Peter B. Kronheimer
Encyclopedia
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
.
Kronheimer has frequently collaborated with Tomasz Mrowka
of MIT. One of their first important results was a structure theorem for Donaldson's polynomial invariants and applications to minimal genus problems of embedded surfaces in 4-manifolds. After the arrival of Seiberg–Witten theory their work on embedded surfaces culminated in a proof of the Thom conjecture
—which had been outstanding for several decades. Another of Kronheimer and Mrowka's results was a proof of the Property P conjecture
for knots.
Kronheimer attended the City of London School
. He completed his PhD at Oxford University under the direction of Michael Atiyah
. He has had a long association with Merton College, the oldest of the constituent colleges of Oxford University, being an undergraduate, graduate, and full fellow of the college.
Besides his research articles, his writings include a book, with Simon Donaldson
, on 4-manifolds, and a book with Mrowka on Seiberg–Witten–Floer homology, entitled "Monopoles and Three-Manifolds".
His PhD students have included Ian Dowker, Jacob Rasmussen, Ciprian Manolescu
, and Olga Plamenevskaya.
On a lighter note, Kronheimer is known for having made small changes to a slightly obscure calligraphic font (Ralph Smith's formal font) for use in his mathematical papers, having been unable to find a generic font that was sufficiently to his taste.
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....
, known for his work on 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...
and its applications to 3-
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
and 4-dimensional
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...
topology. He is presently William Casper Graustein Professor of Mathematics at Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
.
Kronheimer has frequently collaborated with Tomasz Mrowka
Tomasz Mrowka
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...
of MIT. One of their first important results was a structure theorem for Donaldson's polynomial invariants and applications to minimal genus problems of embedded surfaces in 4-manifolds. After the arrival of Seiberg–Witten theory their work on embedded surfaces culminated in a proof of the 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,...
—which had been outstanding for several decades. Another of Kronheimer and Mrowka's results was a proof of 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.
Kronheimer attended the City of London School
City of London School
The City of London School is a boys' independent day school on the banks of the River Thames in the City of London, England. It is the brother school of the City of London School for Girls and the co-educational City of London Freemen's School...
. He completed his PhD at Oxford University under the direction of Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
. He has had a long association with Merton College, the oldest of the constituent colleges of Oxford University, being an undergraduate, graduate, and full fellow of the college.
Besides his research articles, his writings include a book, with Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
, on 4-manifolds, and a book with Mrowka on Seiberg–Witten–Floer homology, entitled "Monopoles and Three-Manifolds".
His PhD students have included Ian Dowker, Jacob Rasmussen, Ciprian Manolescu
Ciprian Manolescu
Ciprian Manolescu is a Romanian mathematician. He is presently an Associate Professor in the mathematics department at the University of California, Los Angeles....
, and Olga Plamenevskaya.
On a lighter note, Kronheimer is known for having made small changes to a slightly obscure calligraphic font (Ralph Smith's formal font) for use in his mathematical papers, having been unable to find a generic font that was sufficiently to his taste.