Moduli Spaces and Integrable Systems

我建议的研究中心的应用技术的代数几何，辛几何，微分几何的研究交织的主题的模空间和可积系统。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强烈交织，两者为空间的结构提供了互补的见解。模空间的几何与可积系统的几何紧密相连，近年来随着模的上同调不变量的发展，模空间的几何与可积系统的几何联系更加紧密，模空间的上同调不变量通常作为生成函数捆绑在一起，伪装成τ函数。这些函数，更确切地说是无限维格拉斯曼行列式丛的部分，在过去的二十年里一直是可积系统理论的支柱。具体问题：*A）模 *1）具有狄拉克型奇点的单极，如维滕和卡普斯廷在几何朗兰兹理论的研究中所考虑的;他们调解赫克变换。*2）通过Nahm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），Nahm变换允许模的描述。3)G-丛与紧化这与理解这些物体在极限下的行为有关;人们希望得到一个允许良好变形理论的描述。4)真实的模量。我最近研究了真实的（即复杂的，但共轭不变的）几何对象的模空间的一些例子;一个感兴趣的领域是特征多样性。B）可积系统。* 1)奇异联络与等单径。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Poisson几何泊松空间可以与各种图相关联，并且与簇代数有有趣的联系。3)τ函数的一般理论：这些函数，与格拉斯曼上的行列式丛相关联，允许几个有趣的概括，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证明了维滕猜想以来，一个永恒的谜团就是Tau函数（以及可积系统）在各种计数问题中的作用。5)Tau函数和Eynard Orantin不变量：对前一个问题的理解似乎要通过更好地理解这些相当引人注目的不变量来进行。

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.**