Michael J. Hopkins: Difference between revisions

Michael J. Hopkins
Michael J. Hopkins
	Michael J. Hopkins, 2009
Born	April 18, 1958 (age 66)
Nationality	American
Alma mater	Northwestern University
Known for	Nilpotence theorem in Mathematics ; Topological modular forms ; Kervaire invariant problem
Awards	Veblen Prize (2001); NAS Award in Mathematics (2012); Nemmers Prize (2014); Senior Berwick Prize (2014); Veblen Prize (2022)
	Scientific career
Fields	Mathematics
Institutions	Harvard University
Doctoral advisors	Mark Mahowald; Ioan James
Doctoral students	Daniel Biss; Jacob Lurie; Charles Rezk; Reid Barton

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

Life

He received his PhD from Northwestern University in 1984 under the direction of Mark Mahowald, with thesis Stable Decompositions of Certain Loop Spaces.^[1] Also 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 the Massachusetts Institute of Technology, 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.

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).^[2] The rest of the Ravenel conjectures (except for the telescope conjecture) were proven by Hopkins and Smith soon after (published in 1998).^[3] 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 $A_{\infty }$ -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 $E_{\infty }$ -ring spectra.^[4] This was later used in the Hopkins–Miller construction of topological modular forms.^[5] 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).^[6]^[7]

The Kervaire invariant problem

On April 21, 2009, Hopkins announced the solution of the Kervaire invariant problem, in joint work with Mike Hill and Douglas Ravenel.^[8] 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.^[9]

Work connected to geometry/physics

This includes papers on smooth and twisted K-theory and its relationship to loop groups^[10] and also work about (extended) topological field theories,^[11] joint with Daniel Freed, Jacob Lurie, and Constantin Teleman.

Recognition

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,^[12] and was a plenary speaker at the 2002 International Congress of Mathematicians in Beijing.^[13] 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,^[14]^[15] 2012 the NAS Award in Mathematics, 2014 the Senior Berwick Prize and also in 2014 the Nemmers Prize in Mathematics. He was named to the 2021 class of fellows of the American Mathematical Society "for contributions to algebraic topology and related areas of algebraic geometry, representation theory, and mathematical physics".^[16] In 2022 he received for the second time the Oswald Veblen Prize in Geometry.^[17]

Notes

^ Michael J. Hopkins at the Mathematics Genealogy Project
^ 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
^ 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
^ Moduli spaces of commutative ring spectra (PDF)
^ Goerss – Topological Modular Forms (PDF)
^ Ando, Matthew; Hopkins, Michael J.; Strickland, Neil 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, S2CID 119932563
^ Multiplicative orientations of KO-theory and of the spectrum of topological modular forms, CiteSeerX 10.1.1.128.1530
^ Geometry and Physics: Atiyah80
^ 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]
^ Freed, Daniel S.; Hopkins, Michael J.; Teleman, Constantin (2003), "Twisted K-theory and loop group representations", arXiv:math/0312155
^ Freed, Daniel S.; Hopkins, Michael J.; Lurie, Jacob; Teleman, Constantin (2010), "Topological quantum field theories from compact Lie groups", A celebration of the mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, Providence, RI: American Mathematical Society, pp. 367–403, arXiv:0905.0731, MR 2648901
^ 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.
^ Hopkins, M. J. (2002). "Algebraic topology and modular forms". Proceedings of the ICM, Beijing. 1: 283–309. arXiv:math/0212397. Bibcode:2002math.....12397H.
^ Mike Hopkins – Biographical Sketch (PDF)
^ Veblen Prize 2001 (PDF)
^ 2021 Class of Fellows of the AMS, American Mathematical Society, retrieved November 2, 2020
^ Oswald Veblen Prize in Geometry 2022

External links

2001 Veblen Prize

[1] Michael J. Hopkins at the Mathematics Genealogy Project

[Nilpotence1-2] 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

[Nilpotence2-3] 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

[Goerss1-4] Moduli spaces of commutative ring spectra (PDF)

[Goerss2-5] Goerss – Topological Modular Forms (PDF)

[Ando1-6] Ando, Matthew; Hopkins, Michael J.; Strickland, Neil 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, S2CID 119932563

[Ando2-7] Multiplicative orientations of KO-theory and of the spectrum of topological modular forms, CiteSeerX 10.1.1.128.1530

[Kervaire1-8] Geometry and Physics: Atiyah80

[Kervaire2-9] 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]

[loop-10] Freed, Daniel S.; Hopkins, Michael J.; Teleman, Constantin (2003), "Twisted K-theory and loop group representations", arXiv:math/0312155

[top-11] Freed, Daniel S.; Hopkins, Michael J.; Lurie, Jacob; Teleman, Constantin (2010), "Topological quantum field theories from compact Lie groups", A celebration of the mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, Providence, RI: American Mathematical Society, pp. 367–403, arXiv:0905.0731, MR 2648901

[12] 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.

