CAREER: Random Surfaces and Conformal Probability

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，如平面布朗运动、高斯自由场、Schramm-Loewner演化和共形环系综。除了它们内在的美，这些物体在量子场论、统计物理和随机表面理论中也有应用。该提议者研究的指导原则是，随机几何中的许多问题最好是根据随机曲面和高度函数（在离散设置中）和随机分布(如高斯自由场（在连续设置中）来理解的。二维随机几何之所以重要，部分原因是统计物理中的许多问题（如晶体表面波动的方式）本质上是二维的。自从Belavin， Polyakov和Zamolodchikov在20世纪70年代和80年代的开创性工作以来，人们已经理解-至少是启发性地-这些系统的某些宏观可观测值在从一个平面域到另一个平面域的保角映射下应该是不变的。在过去的二十年里，物理学家开发了复杂的非严格技术来理解具有保形对称性的随机物体的性质。在过去的几年里，一些数学家已经开始严格地证明一些来自物理文献的预测，以及许多额外的结果。这些问题中的许多都有自然的高维类比，我们才刚刚开始理解。研究者提议的活动包括努力将新的研究生引入这一新兴领域，并帮助他们学习和就业。

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.