Over the past 30 years, geometry in 3 and 4 dimensions has been totally revolutionized by new ideas emerging from theoretical physics. Old problems have been solved but, more importantly, new vistas have been opened up which will keep mathematicians busy for decades to come.
While the initial spark has come from physics (where it was extensively pursued by Edward Witten), the detailed mathematical development has required the full armoury of non-linear analysis, where deep technical arguments have to be carefully guided by geometric insight and topological considerations.
The two main pioneers who both initiated and developed key aspects of this new field are Simon K Donaldson and Clifford H Taubes. Together with their students, they have established an active school of research which is both wide-ranging, original and deep. Most of the results, including some very recent ones, are due to them.
To set the scene, it is helpful to look back over the previous two centuries. The 19th century was dominated by the geometry of 2-dimensional surfaces, starting with the work of Abel on algebraic functions, and developing into the theory of complex Riemann surfaces. By the beginning of the 20th century, Poincare had introduced topological ideas which were to prove so fruitful, notably in the work of Hodge on higher dimensional algebraic geometry and also in the global analysis of dynamical systems.
In the latter half of the 20th century there was spectacular progress in understanding the topology of higher dimensional manifolds and fairly complete results were obtained in dimensions 5 or greater. The two “low dimensions” of 3 and 4, arguably the most important for the real physical world, presented serious difficulties but these were expected to be surmounted, along established lines, in the near future.
In the 1980’s this complacent view was shattered by the impact of new ideas coming from physics. The first breakthrough was made by Simon K Donaldson in his PhD thesis where he used the Yang-Mills equations of SU(2)-gauge theory to study 4-dimensional smooth (differentiable) manifolds. Specifically, Donaldson studied the moduli (or parameter) space of all SU(2)-instantons, solutions of the self-dual SU(2) Yang-Mills equations (which minimize the Yang-Mills functional), and used it as a tool to derive results about the 4-manifold. This instanton moduli space depends on a choice of Riemannian metric on the 4-manifold but Donaldson was able to produce results which were independent of the metric.
There are serious analytical difficulties in carrying out this programme and Donaldson had to rely on the earlier work of Karen Uhlenbeck and Clifford H Taubes. As these new ideas were developed and expanded by Donaldson, Taubes and others, spectacular results came tumbling out. Here is an abbreviated list, which shows the wide and unexpected gulf between topological 4-manifold (where the problems had just been solved by Michael Freedman) and smooth 4-manifold:
(1) Many compact topological 4-manifold which have no smooth structure.
(2) Many inequivalent smooth structures on compact 4-manifold.
(3) Uncountably many inequivalent smooth structures on Euclidean 4-space.
(4) New invariants of smooth structures.
The invariants in (4) were first introduced by Donaldson using his instanton moduli space. Subsequently, an alternative and somewhat simpler approach emerged, again from physics, in the form of Seiberg-Witten theory. Here, one just counted the finite number of solutions of the Seiberg-Witten equations (i.e., the moduli space was now zero dimensional).
One of Taubes’ great achievements was to relate Seiberg-Witten invariants to those introduced earlier by Gromov for symplectic manifolds. Such manifolds occur both as phase spaces in classical mechanics and in complex algebraic geometry, through the Kahler metrics inherited from projective space and exploited by Hodge. Although symplectic manifolds need not carry a complex structure, they always carry an almost (i.e., non-integrable) complex structure. Gromov introduced the idea of “pseudo-holomorphic curves” on symplectic manifolds and obtained invariants by suitably counting such curves. Taubes, in a series of long and difficult papers, proved that, for a symplectic 4-manifold, the Seiberg-Witten invariants essentially coincide with the Gromov-Witten invariants (an extension of the Gromov invariants). The key step in the work of Taubes is the construction of a pseudo-holomorphic curve from a solution of the Seiberg-Witten equations. This is fundamental since it connects gauge theory (a theory of potentials and fields) to sub-varieties (curves). Roughly, it represents a kind of non-linear duality.
In fact, extending complex algebraic geometry to symplectic manifolds (of any even dimension) was again pioneered by Donaldson who proved various existence theorems such as the existence of symplectic submanifolds. In the apparently large gap between algebraic geometry and theoretical physics, symplectic manifolds form a natural bridge and the recent results of Donaldson, Taubes and others provide, so to speak, a handrail across the bridge.
All this work in 4 dimensions has an impact on 3 dimensions, especially through the work of Andreas Floer, and Taubes has made many contributions in this direction. His most outstanding result is his very recent proof, in 3 dimensions, of a long-standing conjecture of Alan Weinstein. This asserts the existence of a closed orbit for a Reeb vector field on a contact 3-manifold. Contact 3-manifold arise naturally as level sets of Hamiltonian functions (energy) on a symplectic 4-manifold, and the Weinstein conjecture now asserts the existence of a closed orbit of the Hamiltonian vector field. This latest tour de force of Taubes exhibits his real power as a geometric analyst.
In recent years Donaldson has turned his attention to the hard problem of finding Hermitian metrics of constant scalar curvature on compact complex manifolds. The famous solution by Yau of the Calabi conjecture is an example of such problems. Donaldson has recast the constant scalar curvature problem in terms of moment maps, an idea derived from symplectic geometry which played a key role in gauge theory. This construction of metrics is a much deeper problem, being extremely non-linear but Donaldson has already made incisive progress on the analytical questions involved. This new work of Donaldson represents an exciting new advance which is currently attracting much attention.
This quick summary of the contributions of both Donaldson and Taubes shows how they have transformed our understanding of 3 and 4 dimensions. New ideas from physics, together with deep and delicate analysis in a topological framework, have been the hallmark of their work. They are fully deserving of the Shaw Prize in Mathematical Sciences for 2009.
Mathematical Sciences Selection Committee
The Shaw Prize
7 October 2009, Hong Kong