[13] Hopkins, M. J. (2002). "Algebraic topology and modular forms". Proceedings of the ICM, Beijing. 1: 283–309. arXiv:math/0212397. Bibcode:2002math.....12397H.

[Bio-14] Mike Hopkins – Biographical Sketch (PDF)

[Veblen-15] Veblen Prize 2001 (PDF)

[16] 2021 Class of Fellows of the AMS, American Mathematical Society, retrieved November 2, 2020

[17] Oswald Veblen Prize in Geometry 2022

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@@ Line 1: / Line 1: @@
+{{Use mdy dates|date=November 2022}}
 {{short description|American mathematician}}
 {{Infobox scientist
@@ Line 9: / Line 10: @@
 | death_date        =
 | death_place       =
-| nationality       = [[Americans|American]]
+| nationality       = American
-| fields            = [[Mathematics]]
+| fields            = Mathematics
 | workplaces        = [[Harvard University]]
 | alma_mater        = [[Northwestern University]]
 | doctoral_advisors = [[Mark Mahowald]]<br>[[Ioan James]]
-| doctoral_students = [[Daniel Biss]]<br>[[Jacob Lurie]]<br>[[Charles Rezk]]
+| doctoral_students = [[Daniel Biss]]<br>[[Jacob Lurie]]<br>[[Charles Rezk]]<br>[[Reid Barton ]]
-| known_for         = [[Nilpotence theorem]] in Mathematics [[Topological modular forms]] <br> [[Kervaire invariant problem]]
+| known_for         = [[Nilpotence theorem]] in Mathematics <br> [[Topological modular forms]] <br> [[Kervaire invariant problem]]
-| awards            = [[Oswald Veblen Prize in Geometry|Veblen Prize]] <small>(2001)</small><br>[[NAS Award in Mathematics]] <small>(2012)</small><br>[[Nemmers Prize in Mathematics|Nemmers Prize]] <small>(2014)</small><br>[[Senior Berwick Prize]] <small>(2014)</small>
+| awards            = [[Oswald Veblen Prize in Geometry|Veblen Prize]] <small>(2001)</small><br>[[NAS Award in Mathematics]] <small>(2012)</small><br>[[Nemmers Prize in Mathematics|Nemmers Prize]] <small>(2014)</small><br>[[Senior Berwick Prize]] <small>(2014)</small><br>[[Oswald Veblen Prize in Geometry|Veblen Prize]] <small>(2022)</small>
 }}
-'''Michael Jerome Hopkins''' (born April 18, 1958) is an American [[mathematician]] known for work in [[algebraic topology]].
+'''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 received his PhD from [[Northwestern University]] in 1984 under the direction of [[Mark Mahowald]], with thesis ''Stable Decompositions of Certain Loop Spaces''.<ref>{{MathGenealogy|name=Michael J. Hopkins|id=5672}}</ref> Also 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 the [[Massachusetts Institute of Technology]], 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 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]],<ref>{{cite book|author=Hopkins, M. J.|year=1994|chapter=Topological modular forms, the Witten genus, and the theorem of the cube|title=''In:'' Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994|volume=Vol. 1|pages=554–565|chapter-url=http://mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf}}</ref> and was a plenary speaker at the 2002 [[International Congress of Mathematicians]] in [[Beijing]].<ref>{{cite journal |author=Hopkins, M. J.|title=Algebraic topology and modular forms|arxiv=math/0212397 |journal=Proceedings of the ICM, Beijing |year=2002 |volume=1 |pages=283–309}}</ref> 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 [[Joseph Ritt|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 [[American Mathematical Society|AMS]] for his work in [[homotopy theory]],<ref name=Bio>{{Citation|url=http://www.zib.de/Euler/2008/biohopkins.pdf|title=Mike Hopkins – Biographical Sketch}}</ref><ref name=Veblen>{{Citation|url=http://www.ams.org/notices/200104/comm-veblen.pdf|title=Veblen Prize 2001}}</ref>  2012 the [[NAS Award in Mathematics]] and 2014 the [[Nemmers Prize in Mathematics]].
 ==Work==
@@ Line 34: / Line 32: @@
