Moduli Spaces and Integrable Systems

点击翻译按钮获取中文摘要

My proposed research centres on the application of techniques of algebraic geometry, symplectic geometry, and differential geometry to the study of the intertwined topics of moduli spaces and integrable systems. These are subjects of both intrinsic and extrinsic interest.******Moduli spaces of various sorts are central not only in geometry, but occur in physics (typically as critical points of an action; as such they have been the focal point for the interaction of geometry and physics), and in areas as wide afield as number theory and combinatorics. One recurrent theme is a strong intertwining of algebraic geometry and symplectic geometry, with the two providing complementary insight into the structure of the spaces.******The geometry of moduli spaces is tied intimately to the geometry of integrable systems, and even more tightly in recent years with the development of cohomological invariants for moduli, typically bundled together as generating functions, in which guise they appear as tau-functions. These functions, more properly sections of a determinant bundle on an infinite dimensional Grassmannian, have been the mainstay of the theory of integrable systems in the last twenty years.******Specific problems:***A) Moduli ***1) Monopoles with Dirac type singularities, as considered Witten and Kapustin in their study of the geometric Langlands theory; they mediate Hecke transforms. ***2) Linking various gauge theoretical moduli spaces via Nahm transforms: for specific manifolds (R^n, ALE or ALF manifolds), the Nahm transform allows a description of moduli.***3) G-bundles and compactification. This is linked to understanding of how these objects behave in a limit; one wants a description that admits a good deformation theory.***4) Real moduli. I have recently looked at some examples of moduli space of real (i.e. complex, but conjugation invariant) geometric objects; one area of interest is character varieties.***B) Integrable systems.***1) Singular connections and isomonodromy. Following on a recent success in describing deformations of irregular singularities, I would now like to work on their Poisson geometry.***2) Poisson geometry of networks. Poisson spaces can be associated to various graphs, and there are interesting links to cluster algebras.***3) The general theory of tau functions: These functions, associated to determinant bundles over Grassmannians, admit several interesting generalisations which should clarify their nature and their role.***4) Tau functions and counting problems: Since the proof by Kontsevich of the Witten conjecture, one abiding mystery is the role of the tau function (and so integrable systems) in various enumerative problems.***5) Tau functions and the Eynard Orantin invariants: The understanding of the previous problem seems to go through a better understanding of these quite remarkable invariants.**