Stony Brook University and Imperial College London
2015 Breakthrough Prize in Mathematics
For the new revolutionary invariants of four-dimensional manifolds and for the study of the relation between stability in algebraic geometry and in global differential geometry, both for bundles and for Fano varieties.
We experience the world in three dimensions: up and down, left and right, forward and back. But according to Einstein, there are actually four: his theory of general relativity integrates time with the three spatial dimensions. And while we can’t visualize four dimensions, we can analyze them with mathematics. Simon Donaldson has transformed our understanding of four-dimensional shapes, showing which ones can be “tamed” with the kind of equations that mathematicians can solve, and which can’t. In the process, he both provided powerful new tools for physicists and incorporated new ideas from physics into mathematics.
It has been my great good fortune that my career has spanned a period of exceptionally exciting developments in my field. Ideas and techniques from different areas — topology, physics, differential equations, geometry — have become interwoven in ways that no one would have predicted half a century ago. It is a privilege to have been able to witness this and take some part in it. Mathematics has a long time scale. One of the pleasantest things is, looking back in time, to contemplate how the developments we see in our lifetimes fit into the longer term. And, looking forward in time, we have confidence that the problems that seem to us intractable will yield to future advances, invisible to us now. I owe enormous thanks to my advisors, Michael Atiyah and Nigel Hitchin, and to all the mathematicians who made Oxford in the 1980s such a special place. What I learned then has underpinned my whole career. Looking forward, rather than back, it has been my great good fortune to have been able to watch the development of many extraordinary research students. I would like to thank my wife, Nora, my parents, and all my family for their support.