 ===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 spectrum|highly structured ring spectra]]. The first success of this program was the Hopkins–Miller theorem: It is about the action of the [[Jack Morava|Morava]] stabilizer group on Lubin–Tate spectra (arising out of the deformation theory of [[formal group|formal group laws]]) and its refinement to <math>A_\infty</math>-ring spectra – this allowed to take homotopy fixed points of finite subgroups of the Morava stabilizer groups, which led to higher real [[K-theory|K-theories]]. Together with Paul Goerss, Hopkins later set up a systematic obstruction theory for refinements to <math>E_\infty</math>-ring spectra.<ref name=Goerss1>{{Citation|url=http://www.math.northwestern.edu/~pgoerss/papers/sum.pdf|title=Moduli spaces of commutative ring spectra }}</ref>  This was later used in the Hopkins–Miller construction of [[topological modular forms]].<ref name=Goerss2>{{Citation|url=http://www.math.northwestern.edu/~pgoerss/papers/Exp.1005.P.Goerss.pdf|title=Goerss – Topological Modular Forms}}</ref> 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).<ref name=Ando1>{{Citation|title=Elliptic spectra, the Witten genus and the theorem of the cube |journal=[[Inventiones Mathematicae]] |volume=146 |issue=3 |pages=595 |bibcode=2001InMat.146..595A |author1=Ando |first1=Matthew |last2=Hopkins |first2=Michael J. |last3=Strickland |first3=Neil P. |year=2001 |doi=10.1007/s002220100175 |citeseerx = 10.1.1.136.5083}}</ref><ref name=Ando2>{{Citation|citeseerx = 10.1.1.128.1530|title=Multiplicative orientations of KO-theory and of the spectrum of topological modular forms}}</ref>
+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 spectrum|highly structured ring spectra]]. The first success of this program was the Hopkins–Miller theorem: It is about the action of the [[Jack Morava|Morava]] stabilizer group on Lubin–Tate spectra (arising out of the deformation theory of [[formal group|formal group laws]]) and its refinement to <math>A_\infty</math>-ring spectra – this allowed to take homotopy fixed points of finite subgroups of the Morava stabilizer groups, which led to higher real [[K-theory|K-theories]]. Together with Paul Goerss, Hopkins later set up a systematic obstruction theory for refinements to <math>E_\infty</math>-ring spectra.<ref name=Goerss1>{{Citation|url=http://www.math.northwestern.edu/~pgoerss/papers/sum.pdf|title=Moduli spaces of commutative ring spectra }}</ref>  This was later used in the Hopkins–Miller construction of [[topological modular forms]].<ref name=Goerss2>{{Citation|url=http://www.math.northwestern.edu/~pgoerss/papers/Exp.1005.P.Goerss.pdf|title=Goerss – Topological Modular Forms}}</ref> 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).<ref name=Ando1>{{Citation|title=Elliptic spectra, the Witten genus and the theorem of the cube |journal=[[Inventiones Mathematicae]] |volume=146 |issue=3 |pages=595 |bibcode=2001InMat.146..595A |last1=Ando |first1=Matthew |last2=Hopkins |first2=Michael J. |last3=Strickland |first3=Neil P. |year=2001 |doi=10.1007/s002220100175 |citeseerx = 10.1.1.136.5083|s2cid=119932563 }}</ref><ref name=Ando2>{{Citation|citeseerx = 10.1.1.128.1530|title=Multiplicative orientations of KO-theory and of the spectrum of topological modular forms}}</ref>
 ===The Kervaire invariant problem===
-On 21 April 2009, Hopkins announced the solution of the [[Kervaire invariant|Kervaire invariant problem]], in joint work with [[Mike Hill (mathematician)|Mike Hill]] and [[Douglas Ravenel]].<ref name=Kervaire1>{{Citation|url=http://www.maths.ed.ac.uk/~aar/atiyah80.htm|title=Geometry and Physics: Atiyah80}}</ref> This problem is connected to the study of [[exotic sphere]]s, but got transformed by work of [[William Browder (mathematician)|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.<ref name=Kervaire2>{{cite arxiv |eprint=0908.3724|title=On the non-existence of elements of Kervaire invariant one |mode=cs2|last1=Hill |first1=Michael A |last2=Hopkins |first2=Michael J |last3=Ravenel |first3=Douglas C |class=math.AT |year=2009 }}</ref>
+On April 21, 2009, Hopkins announced the solution of the [[Kervaire invariant|Kervaire invariant problem]], in joint work with [[Mike Hill (mathematician)|Mike Hill]] and [[Douglas Ravenel]].<ref name=Kervaire1>{{Citation|url=http://www.maths.ed.ac.uk/~aar/atiyah80.htm|title=Geometry and Physics: Atiyah80}}</ref> This problem is connected to the study of [[exotic sphere]]s, but got transformed by work of [[William Browder (mathematician)|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.<ref name=Kervaire2>{{cite arXiv |eprint=0908.3724|title=On the non-existence of elements of Kervaire invariant one |mode=cs2|last1=Hill |first1=Michael A |last2=Hopkins |first2=Michael J |last3=Ravenel |first3=Douglas C |class=math.AT |year=2009 }}</ref>
 ===Work connected to geometry/physics===
