Moduli Spaces and Integrable Systems

我建议的研究集中在代数几何、辛几何和微分几何技术在模空间和可积系统的相互交织的主题研究中的应用。这些主题既有内在的，也有外在的。******各种各样的模空间不仅在几何中是中心，而且出现在物理学中（通常作为一个动作的关键点；因此它们已经成为几何和物理相互作用的焦点），并且在数论和组合学等广泛的领域。一个反复出现的主题是代数几何和辛几何的强烈交织，两者提供了对空间结构的互补见解。******模空间的几何与可积系统的几何密切相关，近年来随着模的上同调不变量的发展，这种关系甚至更加紧密，模的上同调不变量通常捆绑在一起作为生成函数，在这种情况下它们表现为tau函数。这些函数，更确切地说，是无限维格拉斯曼曲线上的行列式束的部分，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***1)具有Dirac型奇点的单极子，如Witten和Kapustin在研究几何朗兰兹理论时所考虑的；它们调解了赫克变换。***2)通过纳姆变换连接各种规范理论模空间：对于特定的流形（R^n， ALE或ALF流形），纳姆变换允许模的描述。***3) g束与紧化。这与理解这些物体在极限下的行为有关；人们需要一种描述，这种描述允许一个好的变形理论。***4)实模。我最近研究了一些实数（即复，但共轭不变）几何对象的模空间的例子；我们感兴趣的一个领域是角色的多样性。***B)可积系统。***1)奇异连接和同构。在最近成功地描述了不规则奇点的变形之后，我现在想研究它们的泊松几何。***2)网络的泊松几何。泊松空间可以与各种图相关联，并且与聚类代数有有趣的联系。***3) tau函数的一般理论：这些函数，与格拉斯曼函数上的行列式束相关，承认一些有趣的推广，这些推广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Witten猜想以来，一个永恒的谜团是Tau函数（以及可积系统）在各种枚举问题中的作用。***5) Tau函数和Eynard Orantin不变量：对前一个问题的理解似乎是通过对这些非常了不起的不变量的更好理解

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.**