Michael J. Hopkins
Encyclopedia
Michael Jerome Hopkins is an American
mathematician
known for work in algebraic topology
.
in 1984 under the direction of Mark Mahowald
. In 1984 he also received his D.Phil. from the University of Oxford
under the supervision of Ioan James
.
He has been professor of mathematics at Harvard University
since 2005, after fifteen years at MIT, a few years of teaching at Princeton University
, a one-year position with the University of Chicago
, and a visiting
lecturer position at Lehigh University
. He gave invited addresses at the 1990 Winter Meeting of
the American Mathematical Society
in Louisville, Kentucky, and at the 1994 International Congress of Mathematicians
in Zurich
. He presented the 1994 Everett Pitcher Lectures at Lehigh
University, the 2000 Namboodiri Lectures at the University of Chicago, the 2000 Marston Morse
Memorial Lectures at the Institute for Advanced Study
, Princeton, the 2003 Ritt Lectures at Columbia University
and the 2010 Bowen Lectures in Berkeley. In 2001 he was awarded the Oswald Veblen Prize in Geometry
from the AMS
for his work in homotopy theory.
. It can roughly be divided into four parts (while the list of topics below is by no means exhaustive):
very roughly say: complex cobordism
(and its variants) see more in the stable homotopy category than you might think. For example, the nilpotence conjecture states that some suspension
of some iteration of a map between finite CW-complexes is null-homotopic iff it is zero in complex cobordism. This was proven by Devinatz, Hopkins and Jeff Smith (published in 1988). The rest of the Ravenel conjectures (except for the telescope conjecture) were proven by Hopkins and Smith soon after (published in 1998). Another result in this spirit proven by Hopkins and Ravenel is the chromatic convergence theorem, which states that one can recover a finite CW-complex from its localizations with respect to wedges of Morava K-theories
.
. The first success of this program was the Hopkins–Miller theorem: It is about the action of the Morava
stabilizer group on Lubin–Tate spectra (arising out of the deformation theory of formal group laws
) and its refinement to
-ring spectra – this allowed to take homotopy fixed points of finite subgroups of the Morava stabilizer groups, which led to higher real K-theories
. Together with Paul Goerss, Hopkins later set up a systematic obstruction theory for refinements to
-ring spectra. This was later used in the Hopkins–Miller construction of topological modular forms
. Subsequent work of Hopkins on this topic includes papers on the question of the orientability of TMF with respect to string cobordism (joint work with Ando, Strickland and Rezk).
. This problem is connected to the study of exotic sphere
s, but got transformed by work of William Browder
into a problem in stable homotopy theory. The proof by Hill, Hopkins and Ravenel works purely in the stable homotopy setting and uses equivariant homotopy theory in a crucial way.
and its relationship to loop group
s and also work about (extended) topological field theories
, joint with Daniel Freed, Jacob Lurie
and Constantin Teleman.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
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....
known for work in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
.
Life
He received his Ph.D. from Northwestern UniversityNorthwestern University
Northwestern University is a private research university in Evanston and Chicago, Illinois, USA. Northwestern has eleven undergraduate, graduate, and professional schools offering 124 undergraduate degrees and 145 graduate and professional degrees....
in 1984 under the direction of Mark Mahowald
Mark Mahowald
Mark E. Mahowald is an American mathematician known for work in algebraic topology.-Life:He received his Ph.D. from University of Minnesota in 1955 under the direction of Bernard Russel Gelbaum with a thesis on Measure in Groups...
. In 1984 he also received his D.Phil. from the University of Oxford
University of Oxford
The University of Oxford is a university located in Oxford, United Kingdom. It is the second-oldest surviving university in the world and the oldest in the English-speaking world. Although its exact date of foundation is unclear, there is evidence of teaching as far back as 1096...
under the supervision of Ioan James
Ioan James
Ioan Mackenzie James is a British mathematician working in the field of topology particularly in homotopy theory.James was born in Croydon, Surrey, England, and was educated at St Paul's School, London and Queen's College, Oxford. In 1953 He earned a D. Phil...
.
He has been 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...
since 2005, after fifteen years at MIT, a few years of teaching at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, a one-year position with the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
, and a visiting
lecturer position at Lehigh University
Lehigh University
Lehigh University is a private, co-educational university located in Bethlehem, Pennsylvania, in the Lehigh Valley region of the United States. It was established in 1865 by Asa Packer as a four-year technical school, but has grown to include studies in a wide variety of disciplines...
. He gave invited addresses at the 1990 Winter Meeting of
the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
in Louisville, Kentucky, and at the 1994 International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Zurich
Zürich
Zurich is the largest city in Switzerland and the capital of the canton of Zurich. It is located in central Switzerland at the northwestern tip of Lake Zurich...
