Geometry of moduli spaces and of integrable systems

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Geometry has played an ever increasing role in mathematics and theoretical physics over the past few years: encoding into a geometric object a solution to a partial differential equation, or a physical field, highlights its symmetries, shows which operations are natural and which are not, and generally gives one an idea of what is going on. One can think here of the role of a connection and its curvature in encoding electromagnetic fields and Maxwell's equations, or of the role of strings and their world sheet surfaces in understanding particle physics. A more applied set of problems with extensive geometric ramifications have been the various shallow water wave equations and their solutions, linked to infinite dimensional Hamiltonian systems. My research, and so this proposal centres on two interrelated classes of objects that often arise in these contexts. The first class is that of moduli spaces, the spaces which classify or describe the sets of all objects of a given type: the moduli space of all curves (of a given genus), the moduli space of all bundles, the space of all solutions to a given differential equation, and so on. Questions studied include their construction or description of these spaces, the study of their topology, how they behave as one varies natural parameters, and of course the relations between these spaces. Specific projects include instanton moduli on ALF manifolds, compactification of moduli, spectral asymptotics, topological stability, geometry of local systems. Techniques used are essentially algebraic geometry and differential geometry, with a bit of the theory of partial differential equations. The second segment of my proposal concerns integrable systems. The original definition of these systems was as mechanical systems with sufficiently many symmetries to ensure that they could be more or less explicitly solved; this was then extended to infinite dimensions, allowing the study of "solitons", solutions to shallow water wave equations which behave like solitary waves. From there, the notion has become even more flexible, and encompasses amongst many other things, flows of a geometric origin, and the theory of the functions which arise in this context; and more generally, extraction of solutions from algebras of symmetries. Specific projects include the geometry of tau-functions, tau functions and enumerative invariants, determinant bundles and isomonodromy, deformations to toric varieties, and the geometry of discrete lattice systems. Again, the range of technique is mainly geometrical. The impact for Canada is basically in ensuring a Canadian presence in what has become a central domain of research internationally, and of course training a new generation of mathematical scientists in the area.