CAREER: Random Surfaces and Conformal Probability

研究人员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对象，如平面布朗运动，高斯自由场，Schramm-Loewner演化和共形循环系综。 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。 提出者研究的指导原则是，随机几何中的许多问题最好用随机表面和高度函数（离散设置）以及随机分布（如高斯自由场（连续设置））来理解。二维随机几何很重要，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-至少在理论上-这些系统的某些宏观可观测量的定律在从一个平面域到另一个平面域的共形映射下应该是不变的。 在过去的二十年里，物理学家已经发展出复杂的非严格技术来理解具有共形对称性的随机物体的性质。在过去的几年里，几位数学家已经开始严格证明物理学文献中的一些预测，沿着有许多额外的结果。 这些问题中的许多都有我们刚刚开始理解的自然的更高维度的类似物。 调查员提议的活动包括努力使新的研究生进入这一新兴领域，并协助他们的学习和职业。

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.