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

黎曼曲面上的微分和积分算子以及缝纫几何和代数

• 批准号: RGPIN-2021-03351
• 负责人: Schippers, Eric
• 金额: $ 1.31万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758576
• 关键词: Differential; Integral; Operators; Riemann; Surfaces

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.

该提议涉及复数分析、黎曼曲面和泰希米勒理论，而复数分析是复数微积分的研究，是数学、工程和物理学等领域不可或缺的工具。复杂分析的主要对象是具有足够结构来定义角度的二维形状，以及在考虑某些类型的微分方程时自然出现的在非常精细的尺度上保留角度的它们之间的映射，以及泰希米勒理论在密码学和理论物理学中都有应用，它是对黎曼曲面变形以及黎曼曲面集合的几何形状的系统研究。 -尺度重新缩放和旋转。它适用于统计力学和量子场论。共形场论的数学研究涉及建立严格的物理模型的问题，以及探索丰富的数学结果。我的研究涉及嵌套曲面，其中内曲面的边缘是非常粗糙的曲线，称为拟圆，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。许多分形都是拟圆的例子，研究的长期目标是理解和关联这些嵌套曲面的几何性质、代数和分析，曲面本身也具有几何性质。整个无限维曲面的集合来自称为缝合的过程，其中曲面沿其边缘连接；这种结构出现在物理学和泰希米勒理论中。对复杂解析或调和映射的空间以及这些空间上的运算符的研究所有三个方面都相互作用：几何表现在不变量中，这些不变量在缝合产生的代数运算下保持不变；就函数空间上的算子而言；并且代数运算可以根据它们在函数空间和算子上的作用来分析地表达，更技术性地说，目标包括共形不变量的索引定理和全局解析量的构造。 Teichmüller 空间、周期矩阵和行列式线束上的卡勒势。​ 获得的结果将被研究人员用于黎曼曲面的全局分析和几何、Teichmüller 理论、复杂的边值问题。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。