A new approach to conformal invariants in complex function theory

这个建议涉及三个数学领域：复分析，Teichmuller理论和共形场论。 这里是对这些领域的解释，以及我的研究如何适应更大背景的解释。 这是针对外行人的。 简而言之，这个建议包含了一个新的方法，共形不变性（解释如下），统一的想法在三个领域。 复分析是在复数的设置微积分的研究。 像数学的所有分支一样，它对许多领域有许多意想不到的后果。 复分析是特别普遍的：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。 一般来说，纯数学的研究可以而且通常确实不关心这类应用;然而，研究每隔几十年就会产生意想不到的应用。 复分析的许多研究涉及复解析映射族的研究。 Teichmuller理论是黎曼曲面的研究族，黎曼曲面是具有角度定义的沿着二维曲面。 黎曼曲面是理解复分析几何的核心。 我自己的研究是共形不变量。不变量的概念是数学（和物理）中的一个强大工具;它是一个在某些变换下不变的量，通常与几何信息相关联。共形不变量是在复解析变换下不变的量。 这一建议考虑的方法，共形不变性，统一的想法，在复杂的分析，Teichmuller理论，共形领域的理论。 在物理学中，共形场论是研究具有共形不变性的量子和力学系统。 共形场论的数学研究开始于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中。 我研究某些家庭的黎曼曲面出现在共形场论。 大卫拉德内尔和我证明，这些家庭相当于已知的家庭在分支复杂的分析和几何称为泰希穆勒理论。 这使我们能够解决一些突出的分析问题的数学公式的共形场论，也导致了许多新的结果，几何函数理论和Teichmuller理论。 这个建议调查进一步的连接，根据新的方法共形不变性。 这项工作将推进复分析和共形场论的数学表述。 此外，它在这些字段之间创建了新的意想不到的连接。

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.