Differential and Integral Operators on Riemann Surfaces and the Geometry and Algebra of Sewing

这一建议涉及复变分析、黎曼曲面和泰希穆勒理论以及保形场理论。复数分析是对复数演算的研究。它是数学、工程和物理等领域不可或缺的工具。黎曼曲面是复数分析的主要对象，复数分析是具有足够结构来定义角度的二维形状，以及它们之间的映射，它们在非常精细的尺度上保持角度。黎曼曲面是在考虑某些类型的微分方程时自然产生的，在密码学和理论物理中都有应用。TeichMuller理论是系统地研究黎曼曲面的变形，以及黎曼曲面集合作为一个整体的几何。保形场理论是研究在小尺度重标度和旋转下不变的物理系统的理论。它在统计力学和量子场论中都有应用。共形场理论的数学研究既涉及建立严格的物理模型的问题，也涉及探索该理论物理思想的丰富数学后果的问题。我的研究涉及嵌套表面，其中内表面的边缘是非常粗糙的曲线，称为准圆。这在黎曼曲面理论中是不可避免的，并且在某些类型的随机过程中自然发生；例如，渗流和随机游动。许多分形图都是准圆的例子。研究的长期目标是了解和关联这些嵌套曲面的几何、代数和分析。曲面本身具有几何属性，整个无限维曲面集合也是如此。这种代数结构来自一种称为缝纫的过程，在这种过程中，表面沿着它们的边连接在一起；这种结构既出现在物理学中，也出现在泰希穆勒理论中。一般说来，接缝是准圆形的。这种分析源于对复解析映射或调和映射空间以及这些空间上的算子的研究。这三个方面都是相互作用的：几何以不变量的形式表现出来，不变量是在缝纫产生的代数运算下不变的量；不变量可以解析地写成函数空间上的算子；代数运算可以解析地表示为它们对函数空间和算子的作用。从技术上讲，目标包括共形不变量的指数定理和全局解析量的构造，例如Teichmuller空间上的Kahler势、周期矩阵和行列式线丛。所获得的成果将用于黎曼曲面的整体分析和几何、泰希米勒理论、复分析中的边值问题和保形场理论的研究人员。从长远来看，这些领域之间基本联系的建立将激发新的研究和意想不到的见解。

This proposal involves complex analysis, Riemann surfaces and Teichmuller theory, and conformal field theory.    Complex analysis is the study of the calculus of complex numbers. It is an indispensable tool in mathematics, engineering, and physics, among other fields. Riemann surfaces are the primary objects of complex analysis, which are two--dimensional shapes with enough structure to define angles, and maps between them which preserve angles on a very fine scale. Riemann surfaces arise naturally when considering certain kinds of differential equations, and have applications to cryptography and theoretical physics. Teichmuller theory is the systematic study of deformations of Riemann surfaces, as well as the geometry of the collection of Riemann surfaces as a whole.    Conformal field theory is the study of physical systems which are invariant under small--scale re-scalings and rotations. It has applications to statistical mechanics and quantum field theory. The mathematical study of conformal field theory involves both the problem of making a rigorous physical model, as well as exploring the rich mathematical consequences of the physical ideas of the theory.    My research involves nested surfaces, where the edges of the inner surfaces are very rough curves called quasicircles. These are inevitable in the theory of Riemann surfaces, and occur naturally in certain kinds of random processes; for example, percolation and random walks. Many fractals are examples of quasicircles. The long--term aim of the research is to understand and relate the geometry, algebra, and analysis of these nested surfaces. The surfaces themselves have geometric properties, as does the entire infinite-dimensional collection of surfaces. The algebraic structure comes from a procedure called sewing, in which surfaces are joined along their edges; this structure arises both in physics and Teichmuller theory. The seams are, in general, quasicircles. The analysis arises in the study of spaces of complex analytic or harmonic maps and operators on these spaces. All three aspects interact: the geometry manifests itself in invariants, which are quantities unchanged under algebraic operations arising from sewing; the invariants can be written analytically in terms of the operators on function spaces; and the algebraic operations can be expressed analytically in terms of their action on the function spaces and operators. More technically speaking, the goals include index theorems for conformal invariants and construction of global analytic quantities such as a Kahler potential on Teichmuller space, period matrices, and determinant line bundles.   The results obtained will be used by researchers in the global analysis and geometry of Riemann surfaces, Teichmüller theory, boundary value problems in complex analysis, and conformal field theory.  The establishment of fundamental connections between these fields will stimulate new research and unexpected insights in the long term.