional manifolds in the work of Simon Donaldson ( gauge theories and instantons ). Here there is an analogy between the instantons, which minimize the Yang-Mills functional (the “energy”) on four-dimensional manifolds , and pseudo-holomorphic mapping of Riemann surfaces into such symplectic manifolds (with regard to an “almost complex” structure compatible with the symplectic structure ).

Before his death, Floer had written work on the application of his theory in differential topology (cutting of manifolds, "surgery") and in the study of nodes in three dimensions. A whole series of other articles published only posthumously by the co-authors up to the mid-1990s shows that a “school” had already developed around him.

In 1989 he received a research grant from the Alfred P. Sloan Foundation ( Sloan Research Fellowship ).

In December 2011, the Ruhr University Bochum opened the Floer Center for Geometry, named after Andreas Floer .

Quotes

"Andreas Floer's life was tragically interrupted, but his mathematical visions and striking contributions have provided powerful methods which are being applied to problems which seemed to be intractable only a few years ago."

"Andreas Floer's life was tragically cut short, but his mathematical insights and outstanding contributions have provided powerful tools that can be applied to problems that just a few years ago seemed insoluble."

“The concept of Floer homology is one of the most striking developments in differential geometry over the past 20 years. [...] The ideas have led to great advances in the areas of low-dimensional topology and symplectic geometry and are intimately related to developments in Quantum Field Theory [...] the full richness of Floer's theory is only beginning to be explored. "

“The Floer homology design is one of the most significant developments in differential geometry in the last twenty years. ... The ideas have led to great advances in the areas of low-dimensional topology and symplectic geometry; they are closely related to developments in quantum field theory [...] The exploration of the whole richness and abundance of Floer's theory has only just begun. "

- Simon Donaldson

“Since its introduction by Andreas Floer in the late nineteen eighties, Floer theory has had a tremendous influence on many branches of mathematics including geometry, topology and dynamical systems. The development of new Floer theoretic tools continues at a remarkable pace and underlies many of the recent breakthroughs in these diverse fields. "

“Since Andreas Floer introduced it in the late 1980s, Floer's theory has had a tremendous impact on many branches of mathematics, such as geometry, topology, and dynamic systems. The development of new tools based on Floer's theory is proceeding at an amazing pace and is the basis of many new discoveries in these various branches of mathematics. "

Fonts

  • Monopoles on asymptotically euclidean 3-manifolds , Bulletin American Mathematical Society, Vol. 16, 1987, pp. 125–127 (the dissertation originally planned in the USA)
  • Proof of the Arnold conjecture for surfaces and generalizations for certain Kähler-Manifolds , Duke Mathematical Journal Vol. 53, 1981, pp. 1–32 (his dissertation)
  • Morse theory of fixed points of symplectic diffeomorphisms , Bulletin of the American Mathematical Society, Volume 16, 1987, pp. 279-281, Project Euclid
  • An instanton-invariant for 3-manifolds , Communications in Mathematical Physics, Vol. 118, 1988, pp. 215-240. Project Euclid
  • Morse theory for Lagrangian intersections , J. Differential Geometry, Vol. 28, 1988, pp. 513-547.
  • Cuplength estimates on Lagrangian intersections , Comm. Pure Appl. Math., Vol. 42, 1989, pp. 335-356.
  • Witten's complex and infinite dimensional Morse theory , Journal Differential Geometry Vol. 30, 1989, pp. 207-221 (Witten had obtained the Morse theory from supersymmetric quantum mechanics in a sensational work in 1982) Project Euclid
  • Elliptic methods in variational problems , International Congress of Mathematicians, Kyōto 1990
  • Self dual conformal structures on , Journal Differential Geometry, Vol. 33, 1991, pp. 551-574
  • Instanton homology and Dehn surgery , in "Floer memorial volume" 1995
  • with Helmut Hofer Coherent orientation for periodic orbit problems in symplectic geometry , Math. Zeitschrift Bd. 212, 1993, pp. 13-38
  • this. Symplectic homology I: Open sets in , Math. Zeitschrift Vol. 215, 1994, pp. 37-88
  • with Hofer, Wysocki Applications of symplectic homology I , Math. Zeitschrift, Vol. 217, 1994, pp. 577-606
  • with Hofer, Cieliebak Symplectic homology II: A General Construction , Math. Journal Vol. 218, 1995, pp. 103-122
  • with Hofer, Cieliebak, Wysocki Applications of symplectic homology II , Math. Zeitschrift, Vol. 223, 1996, pp. 27-45.
  • with Hofer, Salamon Transversality results in the elliptic Morse theory of the action functional , Duke Mathematical Journal, Vol. 80, 1995, 251–292, online here: http://www.math.nyu.edu/~hofer/publications/ trans.ps

literature

  • Hofer , Taubes , Weinstein , Zehnder (Eds.) The Floer Memorial Volume , Progress in Mathematics, vol. 133, Birkhauser Verlag, 1995.
  • this., Obituary Notices American Mathematical Society, August 1991
  • Simon Donaldson , M. Furuta, Dieter Kotschick Floer Homology Groups in Yang-Mills Theory , Cambridge Tracts in Mathematics, Vol. 147. Cambridge University Press, Cambridge, 2002. ISBN 0-521-80803-0
  • ders., P. Braam Floers work on instanton homology, knots and surgery , in "Floer memorial volume" 1995

Web links

Commons : Andreas Floer  - Collection of images, videos and audio files

Individual evidence

  1. ↑ Andreas Floer in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used
  2. ↑ Dr. Gerd Laures: In the depths of space: RUB mathematicians open new research center . , December 5, 2011. Retrieved December 20, 2011.
  3. ^ Hofer, Weinstein, and Zehnder, Andreas Floer: 1956-1991, Notices Amer. Math. Soc. 38 (8): 910-911
  4. ↑ Simon Donaldson: Floer Homology Groups in Yang-Mills Theory , With the assistance of M. Furuta and D. Kotschick. Cambridge Tracts in Mathematics, 147. Cambridge University Press, Cambridge, 2002. viii + 236 pp. ISBN 0-521-80803-0 ( The above citation is from the front flap.)
  5. ^ Mathematics: frontiers and perspectives . Edited by V. Arnold, M. Atiyah, P. Lax and B. Mazur. American Mathematical Society, Providence, RI, 2000. xii + 459 pp. ISBN 0-8218-2070-2
  6. ↑ From the Press Release to the Workshop: New Applications and Generalizations of Floer Theory of the Banff International Research Station (BIRS) 2007 5 Day Workshop: New Applications and Generalizations of Floer Theory | Banff International Research Station
personal data
SURNAME Floer, Andreas
BRIEF DESCRIPTION German mathematician
DATE OF BIRTH 23rd August 1956
PLACE OF BIRTH Duisburg
DATE OF DEATH May 15, 1991
Place of death Bochum