-This includes papers on smooth and [[twisted K-theory]] and its relationship to [[loop group]]s<ref name=loop>{{cite arxiv|eprint=math/0312155|title=Twisted K-theory and loop group representations|mode=cs2|last1=Freed|first1=Daniel S.|last2=Hopkins|first2=Michael J.|last3=Teleman|first3=Constantin|year=2003}}</ref> and also work about (extended) [[topological quantum field theory|topological field theories]],<ref name=top>{{cite book|eprint=0905.0731|contribution=Topological quantum field theories from compact Lie groups|mode=cs2|last1=Freed|first1=Daniel S.|author1-link=Dan Freed| last2=Hopkins|first2=Michael J.| last3=Lurie|first3=Jacob|author3-link=Jacob Lurie| last4=Teleman|first4=Constantin|year=2010|title=A celebration of the mathematical legacy of Raoul Bott| pages=367–403| series=
+This includes papers on smooth and [[twisted K-theory]] and its relationship to [[loop group]]s<ref name=loop>{{cite arXiv|eprint=math/0312155|title=Twisted K-theory and loop group representations|mode=cs2|last1=Freed|first1=Daniel S.|last2=Hopkins|first2=Michael J.|last3=Teleman|first3=Constantin|year=2003}}</ref> and also work about (extended) [[topological quantum field theory|topological field theories]],<ref name=top>{{cite book|arxiv=0905.0731|contribution=Topological quantum field theories from compact Lie groups|mode=cs2|last1=Freed|first1=Daniel S.|author1-link=Dan Freed| last2=Hopkins|first2=Michael J.| last3=Lurie|first3=Jacob|author3-link=Jacob Lurie| last4=Teleman|first4=Constantin|year=2010|title=A celebration of the mathematical legacy of Raoul Bott| pages=367–403| series= CRM Proc. Lecture Notes|volume= 50 |publisher=American Mathematical Society|location= Providence, RI |mr=2648901}}</ref> joint with [[Dan Freed|Daniel Freed]], [[Jacob Lurie]], and [[Constantin Teleman]].
-CRM Proc. Lecture Notes|volume= 50 |publisher=American Mathematical Society|location= Providence, RI |mr=2648901}}</ref> joint with [[Dan Freed|Daniel Freed]], [[Jacob Lurie]], and [[Constantin Teleman]].
+==Recognition==
+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,<ref>{{cite book|author=Hopkins, M. J.|year=1994|chapter=Topological modular forms, the Witten genus, and the theorem of the cube|title=''In:'' Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994|volume=1|pages=554–565|chapter-url=http://mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf}}</ref> and was a plenary speaker at the 2002 [[International Congress of Mathematicians]] in Beijing.<ref>{{cite journal |author=Hopkins, M. J.|title=Algebraic topology and modular forms|arxiv=math/0212397 |journal=Proceedings of the ICM, Beijing |year=2002 |volume=1 |pages=283–309|bibcode=2002math.....12397H}}</ref> 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 [[Joseph Ritt|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 [[American Mathematical Society|AMS]] for his work in [[homotopy theory]],<ref name=Bio>{{Citation|url=http://www.zib.de/Euler/2008/biohopkins.pdf|title=Mike Hopkins – Biographical Sketch}}</ref><ref name=Veblen>{{Citation|url=https://www.ams.org/notices/200104/comm-veblen.pdf|title=Veblen Prize 2001}}</ref>  2012 the [[NAS Award in Mathematics]], 2014 the [[Senior Berwick Prize]] and also in 2014 the [[Nemmers Prize in Mathematics]]. He was named to the 2021 class of fellows of the American Mathematical Society "for contributions to algebraic topology and related areas of algebraic geometry, representation theory, and mathematical physics".<ref>{{citation|url=https://www.ams.org/cgi-bin/fellows/fellows_by_year.cgi?year=2021|title=2021 Class of Fellows of the AMS|publisher=American Mathematical Society|access-date=November 2, 2020}}</ref> In 2022 he received for the second time the [[Oswald Veblen Prize in Geometry]].<ref>[http://www.ams.org/news?news_id=6849 Oswald Veblen Prize in Geometry 2022]</ref>
 ==Notes==
@@ Line 47: / Line 48: @@
 ==External links==
-*[http://www.ams.org/notices/200104/comm-veblen.pdf 2001 Veblen Prize]
+*[https://www.ams.org/notices/200104/comm-veblen.pdf 2001 Veblen Prize]
-* {{mathgenealogy|name=Michael J. Hopkins|id=5672}}
 {{Veblen Prize recipients}}
@@ Line 63: / Line 63: @@
 [[Category:Lehigh University faculty]]
 [[Category:Massachusetts Institute of Technology faculty]]
-[[Category:Harvard University faculty]]
+[[Category:Harvard University Department of Mathematics faculty]]
 [[Category:Living people]]
 [[Category:Members of the United States National Academy of Sciences]]
+[[Category:Fellows of the American Mathematical Society]]

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