Conformal Invariance and Restriction in Multiply Connected Domains and Riemann Surfaces

侦探他的研究计划是先研究多连通域中的共形不变测度，然后再研究黎曼曲面上的共形不变测度。特别地：在满足限制性质的曲线上，识别适当模空间上的扩散，产生保形不变测度；测量其他扩散的限制缺陷，并计算或表征作为模的函数的相交概率；将测量值及其在共形结构摄动下的变化与最高权值表示联系起来；研究多连通域上多米诺骨牌平铺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，如果一个很浅的盘子里装满了水和油，我们可以考虑将这两种液体分开的曲线。另一个例子是海岸线，地图上的国家边界，或者从飞机上看到的河流。通常，这些曲线是非常不规则的。但究竟什么是“典型”，这种不规律性又如何量化呢？因为所有的海岸线都是不同的，所以寻找一条自我回避的曲线作为海岸线的数学模型是没有意义的。另一方面，在所有自我回避曲线的集合（“可能的”海岸线）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，那么统计数据与实际海岸线的统计数据是一致的。P.I.建议在以下情况下研究这些概率分布：几何形状因孔洞而变得复杂（想象浅板上的鹅卵石带有水和油），或者曲线位于曲面上，如球体（球体）、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.