The model of normal matrices will be studied from a rigorous analytical point of view, beyond the realm of formal power series used in the existing literature, introducing suitable regularizations of the (ill-defined) integrals over matrices. This approach is supposed to give insight in the range of times where solutions of dispersionless hierarchies exist (beyond formal power series) and in the geometric meaning of the singularities of solutions.
Preliminary results indicate that most of the analytic methods of P.Deift and coworkers can be adapted to normal matrices, suggesting that, in a second phase of the project, more detailed statistical questions about the spacing of eigenvalues can be answered.
A related class of matrix integrals (in a certain sense the ``holomorphic square roots'' of normal matrix integrals) have recently made their appearance in the string literature, after the seminal paper of R.Dijkgraaf and C.Vafa,and are expected to be important objects in supersymmetric gauge theory and strings.
Here a formal saddle point calculation suggests that the eigenvalues are asymptotically distributed along certain curves. The determination of these curves and the investigation of their possible relation with integrable models is another part of this project. Again, a rigorous definition of the integrals over matrices is a prerequisite. In this case, the problem is to describe middle dimensional cycles in the space of complex matrices, whose asymptotic expansion is given by the saddle point approximation of Dijkgraaf and Vafa.
Symmetric, Hermitian and symplectic matrix integrals are represented by multiple integrals involving a Vandermonde determinant to the powers 1,2 and 4. The Hermitian matrix integrals are tau-functions for the Toda lattice, whereas Adler-van Moerbeke show the symmetric and symplectic matrix integrals are tau-functions for a new lattice, the Pfaff lattice and also they satisfy a subalgebra of Virasoro constraints.
It is a challenging open problem to understand the case of other powers of the Vandermonde (Coulomb interaction). One expects these integrals to satisfy certain new Fay identities, which might point the way to some new integrable lattices.
It has been shown that the Airy universality class for large random matrices is related to the KdV equation, via the Kontsevich integral (Adler, Shiota and van Moerbeke). Almost nothing of the kind is known about the other universality classes appearing in critical regimes Computing the spacings between eigenvalues near the edge is a challenging problem (Bleher, Kuijlaars, Vanlessen).
Topological strings on Calabi-Yau geometries provide a unifying picture connecting non-critical (super)strings, integrable hierarchies, and various matrix models (Dijkgraaf). In particular, ordinary matrix models, the double scaling limit of matrix models, and Kontsevich-like matrix models are all related and arise from studying branes in specific local Calabi-Yau three-folds. Moreover A-model topological strings on CP1 can be realized as B-model topological strings on Calabi-Yau. The Virasoro and W-constraints in these cases simply reflect the smoothness of Calabi-Yau manifold under complex deformation (Dijkgraaf).
Other matrix models have been introduced in physics and mathematics, and have attracted attention. The 2-hermitian-matrix model and its generalizations (chain of matrices, eigenvalues located on fixed homology classes of contours), play an important role in quantum gravity and (boundary) conformal field theory. Many important results have recently been obtained for the 2-matrix model, and it can be expected that the rigorous proof of asymptotics similar to that obtained in the 1-matrix case by (Bleher-Its, Deift et al.) will be achieved in the coming years, opening the route to many other discoveries. One goal is the computation of the isomonodromic tau functions associated to biorthogonal polynomials, `a la' Miwa-Jimbo (the definition of the Miwa-Jimbo tau function cannot apply in this case, which is too singular).
The 2-matrix model has interesting generalizations, like the chain of matrices, and its continuous limit. Another generalization is to replace the unitary group invariance by orthogonal or symplectic, and further extend the results obtained for the 1-matrix model to all symmetric-spaces ensembles of random matrices. Beyond, there are many other matrix models, like the O(n) model or the Potts-model, whose relationship with integrability needs to be addressed.