Geometry of moduli spaces and of integrable systems

在过去的几年里，几何在数学和理论物理中扮演着越来越重要的角色：将偏微分方程或物理场的解编码到几何对象中，突出其对称性，显示哪些操作是自然的，哪些不是，并通常让人对正在发生的事情有一个了解。在这里，人们可以想到连接及其曲率在编码电磁场和麦克斯韦方程中的作用，或者弦及其世界表面在理解粒子物理中的作用。具有广泛几何分支的一组更实用的问题是与无限维哈密顿系统相联系的各种浅水波方程及其解。我的研究，所以这项提议集中在两类相互关联的物体上，它们经常出现在这些上下文中。第一类是模空间，即分类或描述给定类型的所有对象的集合的空间：(给定亏格的)所有曲线的模空间，所有丛的模空间，给定微分方程的所有解的空间，等等。所研究的问题包括这些空间的结构或描述，它们的拓扑结构的研究，它们如何表现为一个不同的自然参数，当然还有这些空间之间的关系。具体项目包括ALF流形上的瞬子模、模紧化、谱渐近、拓扑稳定性、局部系统的几何。使用的技术基本上是代数几何和微分几何，并带有一点偏微分方程组的理论。我的建议的第二部分涉及可积系统。这些系统最初的定义是具有足够多的对称性的机械系统，以确保它们或多或少可以得到明确的解；然后，这被扩展到无限维，允许研究“孤子”，即表现为孤立波行为的浅水波方程的解。从那时起，这个概念变得更加灵活，其中包括几何起源的流动和在此背景下产生的函数理论；以及更一般地，从对称代数中提取解。具体的项目包括tau函数、tau函数和计数不变量的几何，行列式丛和同步性，环面簇的变形，以及离散格子系统的几何。同样，技术的范围主要是几何学的。对加拿大的影响基本上是确保加拿大在已成为国际研究中心领域的领域存在，当然还包括在该领域培养新一代数学科学家。

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.