# Simon K. Donaldson

Wolf Prize Laureate in Mathematics 2020

**Sir Simon Kirwan Donaldson**

#### Affiliation at the time of the award:

**Imperial College London, UK**

**Simons Center, Stony Brook, UK**

#### Award citation:

**“for their contributions to differential geometry and topology”**

#### Prize Share:

**Simon Donaldson **

**Yakov Eliashberg**

Sir Simon Kirwan Donaldson (born 1957, Cambridge, U.K.) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson–Thomas theory.

Donaldson’s passion of youth was sailing. Through this, he became interested in the design of boats, and in turn in mathematics. Donaldson gained a BA degree in mathematics from Pembroke College, Cambridge in 1979, and in 1980 began postgraduate work at Worcester College, Oxford.

As a graduate student, Donaldson made a spectacular discovery on the nature or 4-dimensional geometry and topology which is considered one of the great events of 20th century mathematics. He showed there are phenomena in 4-dlmenslons which have no counterpart in any other dimension. This was totally unexpected, running against the perceived wisdom of the time.

Not only did Donaldson make this discovery but he also produced new tools with which to study it, involving deep new ideas in global nonlinear analysis, topology, and algebraic geometry.

After gaining his DPhil degree from Oxford University in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford, he spent the academic year 1983–84 at the Institute for Advanced Study in Princeton, and returned to Oxford as Wallis Professor of Mathematics in 1985. After spending one year visiting Stanford University, he moved to Imperial College London in 1998. Donaldson is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University and a Professor in Pure Mathematics at Imperial College London.

Donaldson’s work is remarkable in its reversal of the usual direction of ideas from mathematics being applied to solve problems in physics.

A trademark of Donaldson’s work is to use geometric ideas in infinite dimensions, and deep non-linear analysis, to give new ways to solve partial differential equations (PDE). In this way he used the Yang-Mills equations, which has its origin in quantum field theory, to solve problems in pure mathematics (Kähler manifolds) and changed our understanding of symplectic manifolds. These are the phase spaces of classical mechanics, and he has shown that large parts of the powerful theory of algebraic geometry can be extended to them.

Applying physics to problems or pure mathematics was a stunning reversal of the usual interaction between the subjects and has helped develop a new unification of the subjects over the last 20 years, resulting in great progress in both. His use of moduli (or parameter) spaces of solutions of physical equations – and the interpretation of this technique as a form of quantum field theory – is now pervasive throughout many branches of modem mathematics and physics as a way to produce “Donaldson-type Invariants” of geometries of all types. In the last 5 years he has been making great progress with special geometries crucial to string theory in dimensions six (“Donaldson-Thomas theory”), seven and eight.

Professor Simon Donaldson is awarded the Wolf Prize for his leadership in geometry in the last 35 years. His work has been a unique combination of novel ideas in global non-linear analysis, topology, algebraic geometry, and theoretical physics, following his fundamental work on 4-manifolds and gauge theory. Especially remarkable is his recent work on symplectic and Kähler geometry.