A new approach to conformal invariants in complex function theory

复变函数理论中共形不变量的新方法

• 批准号: RGPIN-2015-03681
• 负责人: Schippers, Eric
• 金额: $ 0.8万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=664466
• 关键词: new; approach; conformal; invariants; complex

This proposal involves three fields of mathematics: complex analysis, Teichmuller theory, and conformal field theory.  Here is an explanation of these fields, together with an explanation of how my research fits in the larger context.  This is aimed at a layperson.***In brief, this proposal contains a new approach to conformal invariance (explained below), which unifies ideas in the three fields. ******Complex analysis is the study of calculus in the setting of complex numbers.  Like all branches of mathematics, it has many unexpected consequences for many fields.  Complex analysis is particularly ubiquitous: it is an indispensable tool in engineering, physics, astronomy, and geology among other fields, for example because of its use in the theory of Fourier series and approximations.  Research in pure mathematics in general can and usually does proceed without concern for applications of this kind; however, the research spins off unexpected applications every few decades.  Much of complex analysis research concerns the study of families of complex analytic mappings.  Teichmuller theory is the study families of Riemann surfaces, which are two-dimensional surfaces along with a definition of angle.  Riemann surfaces are central to the geometric understanding of complex analysis.  ***My own research is on conformal invariants. The idea of an invariant is a powerful tool in mathematics (and physics); it is a quantity which is unchanged under some transformation, and is usually associated with geometric information. Conformal invariants are quantities invariant under complex analytic transformations.  This proposal regards an approach to conformal invariance which unifies ideas in complex analysis, Teichmuller theory, and conformal field theory.  ******In physics, conformal field theory is the study of quantum and mechanical systems which possess conformal invariance.  The mathematical study of conformal field theory began in the late 80s and early 90s, and continues to generate exciting new ideas in pure mathematics, especially in algebra and geometry.  I study certain families of Riemann surfaces which arise in conformal field theory.  David Radnell and I proved that these families equivalent to known families in a branch of complex analysis and geometry called Teichmuller theory.  This allowed us to solve some outstanding analytic issues in the mathematical formulation of conformal field theory, and also led to many new results in geometric function theory and Teichmuller theory.  This proposal investigates further connections in light of the new approach to conformal invariance.  ******The work in this proposal will advance both complex analysis and the mathematical formulation of conformal field theory.  Furthermore it creates new unexpected connections between these fields.  **

这一建议涉及到三个数学领域：复分析、Teichmuller理论和共形场理论。以下是对这些领域的解释，以及对我的研究如何适应更大背景的解释。这是针对外行的。简而言之，这个建议包含了一种新的共形不变性方法（如下所述），它统一了三个领域的思想。******复分析是在复数环境下的微积分研究。像数学的所有分支一样，它对许多领域产生了许多意想不到的后果。复分析尤其无处不在：它是工程、物理、天文学和地质学等领域不可或缺的工具，例如，因为它在傅立叶级数和近似理论中的应用。一般来说，纯数学的研究可以而且经常不考虑这类应用而进行；然而，这项研究每隔几十年就会产生意想不到的应用。许多复分析研究涉及复分析映射族的研究。Teichmuller理论是研究黎曼曲面族的理论，黎曼曲面族是具有角度定义的二维曲面。黎曼曲面是复杂分析的几何理解的核心。我自己的研究是关于共形不变量的。不变量的概念在数学（和物理）中是一个强大的工具；它是在某种变换下不变的量，通常与几何信息有关。保形不变量是在复解析变换下不变的量。本文提出了一种将复分析、Teichmuller理论和共形场论相结合的共形不变性方法。******在物理学中，共形场论是对具有共形不变性的量子和力学系统的研究。共形场论的数学研究始于80年代末和90年代初，并继续在纯数学中产生令人兴奋的新思想，特别是在代数和几何方面。我研究了在共形场论中出现的某些黎曼曲面族。David Radnell和我证明了这些族与复分析和几何的一个分支——Teichmuller理论中的已知族是等价的。这使我们在共形场论的数学表述中解决了一些突出的解析问题，并在几何函数理论和Teichmuller理论中得到了许多新的结果。这一建议在新的共形不变性方法的基础上研究了进一步的联系。******本提案中的工作将推进共形场理论的复杂分析和数学公式。此外，它在这些领域之间创造了意想不到的新联系。**