April 23, 24 and 25, 2024: Join us for the Myhill Lecture Series: Forty Years of Four Manifolds, featuring Tomasz Mrowka (MIT). Since the twin breakthroughs in 1982-83 by Freedman and Donaldson the study of four manifolds has been developing rapidly. Freedman’s work showed that the homeomorphism problem for 4-dimensional manifolds was largely under control provided the fundamental group of the 4-manifold was not too complicated. Donaldson’s surprising applications of the Yang-Mills equations hinted that the situation for smooth structures was more complicated and that the tools for studying 4-manifolds would come from diverse parts of mathematics. These three lectures will survey a number of the developments that occurred in the ensuing 40 years.

Many problems have been resolved during this time. An few of highlights include; existence of exotic differentiable structures on ℝ4, the failure of the h-cobordism theorem in dimension four, the Thom conjecture on minimal genus of surfaces in the complex projective plane, the Weinstein conjecture on existence of closed Reeb orbits in dimension three, the disproof of the triangulation conjecture, .... 

Lecture 1 will focus on setting up the problems and basic questions in 4-manifold topology.

Lecture 2 will discuss the many tools that have been developed to aid in this study: the Yang-Mills and Seiberg-Witten equations, Ozsváth and Szabó’s Heegaard Floer theory, Embedded Contact Homology and their applications to question in 3- and 4- dimensional topology.

Lecture 3 will try to sketch where the theory is headed, including the study the diffeomorphism groups of four dimensional manifolds. I hope to make the lectures independent but some things may flow from one lecture to another.

SPEAKER BIO: 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 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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