Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.
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.
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act.
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.
The mathematical term perverse sheaves refers to the objects of certain abelian categories associated to topological spaces, which may be a real or complex manifold, or more general topologically stratified spaces, possibly singular.
Richard Melvin Schoen is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
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.
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 is the journalist Gary Taubes.
In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality
Sylvain Edward Cappell, a Belgian American mathematician and former student of William Browder at Princeton University, is a topologist who has spent most of his career at the Courant Institute of Mathematical Sciences at NYU, where he is now the Silver Professor of Mathematics.
James W. Cannon is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.
Robert Leamon Bryant is an American mathematician. He works at Duke University and specializes in differential geometry.
Carolyn S. Gordon is a mathematician and Benjamin Cheney Professor of Mathematics at Dartmouth College. She is most well known for giving a negative answer to the question "Can you hear the shape of a drum?" in her work with David Webb and Scott A. Wolpert. She is a Chauvenet Prize winner and a 2010 Noether Lecturer.
Mikhail "Mischa" Gershevich Katz is an Israeli mathematician, a professor of mathematics at Bar-Ilan University. His main interests are differential geometry, geometric topology and mathematics education; he is the author of the book Systolic Geometry and Topology, which is mainly about systolic geometry. The Katz–Sabourau inequality is named after him and Stéphane Sabourau.
Lawrence David Guth is a professor of mathematics at the Massachusetts Institute of Technology.
In mathematics, the Atiyah–Jones conjecture is a conjecture about the homology of the moduli spaces of instantons. The original form of the conjecture considered instantons over a 4-dimensional sphere. It was introduced by Michael Francis Atiyah and John D. S. Jones (1978) and proved by Charles P. Boyer, Jacques C. Hurtubise, and Benjamin M. Mann et al.. The more general version of the Atiyah–Jones conjecture is a question about the homology of the moduli spaces of instantons on any 4-dimensional real manifold, or on a complex surface. The Atiyah–Jones conjecture has been proved for ruled surfaces by R. J. Milgram and J. Hurtubise, and for rational surfaces by Elizabeth Gasparim. The conjecture remains unproved for other types of 4 manifolds.
Søren Galatius is a Danish mathematician who works as a professor of mathematics at the University of Copenhagen. He works in algebraic topology, where one of his most important results concerns the homology of the automorphisms of free groups. He is also known for his joint work with Oscar Randal-Williams on moduli spaces of manifolds, comprising several papers.
John Vincent Pardon is an American mathematician who works on geometry and topology. He is primarily known for having solved Gromov's problem on distortion of knots, for which he was awarded the 2012 Morgan Prize. He is currently a permanent member of the Simons Center for Geometry and Physics and a full professor of mathematics at Princeton University.
Robert Clark Penner is an American mathematician whose work in geometry and combinatorics has found applications in high-energy physics and more recently in theoretical biology. He is the son of Sol Penner, an aerospace engineer.
Dan Burghelea is a Romanian-American mathematician, academic, and researcher. He is an Emeritus Professor of Mathematics at Ohio State University.
