I have been working in the development of a random matrix formulation of Chern-Simons theory on certain manifolds. Chern-Simons theory is a purely topological three dimensional quantum field theory with a gauge symmetry. It had quite an impact in topology as it provided a physical way to obtain novel 3d topological invariants. Nowadays, people in physics is interested in Chern-Simons theory due to its role in topological strings and also in condensed matter physics applications, such as the fractional quantum Hall effect.

 

The corresponding models of random matrices are also interesting by themselves and I have paid quite a lot of attention to the study of their mathematical properties. They are characterized by very weakly confining potentials and hence, exhibit different behavior from the usual Wigner-Dyson paradigm of random matrix theory. Concepts from the classical moment problem, fractal geometry and q-deformations appear in a natural way. Another development (related to the previous one) includes a description of several aspects of topological (or quasi-topological) gauge theories, such as Chern-Simons or 2d Yang-Mills theory, in terms of non-intersecting Brownian motion.

 

Many of the structures that appear in the previous lines of research are useful in other topics. For example, we are exploring the relationship with long-standing one dimensional exactly solvable models (like Calogero-Sutherland models) and with the fractional quantum Hall effect and Bose-Einstein condensation in cylindrical traps.

I am also working on problems in theoretical and computational neuroscience, studying neuronal models of  short-term memory and also models of spike-timing dependent synaptic plasticity.