Moduli Spaces and Integrable Systems

我提议的研究中心是应用代数几何、辛几何和微分几何技术来研究模空间和可积系统的相互交织的主题。这些都是内在和外在利益的主题。 各种模空间不仅是几何学的中心，而且出现在物理学中（通常作为作用的临界点；因此它们一直是几何学和物理学相互作用的焦点），以及数论和组合学等广泛领域。一个反复出现的主题是代数几何和辛几何的紧密交织，两者提供了对空间结构的互补洞察。 模空间的几何形状与可积系统的几何形状密切相关，近年来，随着模上同调不变量的发展，模空间的几何关系更加紧密，通常作为生成函数捆绑在一起，以 tau 函数的形式出现。这些函数，更准确地说是无限维格拉斯曼函数上的行列式丛的部分，在过去二十年中一直是可积系统理论的支柱。 具体问题： A) 模数 1) 具有狄拉克型奇点的单极子，正如 Witten 和 Kapustin 在几何朗兰兹理论研究中所考虑的那样；它们介导赫克变换。 2) 通过 Nahm 变换链接各种规范理论模量空间：对于特定流形（R^n、ALE 或 ALF 流形），Nahm 变换允许描述模量。 3）G-丛和紧化。这与理解这些对象在极限条件下的行为有关；人们想要一种承认良好变形理论的描述。 4) 实模。我最近研究了真实（即复杂但共轭不变）几何对象的模空间的一些例子；人们感兴趣的领域之一是性状品种。 B) 可集成系统。 1）奇异连接和等单性。继最近成功描述不规则奇点的变形之后，我现在想研究他们的泊松几何。 2）网络的泊松几何。泊松空间可以与各种图相关联，并且有与聚类代数的有趣链接。 3）tau函数的一般理论：这些函数与格拉斯曼函数的行列式束相关，承认一些有趣的概括，这些概括应该澄清它们的性质和作用。 4）tau 函数和计数问题：自从 Kontsevich 证明了 Witten 猜想以来，tau 函数（以及可积系统）在各种计数问题中的作用一直是个谜。 5）Tau函数和Eynard Orantin不变量：对上一个问题的理解似乎是通过对这些相当显着的不变量的更好理解而实现的。

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.