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

这个建议涉及复分析，黎曼曲面和Teichmuller理论，共形场论。 复分析是研究复数的微积分。它是数学、工程和物理等领域不可或缺的工具。黎曼曲面是复分析的主要对象，它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，这些映射在非常精细的尺度上保持角度。黎曼曲面在考虑某些类型的微分方程时自然出现，并在密码学和理论物理学中有应用。Teichmuller理论是黎曼曲面变形的系统研究，以及黎曼曲面作为一个整体的几何集合。 共形场论是研究在小尺度变换和旋转下不变的物理系统。它可以应用于统计力学和量子场论。共形场论的数学研究既涉及建立严格的物理模型的问题，也涉及探索该理论的物理思想的丰富数学后果。 我的研究涉及嵌套曲面，其中内部曲面的边缘是非常粗糙的曲线，称为准圆。这些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。许多分形都是准圆的例子。研究的长期目标是理解和联系这些嵌套曲面的几何、代数和分析。曲面本身具有几何性质，整个无限维曲面集合也是如此。代数结构来自一种称为缝合的过程，在这种过程中，表面沿着它们的边缘沿着;这种结构出现在物理学和Teichmuller理论中。接缝一般是准圆。分析出现在空间的研究复杂的解析或调和映射和运营商对这些空间。所有这三个方面相互作用：几何学以不变量的形式表现出来，这些不变量是在由缝合产生的代数运算下不变的量; 2不变量可以用函数空间上的算子来解析地表示; 3代数运算可以用它们对函数空间和算子的作用来解析地表示。从技术上讲，这些目标包括共形不变量的指数定理和全局解析量的构建，如Teichmuller空间上的Kahler势，周期矩阵和行列式线丛。 所获得的结果将被研究人员用于Riemann曲面的全局分析和几何、Teichmüller理论、复分析中的边值问题和共形场论。 从长远来看，这些领域之间建立基本联系将刺激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.