John Morgan (mathematician)

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. 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. 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." The level of detail in Morgan and Tian's work was criticized in 2015 by mathematician Abbas Bahri, who found a counterexample to one of their claims corresponding to Perelman's third paper. The error, originating in the incorrect calculation of a geometric evolution equation, was thereafter fixed by Morgan and Tian.

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. 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. Many mathematicians found Perelman's work to be hard to follow due to a lack of detail on a number of technical points.

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

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. 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 simply-connected 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.

Morgan received his B.A. in 1968 and Ph.D. in 1969, both from Rice University. His Ph.D. thesis, entitled Stable tangential homotopy equivalences, was written under the supervision of Morton L. Curtis. He was an instructor at Princeton University from 1969 to 1972, and an assistant professor at MIT from 1972 to 1974. 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. 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.

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.