University of Miami
Department of Mathematics
Professor of Mathematics, MIT
Member of the National Academy of Sciences
Fellow of the American Academy of Arts & Sciences
2007 Veblen Prize in Geometry Recipient
will present
Instantons, Knots and Knotted Graphs
Thursday, February 27, 2020, 5:30pm
Newman Alumni Center, 6200 San Amaro Drive
Reception to follow the lecture
All interested persons are welcome to attend.
Abstract: Over the past four decades, input from geometry and analysis has been central to progress in the field of low-dimensional topology. This talk will focus on one aspect of these developments, namely, the use of Yang-Mills theory, or gauge theory. These techniques were pioneered by Simon Donaldson in his work on 4-manifolds beginning in 1982. The past ten years have seen new applications of gauge theory and new interactions with more recent threads in the subject, particularly in 3-dimensional topology and knot theory.
In our exploration of this subject, a recurring question will be, "How can we detect knottedness?" Many mathematical techniques have found application to this question, but gauge theory in particular has provided its own collection of answers, both directly and through its connection with other tools. Beyond classical knots, we will also take a look at the nearby but less-explored world of spatial graphs.
Tomasz Mrowka is Professor of Mathematics at MIT. He received his Ph.D. from U.C. Berkeley in 1988 under the direction of Clifford Taubes and Robion Kirby. He held faculty appointments at Stanford and Caltech before joining MIT mathematics faculty as professor in 1996. He served as Department Head from 2014 to 2017.
Mrowka's research interests are focused on the topological implications of the differential equations of high energy physics, in particular, the Yang-Mills and Seiberg-Witten equations. With Peter Kronheimer, he gave deep insights into the structure of Donaldson's invariants. Together they developed many tools for the study of three and four dimensional manifolds and knots. They used these tools to resolve a series of long-standing conjectures due to among others Bing, Milnor, and Thom. More recently, they showed that a combinatorially defined knot invariant, Khovanov Homology, was able to distinguish the unknot.
Mrowka is the winner of the 2007 Veblen Prize and 2011 Doob Prize, both jointly with Peter Kronheimer. He was a Sloan Foundation Fellow and a Radcliffe Fellow. In 2017, Mrowka received a Simons Fellowship in Mathematics. In 2018, he gave a plenary address at ICM18 in Rio de Janeiro. He is a Fellow of the American Academy of Arts & Sciences, class of 2007, and Member of the National Academy of Sciences, class of 2015.
The McKnight-Zame Distinguished Lecture Series is made possible by a generous donation from Dr. Jeffry Fuqua (Ph.D., UM, 1972). These annual lectures are named in honor of Professor James McKnight, who directed Dr. Fuqua's Ph.D. thesis, and Professor Alan Zame, who was a close mentor of Dr. Fuqua.
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