Conformal Invariance and Restriction in Multiply Connected Domains and Riemann Surfaces

多重连通域和黎曼曲面中的共形不变性和限制

• 批准号: 0604216
• 负责人: Robert Bauer
• 金额: $ 11.89万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604216&HistoricalAwards=false
• 关键词: Conformal; Invariance; Restriction; Multiply; Connected

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.

P.I.S的研究计划是首先在多连通区域上研究共形不变测度，然后在黎曼曲面上研究共形不变测度。特别是：在适当的模空间上识别出在满足约束性质的曲线上产生共形不变测度的扩散；测量其他扩散的约束缺陷，并计算或刻画作为模的函数的相交概率；将该测度及其在共形结构扰动下的变化与最高权表示联系起来；研究多米诺骨牌平铺和多重连通区域中的高斯自由场的共形不变量。自回避平面曲线在许多自然环境中都会出现。例如，如果一个很浅的盘子里装满了水和油，我们可以考虑分开这两种液体的曲线。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“典型”到底是什么？这种不规则性如何量化？由于所有的海岸线都是不同的，所以寻找一条自回避曲线作为海岸线的数学模型是没有意义的。另一方面，在所有自我回避曲线(“可能”的海岸线)的集合上寻找概率分布是有意义的，这样，如果我们根据该分布从这些曲线集合中进行抽样，则统计数据与真实海岸线的统计数据是一致的。P.I.建议在几何形状复杂的情况下研究这些概率分布(想想有水和油的浅盘中的鹅卵石)，或者曲线位于球体(球体)、甜甜圈或椒盐卷饼等曲面上。