Andreas Floer in 1975
|Born||August 23, 1956|
|Died||May 15, 1991 34)(aged|
|Alma mater||Ruhr-Universität Bochum|
|Known for||Floer homology|
|Institutions|| Ruhr-Universität Bochum |
University of California, Berkeley
|Doctoral advisor|| Eduard Zehnder |
Andreas Floer (German: [ˈfløːɐ] ; 23 August 1956 – 15 May 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.
Germany, officially the Federal Republic of Germany, is a country in Central and Western Europe, lying between the Baltic and North Seas to the north, and the Alps to the south. It borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, France to the southwest, and Luxembourg, Belgium and the Netherlands to the west.
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.
He was an undergraduate student at the Ruhr-Universität Bochum and received a Diplom in mathematics in 1982. He then went to the University of California, Berkeley and undertook PhD work on monopoles on 3-manifolds, under the supervision of Clifford Taubes; but he did not complete it when interrupted by his obligatory alternative service in Germany. He received his PhD (Dr. phil.) at Bochum in 1984, under the supervision of Eduard Zehnder.
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 Bulgaria, Belarus, Bosnia and Herzegovina, Croatia, Estonia, Finland, Poland, Russia, and Ukraine and only for engineers in France, Greece, Hungary, Romania, Serbia, Macedonia, Slovenia, and Brazil.
The University of California, Berkeley is a public research university in the United States. Located in the city of Berkeley, it was founded in 1868 and serves as the flagship institution of the ten research universities affiliated with the University of California system. Berkeley has since grown to instruct over 40,000 students in approximately 350 undergraduate and graduate degree programs covering numerous disciplines.
In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle.
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 mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation.
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
Kyoto, officially Kyoto City, is the capital city of Kyoto Prefecture, located in the Kansai region of Japan. It is best known in Japanese history for being the former Imperial capital of Japan for more than one thousand years, as well as a major part of the Kyoto-Osaka-Kobe metropolitan area.
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.
"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 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".
Sir Simon Kirwan Donaldson, is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson–Thomas theory. He is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University and a Professor in Pure Mathematics at Imperial College London.
"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."
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analog of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Hamiltonian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.
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In mathematics, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows. More specifically, the conjecture claims that on a compact contact manifold, its Reeb vector field should carry at least one periodic orbit.
In the mathematical field of symplectic topology, Gromov's compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have a subsequence which limits to a pseudoholomorphic curve which may have nodes or "bubbles". A bubble is a holomorphic sphere which has a transverse intersection with the rest of the curve. This theorem, and its generalizations to punctured pseudoholomorphic curves, underlies the compactness results for flow lines in Floer homology and symplectic field theory.
Tian Gang is a Chinese mathematician. He is an academician of the Chinese Academy of Sciences and of the American Academy of Arts and Sciences. He is known for his contributions to geometric analysis and quantum cohomology especially Gromov-Witten invariants, among other fields. He has been Vice President of Peking University since February 2017.
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. His brother, Gary Taubes, is a science writer.
Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold. Important consequences of this theorem include the existence of an Exotic R4 and the failure of the smooth h-cobordism theorem in 4 dimensions. The results of Donaldson theory depend therefore on the manifold having a differential structure, and are largely false for topological 4-manifolds.
Katrin Wehrheim is an associate professor of mathematics at the University of California, Berkeley. Her research centers around symplectic topology and gauge theory. She is known for her work on pseudoholomorphic quilts. With Dusa McDuff, she has challenged the foundational rigor of a classic proof in symplectic geometry.
The Geometry Festival is an annual mathematics conference held in the United States.
Tomasz Mrowka is an American mathematician specializing in differential geometry and gauge theory. He is the Singer Professor of Mathematics and head of the Department of Mathematics at the Massachusetts Institute of Technology.
Helmut Hermann W. Hofer is a German-American mathematician, one of the founders of the area of symplectic topology. He is a member of the National Academy of Sciences, and the recipient of the 1999 Ostrowski Prize and the 2013 Heinz Hopf Prize. Since 2009, he is a faculty member at the Institute for Advanced Study in Princeton. He currently works on symplectic geometry, dynamical systems, and partial differential equations. His contributions to the field include Hofer geometry.
In mathematics, the Conley–Zehnder theorem, named after Charles C. Conley and Eduard Zehnder, provides a lower bound for the number of fixed points of Hamiltonian diffeomorphisms of standard symplectic tori in terms of the topology of the underlying tori. The lower bound is one plus the cup-length of the torus (thus 2n+1, where 2n is the dimension of the considered torus), and it can be strengthen to the rank of the homology of the torus (which is 22n) provided all the fixed points are non-degenerate, this latter condition being generic in the C1-topology.
Eduard J. Zehnder is a Swiss mathematician, considered one of the founders of symplectic topology.
Kenji Fukaya is a Japanese mathematician known for his work in symplectic geometry and Riemannian geometry. His many fundamental contributions to mathematics include the discovery of the Fukaya category. He is a permanent faculty member at the Simons Center for Geometry and Physics and a professor of mathematics at Stony Brook University.
In symplectic geometry, the spectral invariants are invariants defined for the group of Hamiltonian diffeomorphisms of a symplectic manifold, which is closed related to Floer theory and Hofer geometry.
François Lalonde is a Canadian mathematician, specializing in symplectic geometry and symplectic topology.
Kaoru Ono is a Japanese mathematician, specializing in symplectic geometry. He is a professor at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University.
Ivan Smith is a British mathematician who deals with symplectic manifolds and their interaction with algebraic geometry, low-dimensional topology, and dynamics. He is a professor at the University of Cambridge.