John Morgan (mathematician)

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John Morgan
Born (1946-03-21) March 21, 1946 (age 78)
Nationality American
Alma mater Rice University
Awards Sloan Research Fellow (1974)
Gauss Lectureship (2008)
Member of the National Academy of Sciences (2009)
Fellow of the American Mathematical Society (2012)
Scientific career
Fields Mathematics
Institutions Stony Brook University
Columbia University
Doctoral advisor Morton L. Curtis
Doctoral students Sadayoshi Kojima  [ ja ]
Peter Ozsváth
Zoltán Szabó

John Willard Morgan (born March 21, 1946) is an American mathematician known for his contributions to topology and geometry. He is a Professor Emeritus at Columbia University and a member of the Simons Center for Geometry and Physics at Stony Brook University.

Contents

Life

Morgan received his B.A. in 1968 and Ph.D. in 1969, both from Rice University. [1] [2] [3] His Ph.D. thesis, entitled Stable tangential homotopy equivalences, was written under the supervision of Morton L. Curtis. [1] [2] He was an instructor at Princeton University from 1969 to 1972, and an assistant professor at MIT from 1972 to 1974. [1] [3] [4] He has been on the faculty at Columbia University since 1974, serving as the Chair of the Department of Mathematics from 1989 to 1991 and becoming Professor Emeritus in 2010. [1] [3] [4] Morgan is a member of the Simons Center for Geometry and Physics at Stony Brook University and served as its founding director from 2009 to 2016. [3] [4]

From 1974 to 1976, Morgan was a Sloan Research Fellow. [1] In 2008, he was awarded a Gauss Lectureship by the German Mathematical Society. In 2009 he was elected to the National Academy of Sciences. [4] In 2012 he became a fellow of the American Mathematical Society. [5] Morgan is a Member of the European Academy of Sciences. [1]

Mathematical contributions

Morgan's best-known work deals with the topology of complex manifolds and algebraic varieties. In the 1970s, Dennis Sullivan developed the notion of a minimal model of a differential graded algebra. [6] One of the simplest examples of a differential graded algebra is the space of smooth differential forms on a smooth manifold, so that Sullivan was able to apply his theory to understand the topology of smooth manifolds. In the setting of Kähler geometry, due to the corresponding version of the Poincaré lemma, this differential graded algebra has a decomposition into holomorphic and anti-holomorphic parts. In collaboration with Pierre Deligne, Phillip Griffiths, and Sullivan, Morgan used this decomposition to apply Sullivan's theory to study the topology of compact Kähler manifolds. Their primary result is that the real homotopy type of such a space is determined by its cohomology ring. Morgan later extended this analysis to the setting of smooth complex algebraic varieties, using Deligne's formulation of mixed Hodge structures to extend the Kähler decomposition of smooth differential forms and of the exterior derivative. [7]

In 2002 and 2003, Grigori Perelman posted three papers to the arXiv which purported to use Richard Hamilton's theory of Ricci flow solve the geometrization conjecture in three-dimensional topology, of which the renowned Poincaré conjecture is a special case. [8] Perelman's first two papers claimed to prove the geometrization conjecture; the third paper gives an argument which would obviate the technical work in the second half of the second paper in order to give a shortcut to prove the Poincaré conjecture.

Starting in 2003, and culminating in a 2008 publication, Bruce Kleiner and John Lott posted detailed annotations of Perelman's first two papers to their websites, covering his work on the proof of the geometrization conjecture. [9] In 2006, Huai-Dong Cao and Xi-Ping Zhu published an exposition of Hamilton and Perelman's works, also covering Perelman's first two articles. [10] In 2007, Morgan and Gang Tian published a book on Perelman's first paper, the first half of his second paper, and his third paper. As such, they covered the proof of the Poincaré conjecture. In 2014, they published a book covering the remaining details for the geometrization conjecture. In 2006, Morgan gave a plenary lecture at the International Congress of Mathematicians in Madrid, saying that Perelman's work had "now been thoroughly checked. He has proved the Poincaré conjecture." [11]

Selected publications

Articles.

Survey articles.

Books.

Related Research Articles

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References

  1. 1 2 3 4 5 6 "Biographical Sketch: John Morgan" (PDF). Chinese University of Hong Kong . Retrieved January 27, 2021.
  2. 1 2 John Morgan at the Mathematics Genealogy Project
  3. 1 2 3 4 "John Morgan". Simons Center for Geometry and Physics at Stony Brook University . Retrieved January 27, 2021.
  4. 1 2 3 4 "The Founding Director". Simons Center for Geometry and Physics at Stony Brook University . Retrieved January 27, 2021.
  5. List of Fellows of the American Mathematical Society, retrieved 2013-02-10.
  6. Dennis Sullivan. Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 269–331
  7. Pierre Deligne. Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. No. 40 (1971), 5–57.
  8. Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications. arXiv : math/0211159
    Grisha Perelman. Ricci flow with surgery on three-manifolds. arXiv : math/0303109
    Grisha Perelman. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv : math/0307245
  9. Bruce Kleiner and John Lott. Notes on Perelman's papers. Geom. Topol. 12 (2008), no. 5, 2587–2855.
  10. Huai-Dong Cao and Xi-Ping Zhu. A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10 (2006), no. 2, 165–492.
  11. John Morgan. The Poincaré Conjecture (special lecture). Minute 43:40.
  12. Chen, Kuo-Tsai (1983). "Review: Rational homotopy theory and differential forms, by P. A. Griffiths and J. W. Morgan". Bull. Amer. Math. Soc. (N.S.). 8 (3): 496–498. doi: 10.1090/s0273-0979-1983-15135-2 .