Geometry of Measures

测量几何

• 批准号: 0856687
• 负责人: Tatiana Toro
• 金额: $ 60.77万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0856687&HistoricalAwards=false
• 关键词: Geometry; Measures

AbstractToroThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The theme of this proposal is the strong relationship that exists between the questions of how the geometry of a domain can be recovered from the regularity of its harmonic measure, and free boundary regularity problems. Remarkably the analogies become more apparent when examined under a Geometric Measure Theory (GMT) magnifying glass. The core of this proposal addresses three questions. The first one aims to understand domains in higher dimensional Euclidean spaces in terms of their harmonic measure as it has been done in 2 dimensions with great success. The underlying thesis is that in higher dimensions GMT plays the role that complex analysis does in 2 dimensions. The second question is that of the existence and regularity of minimizers for variational problems stated in terms of H\"older continuous metrics rather than smooth metrics. This problem includes the understanding of the structure of the corresponding free boundary. A by-product of this, is a question concerning the regularity of quasi-minimizers of the functional studied by Alt and Caffarelli. The third question goes back to a long term interest of the PI concerning the existence of good parameterization for subsets of Euclidean space. A remarkable feature is that this last project, which is purely in geometry, was motivated by an attempt to answer a question in potential theory. The cross-pollenization between harmonic analysis and GMT has been clearly beneficial to both areas.The theory of calculus of variations has been the main theoretical tool used in the study of variational problems often concerning energy minimization. Energy minimization methods are used to understand the equilibrium configuration of molecules. The basic idea is that a stable state of a molecular system should correspond to a local minimum of their potential energy. The proposed research provides new outlets for GMT, a field of Mathematics that has contributed greatly to the development of the calculus of variations and geometric analysis. The transformative aspect of this grant is the invigoration of this fundamental area of Mathematics. In the last few years, the number of students going into GMT in the US, has greatly diminished while it has increased in Europe. An important feature of the proposed work is that, while some results have already been obtained, there is great potential for expansion. In particular, we expect the active participation of graduate students and junior mathematicians. The field, which has been one of the pillars upon of which some areas of geometric analysis have been built, offers the theoretical framework to study a wide array of variational problems coming from different venues of science.

AbstractTorothis奖是根据2009年的《美国复苏与重新投资法》（公法111-5）资助的。该提案的主题是如何从其谐调措施的规律性中恢复领域几何形状的问题之间存在牢固的关系，以及自由边界规律性问题。值得注意的是，当根据几何测量理论（GMT）放大玻璃进行检查时，类比变得更加明显。该提案的核心解决了三个问题。第一个旨在从较高维度的欧几里得空间中了解域中的谐波度量，就像在2个维度上完成的谐波度量。基本的论点是，在较高的维度下，GMT在两个维度中的复杂分析起着作用。第二个问题是根据“较旧的连续指标而不是平稳的度量”所述的各种问题的存在和规律性的存在。这个问题包括对相应的自由边界结构的理解。该问题的副产品是一个问题，这是一个问题，是Quasi-Mintimize firctiment firptiment firptiment firptiment firptiment firptiment firptiment firpe ter Alt和caffarelli的问题。欧几里得空间的良好参数化的存在。了解分子的平衡构型。基本思想是，分子系统的稳定状态应对应于其势能的局部最小值。拟议的研究为GMT提供了新的渠道，GMT是一个数学领域，这对变异和几何分析的计算产生了很大的贡献。这笔赠款的变革性方面是对数学基本领域的启动。在过去的几年中，进入美国GMT的学生人数在欧洲有所增加时大大减少。拟议的工作的一个重要特征是，尽管已经获得了一些结果，但扩展潜力很大。特别是，我们期望研究生和初级数学家积极参与。该领域已经建立了一些几何分析领域之一，它提供了理论框架，可以研究来自不同科学场所的各种变异问题。