Measures on function spaces, statistical mechanics and the rigorous renormalization group

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。泛函积分是由费曼引入的，作为理解基本粒子物理量子场论的一种概念工具。它们提供了从经典力学和场论到量子力学的桥梁。这些积分在物理学中已经无处不在，并为我们目前对一系列学科的理解提供了基础：量子电动力学和色动力学，基本粒子物理的标准模型，超导理论，相变的统计力学，湍流，金融衍生品等等。从数学的角度来看，它们提出的主要问题是在函数或分布的空间中构造和研究无限维概率测度。过去几十年在这一领域取得了令人印象深刻的成果和成功，而这些成功的关键是重整组。后者可以用非技术术语描述如下。考虑到一个非常大的选民群体，他们必须在两个政党之间做出选择。人们可以根据越来越大的地理区域（例如县、州、民族等）的层次结构对它们进行分组。人们还可以为每个这样的地区指派一名“超级选举人”，他的选票由下一个更详细的代表级别的多数票决定。简而言之，重整化组是从一个表示级别上的投票几率配置到下一个更粗的表示级别的转换。这种转换的迭代允许人们从微观行为（个人层面上的概率的数学模型）开始，对这个复杂的相互作用系统的宏观行为（比如国家层面上的多数投票的概率）做出预测。以上与投票的类比仅仅是为了说明。在物理学中，人们会考虑原子水平上的自旋或磁矩，并尝试使用重整化群机制来推断一块材料（如铁）的总体磁化强度，而不是选民。重整化群在物理学中的重要性并不仅仅在于上面给出的定性描述，而在于基于这种直觉而发展起来的定量近似方案。提出的活动的主要目标是开发数学工具，允许严格控制该近似方案中的误差项。PI将特别关注两个不动点之间的完全重整化群轨迹的研究，即需要在从无限小到无限大的整个尺度范围内控制系统的情况。PI将在各种模型上研究这些轨迹，例如在三维中带有修改传播子的PI - 4，以及在二维中使用Gross-Neveu模型。拟议活动的一个重要组成部分是研究生教育。PI将促进研究生参与拟议的研究工作，并为他们提供建设性量子场论和重整化群论所需的数学工具的培训，以解决该领域的研究问题。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Functional integrals were introduced by R. P. Feynman as a conceptual tool for the understanding of the quantum field theory of elementary particle physics. They provide a bridge from classical mechanics and field theory to their quantum counterparts.These integrals have become ubiquitous in physics and provide a foundation for our current understanding of a wide array of subjects: quantum electro and chromodynamics, the standard model of elementary particle physics, the theory of superconductivity, the statistical mechanics of phase transitions, turbulence, financial derivatives, and much more. From a mathematical standpoint the main question they pose is that of the construction and study of infinite dimensional probability measures in spaces of functions or distributions.The last decades have seen impressive results and successes in this area, and key to these successes is the renormalization group. The later can be described in nontechnical terms as follows. Consider a very large population of voters which have to opt between say two political parties. One can group them according to a hierarchy of larger and larger geographical regions, for instance county, state, nation, etc. One can also assign a `super elector' for each such region whose vote is determined by the majority at the next more detailed level of representation. In a nutshell the renormalization group is the transformation from the configurations of voting odds at one representation level to the next coarser one.The iteration of this transformation allows one to make predictions about the macroscopic behavior of this complex interacting system (the odds on say the majority vote at the national level) starting from the microscopic behavior (a mathematical model for the odds at the level of individuals). The above analogy with voting is given simply for the sake of exposition. In physics, one would, instead of voters, consider for instance spins or magnetic moments at the atomic level, and try to use the renormalization group machinery in order to infer the overall magnetization of a piece of material such as iron.The importance of the renormalization group in physics lies not in the mere qualitative description given above, but rather in the quantitative approximation scheme which has been developed on the basis of this intuition. The main goal of the proposed activity is to develop mathematical tools which allow a rigorous control of the error terms in this approximation scheme. The PI will in particular focus on the study of complete renormalization group trajectories between two fixed points, i.e., situations where one needs to control a system over the full range of scales from the infinitely small to the infinitely large. The PI will study such trajectories on a variety of models such as phi-four with a modified propagator in three dimensions, and the Gross-Neveu model in two dimensions.An important component of the proposed activity is graduate education. The PI will foster the involvementof graduate students in the proposed research work and will provide them with training in the mathematical tools from constructive quantum field theory and renormalization group theory needed in order to tackle research problems in the area.