Mathematics that unifies

The unity of mathematics is a discovery rather than an assumption. Subjects that developed independently have often proved to share the same structures and phenomena, and when that happens each gains a vocabulary it lacked; what resisted proof in one can become almost easy in the other.

The deepest correspondences of this kind are still being worked out. The Langlands programme relates number theory to analysis and geometry; monstrous moonshine tied the largest sporadic simple group to modular functions; one short list of symmetries classifies objects in fields that otherwise have nothing in common. Physics supplies more, since a duality between two quantum field theories is a mathematical identity waiting to be proved. We pursue these correspondences and look for others.

Some of our own work runs in the other direction, taking classical questions and attacking them with imported tools. We study how the roots of equations permute, a question as old as Galois, using the topology of braids and the geometry of polytopes. We study symmetry in infinite dimensions, where the familiar theory breaks down, and find representation theory reaching into number theory. We work on the complex geometries that string theory needs and mathematics has yet to classify, on the number theory that appears unbidden inside quantum field theory, and on the boundary between the discrete and the continuous, where combinatorial graphs acquire the geometry of smooth spaces. We study not only the interplay between different branches of mathematics — algebra, geometry, and number theory — but also how the same structures appear in other fields. That is why we are always looking not only to strengthen but to broaden and complement our thematic coverage.

There is another reason to range widely. Machines are changing what a mathematician is for. As the technical work becomes cheaper, breadth and judgement become scarce. We use machines to find patterns in mathematical data, and then do the harder part: turning a pattern into a theorem.

Judgement of that kind is learned by proximity. At the London Institute, mathematics is not an adjunct to the science but part of it. Our mathematicians sit among physicists, which keeps both abreast of the other's tools and occasionally reveals that they were working on the same problem.