. He presented the 1994 Everett Pitcher Lectures at Lehigh
University, the 2000 Namboodiri Lectures at the University of Chicago, the 2000 Marston Morse
Memorial Lectures at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
, Princeton, the 2003 Ritt Lectures at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
and the 2010 Bowen Lectures in Berkeley. In 2001 he was awarded 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...
for his work in homotopy theory.
Work
Hopkins' work concentrates on algebraic topology, especially stable homotopy theoryStable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
. It can roughly be divided into four parts (while the list of topics below is by no means exhaustive):
The Ravenel conjectures
The Ravenel conjecturesRavenel conjectures
The Ravenel conjectures are a set of mathematical conjectures in the field of stable homotopy theory posed by Douglas Ravenel at the end of a paper published in 1984. It was earlier circulated in preprint. The problems involved have largely been resolved, with all but the "telescope conjecture"...
very roughly say: complex cobordism
Complex cobordism
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories...
(and its variants) see more in the stable homotopy category than you might think. For example, the nilpotence conjecture states that some suspension
Suspension
Suspension may refer to:In science and engineering:*Suspension , in mathematics*Suspension , in mathematics*Suspension , mixture of two chemicals with the property that one does not rapidly settle out...
of some iteration of a map between finite CW-complexes is null-homotopic iff it is zero in complex cobordism. This was proven by Devinatz, Hopkins and Jeff Smith (published in 1988). The rest of the Ravenel conjectures (except for the telescope conjecture) were proven by Hopkins and Smith soon after (published in 1998). Another result in this spirit proven by Hopkins and Ravenel is the chromatic convergence theorem, which states that one can recover a finite CW-complex from its localizations with respect to wedges of Morava K-theories
Morava K-theory
In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s...
.
Hopkins–Miller theorem and topological modular forms
This part of work is about refining a homotopy commutative diagram of ring spectra up to homotopy to a strictly commutative diagram of highly structured ring spectraHighly structured ring spectrum
In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an A_\infty-ring is called an E_\infty-ring...
. The first success of this program was the Hopkins–Miller theorem: It is about the action of the Morava
Jack Morava
Jack Johnson Morava is an American topologist.Of Czech and Appalachian descent, he was raised in Mercedes, Texas ; an early interest in topology was strongly encouraged by his parents...
stabilizer group on Lubin–Tate spectra (arising out of the deformation theory of formal group laws
Formal group
In mathematics, a formal group law is a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between...
) and its refinement to
-ring spectra – this allowed to take homotopy fixed points of finite subgroups of the Morava stabilizer groups, which led to higher real K-theoriesK-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
. Together with Paul Goerss, Hopkins later set up a systematic obstruction theory for refinements to
-ring spectra. This was later used in the Hopkins–Miller construction of topological modular formsTopological modular forms
In mathematics, the spectrum of topological modular forms describes a generalized cohomology theory whose coefficient ring is related to the graded ring of holomorphic modular forms with integral cusp expansions...
. Subsequent work of Hopkins on this topic includes papers on the question of the orientability of TMF with respect to string cobordism (joint work with Ando, Strickland and Rezk).
The Kervaire invariant problem
On 21 April 2009, Hopkins announced the solution of the Kervaire invariant problem, in joint work with Mike Hill and Douglas RavenelDouglas Ravenel
Douglas Conner Ravenel is an American mathematician known for work in algebraic topology.-Life:He received his Ph.D. from Brandeis University in 1972 under the direction of Edgar H. Brown, Jr. with a thesis on exotic characteristic classes of spherical fibrations. From 1971 to 1973 he was...
. This problem is connected to the study of exotic sphere
Exotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
s, but got transformed by work of William Browder
William Browder (mathematician)
William Browder is an American mathematician, specializing in algebraic topology, differential topology and differential geometry...
into a problem in stable homotopy theory. The proof by Hill, Hopkins and Ravenel works purely in the stable homotopy setting and uses equivariant homotopy theory in a crucial way.
Work connected to geometry/physics
This includes papers on smooth and twisted K-theoryTwisted K-theory
In mathematics, twisted K-theory is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory....
and its relationship to loop group
Loop group
In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. Specifically, letLG \,denote the space of continuous mapsS^1 \to G...
s and also work about (extended) topological field theories
Topological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
, joint with Daniel Freed, Jacob Lurie
Jacob Lurie
Jacob Alexander Lurie is an American mathematician, who is currently a professor at Harvard University.-Life:While in school, Lurie took part in the International Mathematical Olympiad, where he won a gold medal with a perfect score in 1994...
and Constantin Teleman.