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Andreas Floer | |
---|---|

Andreas Floer in 1975 | |

Born | August 23, 1956 |

Died | May 15, 1991 34) | (aged

Nationality | German |

Alma mater | Ruhr-Universität Bochum |

Known for | Floer homology |

Scientific career | |

Fields | Mathematics |

Institutions | Ruhr-Universität Bochum University of California, Berkeley |

Doctoral advisor | Eduard Zehnder Ralf Stöcker |

**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".^{ }

"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."^{ }

- 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. 42 (1989), no. 4, 335–356.

- 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.

**^**Hofer, Weinstein, and Zehnder,*Andreas Floer: 1956-1991,*Notices Amer. Math. Soc.**38**(8), 910-911**^**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.)**^***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 (Amazon search)**^**From the Press Release to the Workshop*New Applications and Generalizations of Floer Theory*of the Banff International Research Station (BIRS), May 2007 ()

- 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

- O'Connor, John J.; Robertson, Edmund F., "Andreas Floer",
*MacTutor History of Mathematics archive*, University of St Andrews . - Andreas Floer at the Mathematics Genealogy Project
- 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
- Video of Andreas Floer giving a lecture on the Arnold Conjecture, Stony Brook 1986
- Introduction to Floer Theory. Lecture by Dusa McDuff 2010
- Introduction to Floer Theory? at mathoverflow.net

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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.

**Ciprian Manolescu** is a Romanian-American mathematician, working in gauge theory, symplectic geometry, and low-dimensional topology. He is currently a Professor of Mathematics at the University of California, Los Angeles.

**Peter Benedict Kronheimer** is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University.

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 R^{4} 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 2^{2n}) provided all the fixed points are non-degenerate, this latter condition being generic in the C^{1}-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.