A new approach to conformal invariants in complex function theory

这一建议涉及三个数学领域：复分析、泰希穆勒理论和保形场理论。这里对这些领域进行了解释，并解释了我的研究如何适应更大的背景。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(下文解释)，它统一了这三个领域的思想。复数分析是在复数的背景下研究微积分。与数学的所有分支一样，它对许多领域都有许多意想不到的结果。复数分析特别普遍：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使用。在一般情况下，纯数学的研究可以而且通常不考虑这类应用；然而，这项研究每隔几十年就会产生意想不到的应用。许多复杂分析研究都与复解析映射族的研究有关。TeichMuller理论是黎曼曲面的研究族，黎曼曲面是带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中心。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；它是一个在某种变换下不变的量，通常与几何信息有关。共形不变量是复解析变换下的量不变量。本文提出了一种将复分析、TeichMuller理论和共形场论中的思想统一起来的共形不变性的方法。在物理学中，共形场理论是对具有共形不变性的量子和力学系统的研究。共形场理论的数学研究始于80年代末和90年代初，并在纯数学中不断产生令人兴奋的新想法。特别是在代数和几何方面。我研究了共形场理论中出现的某些Riemann曲面族。David Radnell和我证明了这些族等价于复分析和几何的一个分支中的已知族，称为TeichMuller理论。这使我们能够解决共形场理论数学公式中的一些突出的分析问题，并在几何函数论和TeichMuller理论中产生了许多新的结果。该建议从共形不变性的新方法出发，研究了进一步的联系。*这项工作将推进复杂的分析和保形场理论的数学表述。此外，它还在这些场之间建立了新的意想不到的联系。**

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.  **