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.

该建议涉及复杂的分析，黎曼表面和Teichmuller理论以及保形场理论。复杂的分析是对复数计算的研究。它是数学，工程和物理学以及其他领域中必不可少的工具。 Riemann表面是复杂分析的主要对象，它们是具有足够结构的二维形状，可以定义角度，并且它们之间的映射在非常细的尺度上保留角度。当考虑某些类型的微分方程时，Riemann会自然地浮出水面，并在密码和理论物理学上应用。 Teichmuller理论是对Riemann表面变形的系统研究，以及整个Riemann表面的几何形状。共形场理论是对物理系统的研究，在小规模的重新缩放和旋转下是不变的。它在统计机械和量子场理论中有应用。共形场理论的数学研究既涉及制作严格的物理模型的问题，也涉及探索理论物理思想的丰富数学后果。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。这些在Riemann表面理论中是不可避免的，并且自然发生在某些随机过程中。例如，渗透和随机步行。许多分形是准圆的示例。这项研究的长期目的是了解和关联这些嵌套表面的几何形状，代数和分析。表面本身具有几何特性，整个无限二维表面集合也是如此。代数结构来自一个称为缝纫的程序，在该过程中，其边缘沿其表面连接；这种结构在物理和Teichmuller理论中均出现。通常，接缝是准圆。分析是在对这些空间上复杂分析或谐波图和运算符的空间的研究中产生的。这三个方面相互作用：几何形状在不变的数量中表现出来，这些数量是在缝纫引起的代数操作下没有变化的。不变性可以根据功能空间的运算符分析书写；代数操作可以根据其对功能空间和运算符的作用进行分析表达。从技术上讲，这些目标包括用于保形不变的索引定理以及在Teichmuller空间上的Kahler潜力等全球分析量的构建，并确定线条捆绑包。研究人员将在Riemann表面的全球分析和几何形状，Teichmüller理论，复杂分析中的边界价值问题和保形场理论中使用。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外见解。