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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 Lagrangian 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, 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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Eduard J. Zehnder is a Swiss mathematician, considered one of the founders of symplectic topology.
Emmy Murphy is an American mathematician and a professor at Princeton University who works in the area of symplectic topology, contact geometry and geometric topology.
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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.
