Andreas Floer
Encyclopedia
Andreas Floer (ˈfløːɐ; 1956–1991) was a German
mathematician
who made seminal contributions to the areas of geometry
, topology
, and mathematical physics
, in particular the invention of Floer homology
.
in mathematics in 1982. He then went to the University of California, Berkeley
and undertook Ph. D. work on monopole
s on 3-manifold
s, under the supervision of Clifford Taubes
; but he did not complete it when interrupted by his obligatory alternative service
in Germany. He received his Ph. D. (Dr. phil.) at Bochum in 1984, under the supervision of Eduard Zehnder.
Floer’s first pivotal contribution was a solution of a special case of Arnold's conjecture on fixed points of a symplectomorphism
. Because of his work on Arnold's conjecture and his development of instanton homology, he achieved wide recognition and was invited as a plenary speaker for the International Congress of Mathematicians
held in Kyoto
in August 1990. He received a Sloan Fellowship
in 1989.
In 1988 he became an Assistant Professor at the University of California, Berkeley and was promoted to Full Professor of Mathematics in 1990. From 1990 he was Professor of Mathematics at the Ruhr-Universität Bochum, until his suicide in 1991.
Simon Donaldson
wrote: "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" and "the full richness of Floer's theory is only beginning to be explored".
"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."
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who made seminal contributions to the areas of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, and mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, in particular the invention of Floer homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
.
Life
He was an undergraduate student at the Ruhr-Universität Bochum and received a DiplomDiplom
A Diplom is an academic degree in the German-speaking countries Germany, Austria, and Switzerland and a similarly named degree in some other European countries including Belarus, Bosnia and Herzegovina, Croatia, Estonia, Finland , Greece, Hungary, Russia, Serbia, Macedonia, Slovenia, and Ukraine...
in mathematics in 1982. He then went to the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
and undertook Ph. D. work on monopole
Monopole (mathematics)
In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle. The connection and Higgs field should satisfy the Bogomolnyi equation and be of finite action....
s on 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s, under the supervision of Clifford Taubes
Clifford Taubes
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology.-Early career:Taubes received his Ph.D...
; but he did not complete it when interrupted by his obligatory alternative service
Zivildienst
Zivildienst is the civilian branch of the national service systems in Austria and Switzerland. In Germany as well Zivildienst was the alternative service to military service until suspension of conscription in 2011...
in Germany. He received his Ph. D. (Dr. phil.) at Bochum in 1984, under the supervision of Eduard Zehnder.
Floer’s first pivotal contribution was a solution of a special case of Arnold's conjecture on fixed points of a symplectomorphism
Symplectomorphism
In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.-Formal definition:A diffeomorphism between two symplectic manifolds f: \rightarrow is called symplectomorphism, iff^*\omega'=\omega,...
. Because of his work on Arnold's conjecture and his development of instanton homology, he achieved wide recognition and was invited as a plenary speaker for the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
held in Kyoto
Kyoto
is a city in the central part of the island of Honshū, Japan. It has a population close to 1.5 million. Formerly the imperial capital of Japan, it is now the capital of Kyoto Prefecture, as well as a major part of the Osaka-Kobe-Kyoto metropolitan area.-History:...
in August 1990. He received a Sloan Fellowship
Sloan Fellowship
The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This is distinct from the Sloan Fellows in business....
in 1989.
In 1988 he became an Assistant Professor at the University of California, Berkeley and was promoted to Full Professor of Mathematics in 1990. From 1990 he was Professor of Mathematics at the Ruhr-Universität Bochum, until his suicide in 1991.
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."Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
wrote: "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" and "the full richness of Floer's theory is only beginning to be explored".
"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."
Selected publications
- Floer, Andreas. An instanton-invariant for 3-manifolds. Comm. Math. Phys. 118 (1988), no. 2, 215–240. Project Euclid
- Floer, Andreas. Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), no. 3, 513–547.
- Floer, Andreas. Cuplength estimates on Lagrangian intersections. Comm. Pure Appl. Math.Communications on Pure and Applied MathematicsCommunications on Pure and Applied Mathematics is a scientific journal which is associated with the Courant Institute of Mathematical Sciences. It publishes original research originating from or solicited by the institute, typically in the fields of applied mathematics, mathematical analysis, or...
42 (1989), no. 4, 335–356.
Posthumous publications
- Hofer, Helmut. Coherent orientation for periodic orbit problems in symplectic geometry (jointly with A. Floer) Math. Zeit. 212, 13–38, 1993.
- Hofer, Helmut. Symplectic homology I: Open sets in C^n (jointly with A. Floer) Math. Zeit. 215, 37–88, 1994.
- Hofer, Helmut. Applications of symplectic homology I (jointly with A. Floer and K. Wysocki) Math. Zeit. 217, 577–606, 1994.
- Hofer, Helmut. Symplectic homology II: A General Construction (jointly with K. Cieliebak and A. Floer) Math. Zeit. 218, 103–122, 1995.
- Hofer, Helmut. Transversality results in the elliptic Morse theory of the action functional (jointly with A. Floer and D. Salamon) Duke Mathematical Journal, Vol. 80 No. 1 , 251–292, 1995. Download from H. Hofer's homepage at NYU
- Hofer, Helmut. Applications of symplectic homology II (jointly with K. Cieliebak, A. Floer and K. Wysocki) Math. Zeit. 223, 27–45, 1996.
Further reading
- Simon Donaldson, On the work of Andreas Floer, Jahresber. Deutsch. Math.-Verein. 95 (3) (1993), 103-120.
- The Floer Memorial Volume (H. Hofer, C. Taubes, A. Weinstein, and E. Zehnder, eds.), Progress in Mathematics, vol. 133, Birkhauser Verlag, 1995.
- 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
External links
- The In Memoriam website of the Department of Mathematics at the University of California, Berkeley
- The In Memoriam website of the University of California, Berkeley
- Elaborate obituary by Addison/Casson/Weinstein at OAC, Online Archive of California, 1992
- For images, see Wikimedia Commons Category:Andreas Floer and a public Andreas Floer photo album on Facebook