Tomasz Mrowka
Encyclopedia
Tomasz Mrowka is a Polish American
mathematician
. He has been the Singer Professor of Mathematics at Massachusetts Institute of Technology
since 2010. A graduate of MIT, he received the Ph.D. from 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 and at Caltech (professor 1994–96).
Mrowka's research is in differential geometry and gauge theory
. A prior Sloan fellow and Young Presidential Investigator, he was selected for a Clay Mathematics Visiting Professorship in 1995. 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." He is a Fellow of the American Academy of Arts & Sciences.
The first paper in 1995 deals with Donaldson's polynomial invariants, which have been used to prove a variety of results about the topology and geometry of 4-manifold
s. According to the citation, the paper "gives a conceptual framework and an organizing principle for some of the disparate observations and calculations of Donaldson invariants that had been made earlier … and it has been the point of departure and the motivating example for important further developments, most spectacularly for Witten's introduction of the so-called Seiberg–Witten invariants".
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.
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
. 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 foliation
s over the past 20 years,".
received 2011 Doob Prize for their book Monopoles and Three-Manifolds (Cambridge University Press
, 2007).
Polish American
A Polish American , is a citizen of the United States of Polish descent. There are an estimated 10 million Polish Americans, representing about 3.2% of the population of the United States...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He has been the Singer Professor of Mathematics at Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
since 2010. A graduate of MIT, he received the Ph.D. from University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
in 1988 under the direction of Clifford Taubes
Clifford Taubes
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.-Early career:Taubes received his Ph.D...
and Robion Kirby
Robion Kirby
Robion Cromwell Kirby is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology...
. He joined the MIT mathematics faculty as professor in 1996, following faculty appointments at Stanford and at Caltech (professor 1994–96).
Mrowka's research is in differential geometry and gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
. A prior Sloan fellow and Young Presidential Investigator, he was selected for a Clay Mathematics Visiting Professorship in 1995. In 2007, he received the Oswald Veblen Prize in Geometry
Oswald Veblen Prize in Geometry
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...
from the AMS
AMS
-Companies:*AMS *Auxiliary Medical Services* Austria Mikro Systeme, the old name of austriamicrosystems*American Management Systems*AMS Pictures*Arbiter Modeling Systems Pty Ltd, Melbourne, Australia*Association of Muslim Schools...
jointly with Peter Kronheimer, "for their joint contributions to both three- and four-dimensional topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
through the development of deep analytical techniques and applications." He is a Fellow of the American Academy of Arts & Sciences.
Oswald Veblen Prize in Geometry
The award citation mentions three papers that Mrowka and Kronheimer wrote together.The first paper in 1995 deals with Donaldson's polynomial invariants, which have been used to prove a variety of results about the topology and geometry of 4-manifold
4-manifold
In mathematics, 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...
s. According to the citation, the paper "gives a conceptual framework and an organizing principle for some of the disparate observations and calculations of Donaldson invariants that had been made earlier … and it has been the point of departure and the motivating example for important further developments, most spectacularly for Witten's introduction of the so-called Seiberg–Witten invariants".
The second paper proves the so-called Thom conjecture
Thom conjecture
In mathematics, a smooth algebraic curve C in the complex projective plane, of degree d, has genus given by the formulag = /2.The Thom conjecture, named after French mathematician René Thom, states that if \Sigma is any smoothly embedded connected curve representing the same class in homology as C,...
and was one of the first deep applications of the then brand new Seiberg–Witten equations to four-dimensional topology.
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
Property P conjecture
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 Dehn surgery on the knot is non-simply-connected...
for knots
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
. The citation says: "The proof is a beautiful work of synthesis which draws upon advances made in the fields of gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
, symplectic and contact geometry
Contact geometry
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'...
, and foliation
Foliation
In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
s over the past 20 years,".
Doob Prize
In 2011 Mrowka with Peter B. KronheimerPeter B. Kronheimer
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 presently William Casper Graustein Professor of Mathematics at Harvard University....
received 2011 Doob Prize for their book Monopoles and Three-Manifolds (Cambridge University Press
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...
, 2007).