Conformal Invariance and Restriction in Multiply Connected Domains and Riemann Surfaces

私家侦探的研究计划是研究共形不变措施首先在多连通域，然后在黎曼曲面。特别是：识别适当的模空间上的扩散，该扩散在满足限制性质的曲线上产生共形不变测度;测量其他扩散的限制缺陷，并计算或表征相交概率作为模的函数;将共形结构扰动下的测度及其变化与最高权重表示联系起来;研究多连通域上多米诺骨牌和高斯自由场的共形不变量。 自回避平面曲线出现在许多自然环境中。例如，如果一个非常浅的板充满了水和油，我们可以考虑分离两种液体的曲线（或曲线）。另一个例子是海岸线，地图上国家的边界，或从飞机上看到的河流。通常，这些曲线非常不规则。但是，究竟什么是"典型的“，以及如何量化这种不规则性？由于所有的海岸线都是不同的，寻找一条自避曲线作为海岸线的数学模型是没有意义的。另一方面，在所有自避曲线（“可能的”海岸线）的集合上寻找概率分布是有意义的，这样，如果我们根据这个分布从这些曲线的集合中抽样，那么统计数据与真实的海岸线的统计数据是一致的。私家侦探建议研究这些概率分布的情况下，几何形状是复杂的孔（认为卵石在浅板与水和油）或曲线位于曲面上，如一个球体（地球仪），一个甜甜圈，或椒盐卷饼。

The P.I.'s research programme is to study conformally invariant measures first in multiply connected domains and then on Riemann surfaces. In particular: Identify diffusions on an appropriate moduli space that give rise to conformally invariant measures on curves satisfying the restriction property; Measuring the restriction defect for other diffusions, and calculating or characterize intersection probabilities as functions of the moduli; Link the measure and its change under perturbations of the conformal structure to highest weight representations; Study conformally invariant quantities for domino tilings and the Gaussian free field in multiply connected domains. Self-avoiding planar curves arise in many natural contexts. For example, if a very shallow plate is filled with water and oil, we can consider the curve (or curves) separating the two liquids. Another example are coastlines, the borders of countries on maps, or rivers seen from an airplane. Typically, these curves are very irregular. But what exactly is ``typical,'' and how can the irregularity be quantified? Since all coastlines are different, it does not make sense to look for one self-avoiding curve as the mathematical model of a coastline. On the other hand, it does make sense to look for a probability distribution on the set of all self-avoiding curves ( the ``possible'' coastlines) so that if we sample from the set of these curves according to this distribution, then the statistics are in agreement with statistics of real coastlines. The P.I. proposes to study these probability distributions in the case when the geometry is complicated by holes (think of pebbles in the shallow plate with water and oil) or the curves are situated on a curved surface such as a sphere (the globe), a donut, or a pretzel.