Michael Jerome Hopkins is a mathematician at Harvard University. He got his PhD from Northwestern University in 1984, advised by Mark Mahowald, and became a world leading researcher in algebraic topology and (stable-)homotopy theory.
Among notable achievements are his work on the Ravenel conjectures, the introduction and discussion of the generalized cohomology theory tmf and its string orientation, a formalization and construction of differential cohomology, the proof of the Kervaire invariant problem. More recently via Jacob Lurie‘s work on the cobordism hypothesis Hopkins participates in work related to the foundations of quantum field theory.
On the stable homotopy theory of the loop spaces of special unitary group and of quaternionic unitary groups (with early discussion of finite-dimensional complex orientation of generalized cohomology theories):
Introducing the nilpotence theorem in stable homotopy theory:
Ethan Devinatz, Michael Hopkins, Jeffrey Smith, Nilpotence and Stable Homotopy Theory I, Annals of Mathematics Second Series 128 2 (1988) 207-241 [jstor:1971440]
Ethan Devinatz, Michael Hopkins, Jeffrey Smith, Nilpotence and Stable Homotopy Theory II, Annals of Mathematics Second Series 148 1 (1998) 1-49 [jstor:120991]
On the Conner-Floyd isomorphism for the Atiyah-Bott-Shapiro orientation of KU and KO (cobordism theory determining homology theory):
The construction of tmf was originally announced, as joint work with Mark Mahowald and Haynes Miller, in
(There the spectrum was still called “” instead of “”.)
On elliptic genera, the Witten genus and the string orientation of tmf:
Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595-687 [doi:10.1007/s002220100175, pdf, MR1869850]
Matthew Ando, Michael Hopkins, Neil Strickland, The sigma orientation is an H-infinity map, American Journal of Mathematics 126 2 (2004) 247-334 [arXiv:math/0204053, doi:10.1353/ajm.2004.0008]
The details of the definition then appeared in
On complex oriented cohomology theory and stacks:
Michael Hopkins: Complex oriented cohomology theories and the language of stacks, lecture notes (1999) [pdf, pdf]
Mike Hopkins (talk notes by Michael Hill), Stacks and complex oriented cohomology theories Talbot lecture notes (2007) [pdf]
On generalized (transchromatic) group characters via complex oriented cohomology theory:
Introducing generalized differential cohomology motivated by the M5-brane partition function:
On twisted equivariant K-theory with an eye towards twisted ad-equivariant K-theory:
On ∞-groups of units, Thom spectra and twisted generalized cohomology:
Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra [arXiv:0810.4535]
Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory, Journal of Topology, 7 4 (2014) [arXiv:1403.4320, arXiv:10.1112/jtopol/jtu009]
Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, An -categorical approach to -line bundles, -module Thom spectra, and twisted -homology, Journal of Topology 7 3 (2014) 869-893 [arXiv:1403.4325, doi:10.1112/jtopol/jtt035]
On topological quantum field theory:
On twisted ad-equivariant K-theory of compact Lie groups and the identification with the Verlinde ring of positive energy representations of their loop group:
Daniel S. Freed, Michael Hopkins, Constantin Teleman,
Loop Groups and Twisted K-Theory I,
J. Topology 4 (2011) 737-789
Loop Groups and Twisted K-Theory II,
J. Amer. Math. Soc. 26 (2013) 595-644
Loop Groups and Twisted K-Theory III,
Annals of Mathematics 174 (2011) 947-1007
On ambidextrous adjunctions in stable homotopy theory
Introducing Hodge-filtered differential cohomology and its specialization to Hodge-filtered complex cobordism theory:
Solving the Arf-Kervaire invariant problem with methods of equivariant stable homotopy theory:
Michael Hill, Michael Hopkins, Douglas Ravenel, Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem, New Mathematical Monographs, Cambridge University Press (2021) [doi:10.1017/9781108917278]
Michael Hill, Michael Hopkins, Douglas Ravenel, On the non-existence of elements of Kervaire invariant one, Annals of Mathematics 184 1 (2016)[doi:10.4007/annals.2016.184.1.1, arXiv:0908.3724, talk slides]
Michael Hill, Michael Hopkins, Douglas Ravenel, The Arf-Kervaire problem in algebraic topology: Sketch of the proof, Current Developments in Mathematics, 2010: 1-44 (2011) (pdf, doi:10.4310/CDM.2010.v2010.n1.a1)
Michael Hill, Michael Hopkins, Douglas Ravenel, The Arf-Kervaire invariant problem in algebraic topology: introduction (2016) [pdf]
On classification of invertible TQFTs via reflection positivity:
On a relation of extended 3d TQFT to lattice models for topological phases of matter (like the 3d toric code):
Last revised on November 23, 2025 at 15:11:57. See the history of this page for a list of all contributions to it.