I have been mainly working on Matrix Model descriptions in gauge theories like Chern-Simons theory and 2D Yang-Mills theory (and its q-deformation), and also in Donaldson-Thomas theory. Chern-Simons theory is a purely topological three dimensional quantum field theory with a gauge symmetry. It has been relevant in topology, as it provided a physical way to obtain novel 3d topological invariants. In physics, Chern-Simons theory appears in the study of 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 per se and I have great curiosity in their mathematical properties. The models 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 and unusual connections with integrable models, six-vertex models and spin chains.

 

There are also interesting connections between the above mentioned results and classical results in condensed matter physics, involving the fractional quantum Hall effect and Luttinger liquids.