Simon Donaldson

Simon Donaldson, 2009

Sir Simon Kirwan Donaldson (born August 20, 1957 in Cambridge ) is an English mathematician and Fields Medalist who is best known for his work on the topology of four-dimensional manifolds . He is a professor at Imperial College London .

Life

Donaldson studied mathematics at Cambridge University , where he received his BA in 1979 . He continued his studies with Nigel Hitchin and Michael Atiyah at Worcester College at the University of Oxford . While still a student, he published Self-dual connections and the topology of smooth 4-manifolds (Bulletin American Mathematical Society 1983) in 1982, which established his reputation. After Michael Freedman had topologically classified smooth (i.e. continuous) manifolds of dimension 4 and at the same time proved the Poincare conjecture for this dimension, Donaldson examined differentiable structures on these manifolds with the help of non-Abelian gauge theories ( Yang-Mills theories , correspondingly non-linear partial differential equations) and special solutions of these theories, the instantons , which are differentiated by topological invariants . He was able to derive the necessary conditions for the intersection form on such manifolds, which severely restricted the possible differentiable structures.

He also found - from gauge theories - new polynomial invariants for characterizing these differentiable structures and was able to show that, in contrast to all other dimensions, there are "exotic" differentiable structures on the , i.e. 4-manifolds that are topologically equivalent (" homeomorph ”) to the 4-dimensional Euclidean space , but not with regard to differentiable structures (not“ diffeomorphic ”). ${\displaystyle d=4}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle \mathbb {R} ^{4}}$

After completing his dissertation in Oxford in 1983, Donaldson became a Fellow at All Souls College and went to the Institute for Advanced Study in Princeton in 1983/84 . In 1985 he was awarded the Valais Professorship in Mathematics at Oxford University . In 1999 he moved to Imperial College London .

In the 1990s he dealt with the application of Floer homology and symplectic topology and geometry.

In 2012, Donaldson announced with Xiuxiong Chen and Song Sun a proof of the conjecture of Donaldson, Shing-Tung Yau and Gang Tian . The conjecture formulates a criterion for the existence of Kähler-Einstein metrics on compact Kähler manifolds with positive first Chern class (Fano manifolds). At about the same time, Gang Tian announced evidence, which led to a priority dispute (Donaldson and colleagues accused him of plagiarism).

His doctoral students include Dominic Joyce , Paul Seidel , Gábor Székelyhidi and Dieter Kotschick .

Awards

In 1986 he was elected to the Royal Society and received the Fields Medal at the ICM 1986 (Lecture: Geometry of 4-Manifolds ). In 1992 he was awarded the Royal Medal of the Royal Society. In 1994 he received the Crafoord Prize and in 2000 he was elected to the National Academy of Sciences . In 2008 he was awarded the Nemmers Prize for Mathematics . In 2009 he received the Shaw Prize for Mathematics together with Clifford Taubes . In 1992 he gave a plenary lecture at the first European Congress of Mathematicians in Paris ( Gauge theory and four manifold topology ). In 1998 he was invited speaker at the International Congress of Mathematicians in Berlin ( Lefschetz fibrations in symplectic geometry ) and also in 1983 in Warsaw ( Gauge theory and topology ). On February 16, 2012, he was beaten to a Knight Bachelor in the New Year Honors . He is a fellow of the American Mathematical Society . In 2014, together with four other mathematicians, he was awarded the Breakthrough Prize in Mathematics for new revolutionary invariants of four-dimensional manifolds and for the study of the relationship between stability in algebraic geometry and in global differential geometry for both fiber bundles and Fano varieties (laudatory speech). In 2018 he was plenary speaker at the ICM in Rio ( Some recent developments in Kähler geometry and exceptional holonomy ). In 2019 he received the Oswald Veblen Prize in Geometry and in 2020 the Wolf Prize in Mathematics.

Fonts

• An application of gauge theory to four-dimensional topology ., Journal of Differential Geometry Vol. 18, 1983, pp. 279-315.
• Self-dual connections and the topology of smooth 4-manifolds , Bulletin American Mathematical Society Vol. 8, 1983, pp. 81-83.
• with Peter Kronheimer : The geometry of four-manifolds . Oxford Mathematical Monographs, New York 1990, Oxford University Press, ISBN 0-19-853553-8 .
• Remarks on gauge theory, complex geometry and 4-manifold topology , in Atiyah, Iagolnitzer: Fields medaillist lectures , World Scientific 1997
• Floer homology in Yang-Mills theory , Cambridge Tracts 2002
• 100 years of manifold topology . In: Ioan James (editor) History of topology , North Holland 1999
• with Chen, Sun: Kähler-Einstein metrics on Fano manifolds I, II, III. J. Amer. Math. Soc. 28 (2015)

literature

• Tian, ​​Stern: Lautatio on the occasion of the Cranford Prize . In “Notices of the AMS”, July 1994
• Atiyah: Fields-Medal laudation . In “International Congress Mathematicians” (ICM) 1986, Berkeley

Web links

Commons : Simon Donaldson  - Collection of images, videos and audio files

• John J. O'Connor, Edmund F. Robertson :  Simon Donaldson. In: MacTutor History of Mathematics archive .
• Simon Donaldson in the Mathematics Genealogy Project (English)
• Homepage of Donaldson at Imperial College

Individual evidence

1. ^ Kahler-Einstein metrics and stability, Preprint 2012
2. ↑ Donaldson, Chen, Sun, Kähler-Einstein metrics on Fano manifolds, part 1, J. AMS, 28, 2015, 183–197, part 2, pp. 199–234, part 3, pp. 235–278
3. ↑ The existence for negative first Chern class was proved by Thierry Aubin and Yau, for vanishing Chern class by Yau. In the case of a positive Chern class there are counterexamples, but it was assumed that it existed if certain stability criteria were met (K stability)
4. ^ Tian, ​​K-stability and Kähler-Einstein metrics, Preprint 2012
5. ↑ Donaldson, Chen, Sun, On some recent developments in Kähler geometry, September 19, 2013, pdf
6. ↑ Knights and Dames: COM-DON at Leigh Rayment's Peerage
7. ↑ Breakthrough Prize 2014 ( Memento of the original from June 24, 2014 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
8. ↑ Oswald Veblen Prize 2019
9. ↑ Wolf Prize 2020