Geometry of moduli spaces and of integrable systems

在过去的几年里，几何学在数学和理论物理学中发挥了越来越重要的作用：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，我们可以在这里思考连接及其曲率在编码电磁波中的作用，场和麦克斯韦方程，或者弦和它们的世界片表面在理解粒子物理学中的作用。具有广泛几何分支的一组更实用的问题是各种浅水波方程及其解，与无限维哈密顿系统有关。 我的研究，因此这个建议集中在两个相互关联的对象类别，经常出现在这些情况下。 第一类是模空间，这种空间对给定类型的所有对象的集合进行分类或描述：所有曲线的模空间（给定亏格的），所有丛的模空间，给定微分方程的所有解的空间，等等。研究的问题包括它们的构造或这些空间的描述，它们的拓扑结构的研究，它们如何作为一个整体而变化，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，模的紧化，谱渐近，拓扑稳定性，局部系统的几何。所使用的技术基本上是代数几何和微分几何，与偏微分方程的理论位。 第二部分是关于可积系统的。这些系统的最初定义是作为机械系统与足够多的对称性，以确保他们可以或多或少明确解决;这是然后扩展到无限维，允许研究“孤子”，解决方案浅水波方程的行为像孤立波。从那里，概念已变得更加灵活，并包括在许多其他的事情，流动的几何起源，理论的职能，出现在这方面;更一般地说，提取解决方案的代数的对称性。 具体项目包括几何的tau函数，tau函数和枚举不变量，行列式束和isomonodromy，变形的环面品种，和几何的离散格系统。同样，技术的范围主要是几何的。 对加拿大的影响基本上是在确保加拿大的存在，在什么已成为一个中心领域的研究国际，当然培训了新一代的数学科学家在该地区。

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.