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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612503
• 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.

该建议涉及数学的三个领域：复杂分析，Teichmuller理论和保形场理论。这是对这些领域的解释，以及我的研究如何在更大的背景下适应的解释。这是针对外行的。 简而言之，该提案包含一种新的保形不变性方法（如下所述），该方法在三个领域中统一了想法。 复杂分析是在复数设置中对演算的研究。像数学的所有分支一样，它对许多领域都产生了许多意想不到的后果。复杂的分析特别普遍：它是工程，物理，天文学和地质和其他领域的必不可少的工具，例如，由于它在傅立叶序列和近似理论中的使用。一般来说，纯数学的研究通常可以并且通常不关心这种应用。但是，该研究每隔几十年就可以消除意外应用。大部分复杂分析研究涉及对复杂分析映射家族的研究。 Teichmuller理论是Riemann表面的研究家族，它们是二维表面以及角度的定义。 Riemann表面对于复杂分析的几何理解至关重要。 我自己的研究是关于保形不变的。不变的想法是数学（和物理）的强大工具。它的数量在某些转换下不变，通常与几何信息有关。在复杂的分析转换下，保形不变是不变的数量。该提案涉及一种结构不变性的方法，该方法在复杂分析，Teichmuller理论和保形场理论中统一了思想。 在物理学中，共形场理论是具有保形不变性的量子和机械系统的研究。共形场理论的数学研究始于80年代末和90年代初，并继续在纯数学中产生令人兴奋的新思想，尤其是在代数和几何学中。我研究了某些在共形场理论中出现的黎曼表面的家庭。戴维·拉德内尔（David Radnell）和我证明了这些家庭在复杂分析和几何学的分支中等同于已知家庭，称为Teichmuller理论。这使我们能够在保形场理论的数学公式中解决一些出色的分析问题，并导致了几何函数理论和Teichmuller理论的许多新结果。根据新的保形不变性方法，该提案研究了进一步的联系。 该提案中的工作将提高复杂分析和保形场理论的数学公式。此外，它在这些字段之间创建了新的意外连接。