Conformally invariant random systems

近年来，人们在理解二维统计物理方面取得了重要进展。Schramm-Loewner演化(SLE)和相关对象的引入极大地促进了对由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中的共形场理论(CFT)预测的结果，大多数情况下不需要匹配技术。P.I.S的研究计划主要是为了从基于SLE的建筑中实现CFT概念。这些方法和问题涉及几何、泛函分析和表示理论方面的相互作用。这项建议的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的情况下发生的急剧的质变，如水的冻结(随着温度的降低，从液态转变为固态)。在相变时，可能会出现一种随机的宏观几何形状，类似于分离油和水等不混合流体的摆动界面。对这些起伏界面的研究既源于概率论，又源于理论物理。将特别关注这两种方法之间的相互作用，以及分析中涉及的其他数学领域。

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.