# Michael J. Hopkins

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Michael J. Hopkins
Michael J. Hopkins, 2009
BornApril 18, 1958 (age 61)
NationalityAmerican
Alma materNorthwestern University
Known forNilpotence theorem in Mathematics Topological modular forms
Kervaire invariant problem
AwardsVeblen Prize (2001)
NAS Award in Mathematics (2012)
Nemmers Prize (2014)
Senior Berwick Prize (2014)
Scientific career
FieldsMathematics
InstitutionsHarvard University
Doctoral advisorMark Mahowald
Ioan James
Doctoral studentsDaniel Biss
Jacob Lurie

Michael Jerome Hopkins (born April 18, 1958) is an American mathematician known for work in algebraic topology.

## Life

He received his Ph.D. from Northwestern University 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, at the 1994 International Congress of Mathematicians in Zurich,[1] and was a plenary speaker at the 2002 International Congress of Mathematicians in Beijing.[2] 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,[3][4] 2012 the NAS Award in Mathematics and 2014 the Nemmers Prize in Mathematics.

## Work

Hopkins' work concentrates on algebraic topology, especially stable homotopy theory. It can roughly be divided into four parts (while the list of topics below is by no means exhaustive):

### The Ravenel conjectures

The Ravenel conjectures 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 Ethan Devinatz, Hopkins and Jeff Smith (published in 1988).[5] The rest of the Ravenel conjectures (except for the telescope conjecture) were proven by Hopkins and Smith soon after (published in 1998).[6] Another result in this spirit proven by Hopkins and Douglas 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.

### 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 spectra. 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 ${\displaystyle A_{\infty }}$-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 ${\displaystyle E_{\infty }}$-ring spectra.[7] This was later used in the Hopkins–Miller construction of topological modular forms.[8] 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).[9][10]

### 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 Ravenel.[11] This problem is connected to the study of exotic spheres, 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.[12]

### Work connected to geometry/physics

This includes papers on smooth and twisted K-theory and its relationship to loop groups[13] and also work about (extended) topological field theories,[14] joint with Daniel Freed, Jacob Lurie and Constantin Teleman.

## Notes

1. ^ Hopkins, M. J. (1994). "Topological modular forms, the Witten genus, and the theorem of the cube" (PDF). In: Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994. Vol. 1. pp. 554–565.
2. ^ Hopkins, M. J. (2002). "Algebraic topology and modular forms". Proceedings of the ICM, Beijing. 1: 283–309. arXiv:math/0212397.
3. ^
4. ^ Veblen Prize 2001 (PDF)
5. ^ Devinatz, Ethan S.; Hopkins, Michael J.; Smith, Jeffrey H. (1988), "Nilpotence and Stable Homotopy Theory I", Annals of Mathematics, 128 (2): 207–241, doi:10.2307/1971440, JSTOR 1971440, MR 0960945
6. ^ Hopkins, Michael J.; Smith, Jeffrey H. (1998), "Nilpotence and Stable Homotopy Theory II", Annals of Mathematics, 148 (1): 1–49, CiteSeerX 10.1.1.568.9148, doi:10.2307/120991, JSTOR 120991
7. ^
8. ^
9. ^ Ando, M; Hopkins, M. J; Strickland, N. P (2001), "Elliptic spectra, the Witten genus and the theorem of the cube", Inventiones Mathematicae, 146 (3): 595, Bibcode:2001InMat.146..595A, CiteSeerX 10.1.1.136.5083, doi:10.1007/s002220100175
10. ^ Multiplicative orientations of KO-theory and of the spectrum of topological modular forms, CiteSeerX 10.1.1.128.1530
11. ^
12. ^ Hill, Michael A; Hopkins, Michael J; Ravenel, Douglas C (2009), "On the non-existence of elements of Kervaire invariant one", arXiv:0908.3724 [math.AT]
13. ^ Freed, Daniel S; Hopkins, Michael J; Teleman, Constantin (2003), "Twisted K-theory and loop group representations", arXiv:math/0312155
14. ^ Freed, Daniel S; Hopkins, Michael J; Lurie, Jacob; Teleman, Constantin (2009), "Topological quantum field theories from compact Lie groups", arXiv:0905.0731 [math.AT]