Tackling big questions about the fundamental forces, symmetry and information, and the intimate interplay between physics and mathematics.

The elegant universe

General relativity and quantum field theory are among physics’ greatest triumphs. One describes gravity as the geometry of spacetime; the other describes matter and the remaining three fundamental forces. Both are confirmed to many decimal places, and each fails where the other cannot be ignored. Reconciling them is the deepest open problem in physics.

The Standard Model, our quantum theory of these particles and forces, leaves some of nature’s most basic patterns unexplained. Why do quarks and leptons come in three families, and why do their masses differ so widely? What is dark matter? What explains the pattern of particles and charges that cancels otherwise fatal gauge anomalies? We explore these questions through connections with [condensed-matter physics](/theme/life-learning-and-emergence/) and the search for a deeper framework that unites matter and gravity in a single quantum description.

Gravity brings its own puzzles, from the singularities predicted by general relativity to the dark energy associated with the universe's accelerating expansion. Black holes need gravity and quantum theory at once, and neither suffices. Their predicted evaporation appears to destroy what quantum evolution must preserve. We study what black-hole thermodynamics reveals about spacetime, and the wider relationship between information and matter—from the physical cost of erasing information to the role of quantum entanglement in the emergence of spacetime itself.

String theory and its proposed extension, M-theory, offer a possible route to a single quantum description of matter and gravity and dualities are central to this picture — two theories that look nothing alike can describe the same world. These surprising equivalences can connect strong interactions to weak ones; holography even relates a world with gravity to a lower-dimensional one without it. They raise a deeper question: could spacetime itself emerge from quantum physics? We explore these connections within exactly solvable models, where special symmetries make exact answers possible—offering rare footholds for testing ideas about nature’s underlying laws.

The search for exact solutions and hidden symmetries also opens new paths in pure mathematics. Wigner noted the unreasonable effectiveness of mathematics in physics. The reverse now holds too, and the hunt for exact solutions and hidden symmetries is driving [pure mathematics](/theme/mathematics-that-unifies/), to the point where ideas from quantum theory now furnish new proofs of classical theorems in geometry. We investigate that convergence and take it as a guide: our aim is a description of nature in which the mathematics is not merely adequate but inevitable.

Developing mathematical foundations for order in matter, organisation in living things, machine intelligence, and other emergent phenomena.

Life, learning and emergence

Many of nature's most striking phenomena are emergent: their behaviour belongs to a system's organisation rather than its substance. Statistical physics makes this precise, and its reach extends from electrons in a solid to the origin of living order and to the nature of learning.

Perhaps the most intriguing emergent phenomenon is life itself. Darwin explained how life evolves, but not how it began. We investigate the thermodynamic basis of self-replication and adaptation, and how flows of energy and matter keep organisms in non-equilibrium states. Which of these principles extend beyond biology, can they guide the creation of artificial life, and can evolution be made predictive?

Living systems also process information. The genome is less a blueprint than a program, run on gene regulatory networks. We study how these networks compute and what their architecture rules out, seeking a theory of cell programming precise enough to steer a cell to a chosen state.

Nature found its computers; we have to design ours. How do we make intelligent machines? We seek mathematical principles for how learning, memory and generalisation arise in biological and artificial networks, and for the part played by causal reasoning, modularity and representations of the environment. [AI-assisted discovery](/theme/ai-assisted-discovery/) treats machine intelligence as an instrument; here it is an object of theory.

Emergence is not confined to things that are alive or that learn. In condensed matter, interactions lead to emergence of new (quasi)particles that are responsible for the remarkable properties of materials. Among the best-known examples are superconductivity, underpinned by Cooper pairing of electrons, and the fractional quantum Hall effect, in which magnetic fields combine to produce excitations carrying a fraction of the electron's charge. But even spacetime — the stage on which these collective phenomena unfold — may itself be an emergent feature of quantum dynamics, rather than a fundamental ingredient of nature. We study phase transitions and the emergence of order in classical and quantum systems, at equilibrium and far from it. Quantum field theory is the common language here, as it is for [the fundamental forces](/theme/the-elegant-universe/), and ideas have long travelled both ways. 

We are drawn, too, to questions still taking shape — quantitative measures of selection and assembly, open-ended evolution, collective intelligence and the emergence of autonomy and agency. Our aim is a mathematics of emergence: laws that turn on organisation rather than substance, that hold for matter, organisms and machines alike, and that do not dissolve when a system is taken apart.

Understanding the relationships that bind different branches of mathematics and physics, and building the overarching theories they demand.

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](/theme/the-elegant-universe/) 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](/theme/ai-assisted-discovery/).

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.

Shaping the partnership between human and machine intelligence to extend the reach of theoretical physics and mathematics.

AI-assisted discovery

New technologies have always reshaped the way we do research and expanded the scope of what we can discover. Artificial intelligence is the latest example, although the transformation it has triggered has been unusually swift. At the London Institute, we do not merely adapt, we help set its direction. We treat AI as an epistemic instrument that amplifies human intention and enlarges the space of ideas we can explore.

Mathematics and theoretical physics are especially fertile ground for human–machine collaboration. Their objects—numbers, series, graphs and geometric shapes—form an immense, exact and highly organised body of Platonic data. Using machine-learning methods, we can search this landscape for regularities and exceptional structures. Among the most evocative examples are the “murmurations” of elliptic curves—intricate, previously unseen patterns in arithmetic data that point to a deep underlying mathematical order. In another line of work, we have developed a framework for the systematic discovery of new [quantum integrable models](/theme/the-elegant-universe/).

Large language models offer a different mode of collaboration. They can survey literature, accelerate calculations, write and debug code, and sustain exploratory dialogue across disciplines. They can help to turn half-formed intuitions into testable results, but mathematical taste—choosing fertile questions, recognising illuminating arguments and inventing the right concepts—remains decisive. The art of discovery increasingly unites machine power with human judgement.

We also turn mathematics back upon AI. Using mathematically controlled toy models, we investigate why learning systems favour simple solutions, how generalisation emerges, and what intelligence owes to data, optimisation and architecture. A science increasingly mediated by machines must also develop a science of those machines and understand their limits.

Our aim is not automation for its own sake, but a broader, faster and more deeply connected science—one in which human purposes continue to guide increasingly powerful forms of machine intelligence.

Our four main research themes are the elegant universe; life, learning and emergence; mathematics that unifies; and AI-assisted discovery.

Themes

All of our research is fundamental rather than applied, and theoretical rather than experimental. Our research activity falls into four main themes, interwoven in method and aim: the elegant universe; life, learning and emergence; mathematics that unifies; and AI-assisted discovery.