Andreas Floer

Andreas Floer [ Flø ː ɐ ] ( born August 23, 1956 in Duisburg, † 15 May 1991, Bochum ) was a German mathematician who made major contributions to the ( symplectic ) topology, differential geometry and mathematical physics made ​​. He developed what is now called Floer homology, which has proven itself as an important mathematical tool.

Life and work

Floer studied at the Ruhr- University Bochum mathematics and received his diploma in 1982. Afterwards he went to the University of California at Berkeley, where he ( in Yang-Mills theories ) worked on three-dimensional manifolds with Alan Weinstein and Clifford Taubes about monopolies. The promotion was interrupted by the performance of military - civilian service, yet he received his doctorate in Bochum with Eduard Zehnder in 1984.

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

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

His theory also has applications in quantum field theory (eg Seiberg -Witten theory ), and vice versa from there, especially in the work of Edward Witten, new methods were incorporated into the differential geometry, especially in the classification of differentiable structures on four-dimensional manifolds in the works by Simon Donaldson ( gauge theories and instantons ). Here is an analogy between the instantons on four-dimensional manifolds, the Yang-Mills functional ( "energy " ) and minimize holomorphic maps of Riemann surfaces in symplectic manifolds such that say " almost complex " structures can be defined.

Before his death Floer had written still working on the application of his theory in differential topology (reshuffling of manifolds, " surgery", engl. Surgery ) and in the study of knots in three dimensions. A whole series of other posthumously published by the co-authors to the mid-1990s essays testifies that about him had already formed a "school ".

In December 2011, the Ruhr- University Bochum opened, named after Andreas Floer Floer center for geometry.

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 Floers life was terminated in a tragic way, but his mathematical insights and outstanding contributions have powerful tools supplied, which are applied to problems which seemed unsolvable a few years ago. "

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

" 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 under read many of the recent breakthroughs in various fields thesis. "

" Since Andreas Floer introduced it in the late eighties, the Floer theory has had a huge impact on many branches of mathematics, such as geometry, topology and dynamical systems. The development of new products based on the Floer theory tools is progressing at an amazing pace and is the basis of many new findings in these various branches of mathematics. "

Writings

  • Monopoles on asymptotically euclidean 3- manifolds, Bulletin of the American Mathematical Society, Bd.16, 1987, p.125 -127 ( originally planned in the U.S. dissertation)
  • Proof of the Arnold conjecture for surfaces and generalizations for certain Kähler manifolds, Duke Mathematical Journal Bd.53, 1981, p.1 -32 ( his dissertation )
  • With Eduard Zehnder Morse theory of fixed points of symplectic diffeomorphisms, Bulletin of the American Mathematical Society 1987
  • An instanton - invariant for 3- manifolds, Communications in Mathematical Physics, Bd.118, 1988, p.215 - 240.
  • Morse theory for Lagrangian intersections, J. Differential Geometry, Vol 28, 1988, p. 513 - 547th
  • Cuplength estimates on Lagrangian intersections, Comm. Pure Appl. Math, Vol 42, 1989, p.335 - 356.
  • Witten's complex and infinite dimensional Morse theory, Journal Differential Geometry Bd.30, 1989, p.207 -221 ( Witten had won in a sensational work 1982, the Morse theory of the supersymmetric quantum mechanics)
  • Elliptic methods in variational problems, International Congress of Mathematicians, Kyoto 1990
  • Self dual conformal structures on, Journal Differential Geometry, Bd.33, 1991, S.551 -574
  • Instanton homology and Dehn surgery, in " Floer memorial volume" in 1995
  • With Helmut Hofer, Coherent orientation for periodic orbit problems in symplectic geometry, Math Journal Vol 212, 1993, p.13 -38
  • This. Symplectic homology I: Open sets in, Math Journal Vol 215, 1994, pp. 37-88
  • By Hofer, Wysocki Applications of symplectic homology I, Math Journal, Vol 217, 1994, S.577 -606
  • With Hofer, Symplectic homology Cieliebak II: A general construction, Math journal Bd.218, 1995, p.103 -122
  • With Hofer, Cieliebak, Wysocki Applications of symplectic homology II, Math Journal, Bd.223, 1996, p.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
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