Research Projects
These are a few of the research projects I am offering for PhB, Summer, Honours and PhD students
- Finite dimensional representations of quantum affine algebras
Quantum affine algebras are deformations of affine Lie algebras (central extensions of loop algebras). Understanding their representation theory is of utmost importance in the study of, e.g., integrable models of statistical mechanics. As yet, however, even the finite dimensional irreducible representations are poorly understood except in the simplest cases. The aim of this project is to formulate a conjecture on the general structure of finite dimensional irreducible representations of quantum affine algebras consistent with what is known already.
- Generalized geometry, T-duality and mirror symmetry
T-duality, in its simplest form, is the R to 1/R symmetry of String Theory compactified on a circle of radius R. It can be generalized to manifolds which admit circle actions (e.g. circle bundles) or, more generally, torus actions. In the case of nontrivial torus bundles, and in the background of H-flux, T-duality relates manifolds of different topology and can even map to noncommutative geometries. The purpose of this project is to study these phenomena (as well as the closely related mirror symmetry) in the context of Hitchin’s generalized geometry.
- Statistics of quasiparticles
Collective excitations in quantum many body systems (as, e.g., in the fractional quantum Hall effect) often exhibit statistics different from the usual boson or fermion statistics. A particular form of these more general exclusion statistics has been introduced by Haldane. The aim of this project is to investigate aspects of these more general exclusion statistics using techniques from algebraic geometry.
- Topics in String Theory
String theory is, at present, the only candidate for a consistent unification of all four fundamental forces. Many important advances in String Theory have been made in the last couple of years. These include: String dualities, a microscopic derivation of the Bekenstein-Hawking black-hole entropy, a concrete manifestation of the holographic principle, the emergence of noncommutative geometry as a candidate for the quantum geometry of spacetime and the classification of D-brane changes in terms of K-theory. This project aims to investigate one of more of these advances in more detail.
