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In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.
Mikhael Leonidovich Gromov is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of Institut des Hautes Études Scientifiques in France and a professor of mathematics at New York University.
Sir Simon Kirwan Donaldson is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. He is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University in New York, and a Professor in Pure Mathematics at Imperial College London.
Andreas Floer was a German mathematician who made seminal contributions to symplectic topology, and mathematical physics, in particular the invention of Floer homology. 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, 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 analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called symplectic 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, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is associated to a contact manifold and one of its Legendrian submanifolds. It is a part of a more general invariant known as symplectic field theory, and is defined using pseudoholomorphic curves.
Peter Steven Ozsváth is a professor of mathematics at Princeton University. He created, along with Zoltán Szabó, Heegaard Floer homology, a homology theory for 3-manifolds.
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Louis Hirsch Kauffman is an American mathematician, mathematical physicist, and professor of mathematics in the Department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago. He does research in topology, knot theory, topological quantum field theory, quantum information theory, and diagrammatic and categorical mathematics. He is best known for the introduction and development of the bracket polynomial and the Kauffman polynomial.
In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into , which is tangent to the standard contact structure on . It is the lowest-dimensional case of a Legendrian submanifold, which is an embedding of a k-dimensional manifold into a (2k+1)-dimensional contact manifold that is always tangent to the contact hyperplane.
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Iakov (Yan) Soibelman born 15 April 1956 is a Russian American mathematician, professor at Kansas State University, member of the Kyiv Mathematical Society (Ukraine), founder of Manhattan Mathematical Olympiad.
Eduard J. Zehnder is a Swiss mathematician, considered one of the founders of symplectic topology.
François Lalonde is a Canadian mathematician, specializing in symplectic geometry and symplectic topology.
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.
Ronald Alan Fintushel is an American mathematician, specializing in low-dimensional geometric topology and the mathematics of gauge theory.
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References
- ↑ "Vitae and Bibliography for Lenhard Ng". Oct 12, 2009. Retrieved 2010-01-21.
- ↑ "Curriculum Vitæ: Lenhard Ng". November 2010. Retrieved 2011-04-13.
- ↑ "Curriculum Vitæ: Yee Jack Ng" . Retrieved 2016-12-31.
- ↑ Muratori, Michelle C. (2006), "Insights From SMPY's Greatest Former Child Prodigies: Drs. Terence ("Terry") Tao and Lenhard ("Lenny") Ng Reflect on Their Talent Development", Gifted Child Quarterly, 50 (4): 307–324, doi:10.1177/001698620605000404, S2CID 145798062
- ↑ "Welcome".
- ↑ "Up at Bat Softball Novice Lenny Ng is Taking His Best Swing at Becoming Tops in the Math World".
- ↑ "Up at Bat Softball Novice Lenny Ng is Taking His Best Swing at Becoming Tops in the Math World".
- ↑ "International Mathematical Olympiad: Lenhard Ng". IMO. Retrieved 2008-10-10.
- ↑ Joseph Gallian. "The Putnam Competition from 1938-2009" (PDF). Archived from the original (PDF) on 2009-10-13. Retrieved 2010-09-25.
- ↑ 2019 Class of the Fellows of the AMS, American Mathematical Society , retrieved 2018-11-07
- Ng, Lenhard. Knot and braid invariants from contact homology I. Geom. Topol. 9 (2005), 247–297.
- Ng, Lenhard. Knot and braid invariants from contact homology II. Geom. Topol. 9 (2005), 1603–1637.
