5 December 2024 | 16:00

Venue: Room G20, Royal School of Mines, Prince Consort Road, Imperial College London, SW7 2BP

Organiser: Steven Sivek (Imperial College)

Forty Years of Four Manifolds

Tomasz Mrowka, Professor of Mathematics, Massachusetts Institute of Technology (MIT), USA

Abstract: Since the twin breakthroughs in 1982-83 by Freedman and Donaldson the study of four manifolds has been developing rapidly.  Freedman’s resolution of the 4d topological Poincaré conjecture and refinements combined with Donaldson’s surprising applications of the Yang-Mills equations to show that the situation for smooth structures in dimension 4 was more complicated than in higher dimensions. Since then  new theories including Seiberg-Witten equations, Ozsvath and Szabó’s Heegaard Floer theory, and Embedded Contact Homology and have led to surprising applications to questions in 3 and 4 dimensional topology.  This talk will survey some of these developments.

About the Speaker: Tomasz Mrowka’s research interests focus on problems in differential geometry and gauge theory. His work combines analysis, geometry, and topology, specializing in the use of partial differential equations, such as the Yang-Mills equations from particle physics to analyze low-dimensional mathematical objects. Jointly with Robert Gompf, he discovered four-dimensional models of space-time topology. 

A graduate of MIT, Mrowka received the Ph.D. from U.C. Berkeley in 1988 under the direction of Clifford Taubes and Robin Kirby. He joined the MIT mathematics faculty as professor in 1996, following faculty appointments at Stanford and at Caltech (professor 1994-96). He chaired the Graduate Student Committee 1999-02, and chaired the Pure Mathematics Committee, 2004-15. From 2014-2017 he served as Department Head. A prior Sloan fellow and Young Presidential Investigator, Mrowka was selected for a Clay Mathematics Visiting Professorship in 1995.

In 2007 he received the Oswald Veblen Prize in Geometry by the AMS, jointly with Peter Kronheimer, “for their joint contributions to both three- and four- dimensional topology through the development of deep analytical techniques and applications.” Their book, Monopoles and Three Manifolds (Cambridge University Press) also garnered the 2011 Joseph Doob Prize of the AMS. He was appointed Singer Professor of Mathematics from 2007 to 2017. In 2017, Mrowka received a Simons Fellowship in Mathematics. In 2018 delivered a plenary address at ICM18 in Rio de Janeiro. He is a Fellow of the American Academy of Arts & Sciences (2007) and Member of the National Academy of Sciences (2015). Most recently, he was awarded the 2023 Leroy P. Steele Prize for Seminal Contribution to Research for his joint paper with Peter Kronheimer, ‘Gauge theory for embedded surfaces, I’ published in 1993 in Topology. 

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