Conformally invariant random systems

近年来，在二维统计物理的理解方面取得了重要进展。Schramm-Loewner演化（SLE）及其相关对象的引入极大地促进了对微观随机输入和相互作用产生的宏观随机几何结构的严格研究。这使得能够建立理论物理学中的共形场论（CFT）预测的结果，大多数情况下没有匹配的技术。私家侦探的研究计划主要是针对实现CFT的概念，从SLE为基础的建设。这些方法和问题涉及到几何、泛函分析和表示论等方面的相互作用，其目的是分析相变物理现象的数学模型。相变描述了物理系统在外部参数变化下的急剧质变，例如水的冻结（随着温度降低从液相转变为固相）。在相变时，可能会出现随机的宏观几何形状，类似于分离非混合流体（如油和水）的摇摆界面。这些起伏界面的研究植根于概率论和理论物理学。特别重点将放在这两种方法之间的相互作用，和其他领域的数学参与分析。

Recent years have seen important progress in the understanding of twodimensional statistical physics. The rigorous study of macroscopicrandom geometric structures generated by microscopic random inputs andinteractions has been greatly stimulated by the introduction ofSchramm-Loewner Evolutions (SLE) and related objects. This enabled toestablish results predicted by Conformal Field Theory (CFT) in TheoreticalPhysics, most often without matching the techniques. The P.I.'s research programis primarily directed at realizing CFT concepts from SLE basedconstructions. The methods and issues involve an interplay ofgeometric, functional analytic and representation theoretic aspects.The goal of this proposal is to analyze mathematical models of thephysical phenomenon of phase transition. A phase transition describes a sharpqualitative change in a physical system under variation of an externalparamater, such as freezing of water (transition from liquid tosolid phase as temperature decreases). At the phase transition, a random macroscopic geometrymay emerge, akin to the wiggly interfaces separating non mixing fluidslike oil and water. The study of those fluctuating interfaces isrooted in both Probability Theory and Theoretical Physics. Special focuswill be placed on the interaction between these two approaches, andother areas of mathematics involved in the analysis.