[[Robion Kirby]]"},"doctoral_students":{"wt":"[[Larry Guth]]
[[Lenhard Ng]]
[[Sherry Gong]]"},"known_for":{"wt":""},"awards":{"wt":"{{plainlist|\n* Fellow, [[American Academy of Arts and Sciences]] (2007)\n* [[Oswald Veblen Prize in Geometry|Veblen Prize]] (2007)\n* [[Joseph L. Doob Prize|Doob Prize]] (2011)\n* Member, [[National Academy of Sciences]] (2015)\n}}"},"thesis_title":{"wt":"A local Mayer-Vietoris principle for Yang-Mills moduli spaces"},"thesis_year":{"wt":"1988"}},"i":0}}]}" id="mwBw">
Tomasz Mrowka (born September 8, 1961) is an American mathematician specializing in differential geometry and gauge theory. He is the Singer Professor of Mathematics and former head of the Department of Mathematics at the Massachusetts Institute of Technology.
Mrowka is married to MIT mathematics professor Gigliola Staffilani.^{ [1] }
A 1983 graduate of the Massachusetts Institute of Technology, he received the Ph.D from the University of California, Berkeley in 1988 under the direction of Clifford Taubes and Robion Kirby. He joined the MIT mathematics faculty as professor in 1996, following faculty appointments at Stanford University and at the California Institute of Technology (professor 1994–96).^{ [2] } At MIT, he was the Simons Professor of Mathematics from 2007–2010. Upon Isadore Singer's retirement in 2010 the name of the chair became the Singer Professor of Mathematics which Mrowka held until 2017. He was named head of the Department of Mathematics in 2014 and held that position for 3 years.^{ [3] }
A prior Sloan fellow and Young Presidential Investigator, in 1994 he was an invited speaker at the International Congress of Mathematicians (ICM) in Zurich. In 2007, he received the Oswald Veblen Prize in Geometry from the AMS jointly with Peter Kronheimer, "for their joint contributions to both three- and four-dimensional topology through the development of deep analytical techniques and applications."^{ [4] } He was named a Guggenheim Fellow in 2010, and in 2011 received the Doob Prize with Peter B. Kronheimer for their book Monopoles and Three-Manifolds (Cambridge University Press, 2007).^{ [5] }^{ [6] } In 2018 he gave a plenary lecture at the ICM in Rio de Janeiro, together with Peter Kronheimer.
He became a fellow of the American Academy of Arts & Sciences in 2007,^{ [7] } and a member of the National Academy of Sciences in 2015.^{ [8] }
Mrowka's work combines analysis, geometry, and topology, specializing in the use of partial differential equations, such as the Yang-Mills equations from particle physics to analyze low-dimensional mathematical objects.^{ [3] } Jointly with Robert Gompf, he discovered four-dimensional models of space-time topology.^{ [9] }
In joint work with Peter Kronheimer, Mrowka settled many long-standing conjectures, three of which earned them the 2007 Veblen Prize. The award citation mentions three papers that Mrowka and Kronheimer wrote together. The first paper in 1995 deals with Donaldson's polynomial invariants and introduced Kronheimer–Mrowka basic class, which have been used to prove a variety of results about the topology and geometry of 4-manifolds, and partly motivated Witten's introduction of the Seiberg–Witten invariants.^{ [10] } The second paper proves the so-called Thom conjecture and was one of the first deep applications of the then brand new Seiberg–Witten equations to four-dimensional topology.^{ [11] } In the third paper in 2004, Mrowka and Kronheimer used their earlier development of Seiberg–Witten monopole Floer homology to prove the Property P conjecture for knots.^{ [12] } The citation says: "The proof is a beautiful work of synthesis which draws upon advances made in the fields of gauge theory, symplectic and contact geometry, and foliations over the past 20 years."^{ [4] }
In further recent work with Kronheimer, Mrowka showed that a certain subtle combinatorially-defined knot invariant introduced by Mikhail Khovanov can detect “knottedness.”^{ [13] }
Edward Witten is an American mathematical and theoretical physicist. He is currently the Charles Simonyi Professor in the School of Natural Sciences at the Institute for Advanced Study. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. In addition to his contributions to physics, Witten's work has significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, awarded for his 1981 proof of the positive energy theorem in general relativity. He is considered to be the practical founder of M-theory.
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 in New York, and a Professor in Pure Mathematics at Imperial College London.
In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing (non-trivial) Dehn surgery on the knot is not simply-connected. The conjecture states that all knots, except the unknot, have Property P.
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique.
In mathematics, Khovanov homology is an oriented link invariant that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial.
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.
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 Stanford University.
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 and former chair of the mathematics department.
In mathematics, a smooth algebraic curve in the complex projective plane, of degree , has genus given by the genus–degree formula
Tian Gang is a Chinese mathematician. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at Princeton University. He is known for contributions to the mathematical fields of Kähler geometry, Gromov-Witten theory, and geometric analysis.
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.
The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was founded in 1961 in memory of Oswald Veblen. The Veblen Prize is now worth US$5000, and is awarded every three years.
In mathematics, and especially gauge theory, 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.
John Willard Morgan is an American mathematician, with contributions to topology and geometry. He is, as of 2020, Professor Emeritus at Columbia University.
In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten (1994), using the Seiberg–Witten theory studied by Nathan Seiberg and Witten during their investigations of Seiberg–Witten gauge theory.
The Geometry Festival is an annual mathematics conference held in the United States.
In mathematics, the Kronheimer–Mrowka basic classes are elements of the second cohomology H^{2}(X) of a simple smooth 4-manifold X that determine its Donaldson polynomials. They were introduced by P. B. Kronheimer and Tomasz S. Mrowka (1994, 1995).
Daniel Bennequin is a French mathematician, known for the Thurston–Bennequin number introduced in his doctoral dissertation.
Ronald Alan Fintushel is an American mathematician, specializing in low-dimensional geometric topology and the mathematics of gauge theory.
Dietmar Arno Salamon is a German mathematician.