Fields medal work Witten was awarded the
Fields Medal by the
International Mathematical Union in 1990. In a written address to the
ICM,
Michael Atiyah said of Witten: In particular, Witten realized that a physical theory now called
Chern–Simons theory could provide a framework for understanding the mathematical theory of
knots and
3-manifolds. Although Witten's work was based on the mathematically ill-defined notion of a
Feynman path integral and therefore not
mathematically rigorous, mathematicians were able to systematically develop Witten's ideas, leading to the theory of
Reshetikhin–Turaev invariants. Another result for which Witten was awarded the Fields Medal was his proof in 1981 of the
positive energy theorem in
general relativity. This theorem asserts that (under appropriate assumptions) the total
energy of a gravitating system is always positive and can be zero only if the geometry of
spacetime is that of flat
Minkowski space. It establishes Minkowski space as a stable ground state of the
gravitational field. While the original proof of this result due to
Richard Schoen and
Shing-Tung Yau used
variational methods, Witten's proof used ideas from
supergravity theory to simplify the argument. A third area mentioned in Atiyah's address is Witten's work relating
supersymmetry and
Morse theory, a branch of mathematics that studies the
topology of
manifolds using the concept of a
differentiable function. Witten's work gave a physical proof of a classical result, the
Morse inequalities, by interpreting the theory in terms of
supersymmetric quantum mechanics. Speaking at
Strings '95 conference at
University of Southern California, Witten made the surprising suggestion that these five string theories were in fact not distinct theories, but different limits of a single theory, which he called
M-theory. Witten's proposal was based on the observation that the five string theories can be mapped to one another by certain rules called
dualities and are identified by these dualities. It led to a flurry of work now known as the
second superstring revolution. Maldacena's discovery has dominated high-energy theoretical physics for the past 15 years because of its applications to theoretical problems in quantum gravity and quantum field theory. Witten's foundational work following Maldacena's result has shed light on this relationship. In collaboration with
Nathan Seiberg, Witten established several powerful results in quantum field theories. In their paper on string theory and
noncommutative geometry, Seiberg and Witten studied certain
noncommutative quantum field theories that arise as limits of string theory. In another well-known paper, they studied aspects of
supersymmetric gauge theory. The latter paper, combined with Witten's earlier work on topological quantum field theory, With
Anton Kapustin, Witten has made deep mathematical connections between S-duality of gauge theories and the
geometric Langlands correspondence. Partly in collaboration with Seiberg, one of his recent interests includes aspects of field theoretical description of topological phases in condensed matter and non-supersymmetric dualities in field theories that, among other things, are of high relevance in condensed matter theory. In 2016, he has also brought tensor models to the relevance of holographic and quantum gravity theories, by using them as a generalization of the
Sachdev–Ye–Kitaev model. In particular, Witten is known for collaborating with
Ruth Britto on a method calculating scattering amplitudes known as the
BCFW recursion relations. ==Awards and honors==