Branes are multidimensional quantum objects in string theory that can be thought of as higher-dimensional versions of the fundamental strings. They are often used to define non-trivial string theory backgrounds whose low-energy spectrum contains gauge fields. This construction of string theory (called Geometric Engineering) allows the study of non-perturbative aspects of quantum gauge theories that are beyond the reach of traditional methods like Feynman-diagrams expansions. My goal is to develop a mathematical framework for the computation of physical observables appearing in this context. This framework, called Algebraic Engineering, is based on advanced algebraic techniques involving the representation theory of quantum groups.

Integrable systems are physical systems that possess hidden symmetries allowing the exact calculation of certain observables. For quantum systems, these symmetries are encoded in a mathematical structure called a quantum group. A number of correspondences have been discovered between quantum integrable systems and quantum field theory/string theory, such as the Bethe/gauge correspondence. The study of such correspondences is a very important aspect of modern theoretical physics because it makes it possible to reformulate difficult problems on one side into (sometimes) simpler problems on the other side.

Quantum groups entering into the description of most quantum integrable systems are based on affine Lie algebras (or Kac-Moody algebras). However, the integrable structures observed in brane constructions are based on toroidal (or twice affine) Lie algebras. Unfortunately, these toroidal quantum groups have been much less studied, and it is part of the mathematical aspects of my work to develop their representation theory. It is really fascinating to understand the connections between these algebras and other branches of mathematics. For instance, they have deep relations with W-algebras which describe the symmetries of 2d Conformal Field Theories.

When electrons are confined to a thin layer and placed in a strong magnetic field, they can form remarkable collective states known as the quantum Hall effect. In some fractional quantum Hall states, the excitations behave like anyons, particles with unusual rules when they move around one another. For non-Abelian anyons, the final state depends on the order of these movements, a property that could be used for topological quantum computing. My goal is to develop mathematical models for the motion of the electrons and relate the braiding of their excitations to symmetries encoded by quantum groups.