CAREER: Random Surfaces and Conformal Probability

职业：随机曲面和共形概率

• 批准号: 0946296
• 负责人: Scott Sheffield
• 金额: $ 41.85万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-05-29 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0946296&HistoricalAwards=false
• 关键词: CAREER; Random; Surfaces; Conformal; Probability

The investigator will study spatial problems of probability theory, with a particular focus on random surfaces and two dimensional random objects with conformal symmetries, such as planar Brownian motion, the Gaussian free field, the Schramm-Loewner evolution, and the conformal loop ensembles. In addition to their intrinsic beauty, these objects find applications in quantum field theory, statistical physics, and the theory of random surfaces. A guiding principle of the proposer's research is that many problems in stochastic geometry are best understood in terms of random surfaces and height functions (in discrete settings) and random distributions such as the Gaussian free field (in continuum settings).Two dimensional random geometries are important in part because many problems in statistical physics (such as the way crystal surfaces fluctuate) are essentially two dimensional. Since the path-breaking work of Belavin, Polyakov, and Zamolodchikov in the 1970's and 1980's, it has been understood---at least heuristically---that the laws of certain macroscopic observables of these systems should be invariant under conformal maps from one planar domain to another. Over the past two decades, physicists have developed sophisticated non-rigorous techniques for understanding the properties of random objects with conformal symmetries. During the past few years, several mathematicians have begun to rigorously prove some of the predictions from the physics literature, along with many additional results. Many of the these problems have natural higher dimensional analogs that we are only beginning to understand. The investigator's proposed activities include efforts to bring new graduate students into this emerging field and to assist them in their studies and careers.

研究者将研究概率论的空间问题，特别关注随机表面和具有共形对称性的二维随机物体，例如平面布朗运动、高斯自由场、施拉姆-洛纳演化和共形环系综。 除了其内在美之外，这些物体还在量子场论、统计物理学和随机表面理论中找到了应用。 提议者研究的指导原则是，随机几何中的许多问题最好通过随机表面和高度函数（在离散设置中）和随机分布（例如高斯自由场（在连续设置中））来理解。二维随机几何是这很重要，部分原因是统计物理学中的许多问题（例如晶体表面波动的方式）本质上是二维的。 自从 Belavin、Polyakov 和 Zamolodchikov 在 1970 年代和 1980 年代的开创性工作以来，人们已经理解（至少是启发式的）这些系统的某些宏观可观测量的定律在共形映射下应该是不变的。平面域到另一个。 在过去的二十年中，物理学家开发了复杂的非严格技术来理解具有共形对称性的随机物体的属性。在过去的几年中，一些数学家已经开始严格证明物理文献中的一些预测以及许多其他结果。 其中许多问题都有自然的高维类似物，而我们才刚刚开始理解这些问题。 研究人员提议的活动包括努力将新研究生带入这一新兴领域并协助他们的学习和职业。