METRIC MEASURE SPACES, EINSTEIN METRICS, SPECTRAL GEOMETRY

公制测量空间、爱因斯坦度量、谱几何

• 批准号: 1005552
• 负责人: Jeff Cheeger
• 金额: $ 24万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005552&HistoricalAwards=false
• 关键词: METRIC; MEASURE; SPACES; EINSTEIN; METRICS

There are three main areas of investigation in the proposal. The first concerns the analytic and geometric structure of metric measure spaces for which the measure is doubling and a Poincare inequality holds, and also the behavior of Lipschitz maps from such spaces to the Banach space $L_1$. This study has connections with and applications to a number of areas of mathematics and to theoretical computer science. The second area of the proposal concerns the degeneration of Einstein metrics, especially in dimension 4, with particular emphasis on the collapsed case. The eventual goal is a complete understanding of all possible degenerations. The third area concerns the spectral geometry approach to obtaining combinatorial formulas for the Pontrjagin classes of a triangulated manifold in terms of $\eta$-invariants of links. A scheme for modifying this formula is proposed, leading to a formula which is local on all iterated links. This comes at the cost of choosing certain additional data on the links; some such choice cannot be avoided if one wants a formula of this type.A basic distinction in mathematics is between objects which are everywhere smooth (like the surface of a sphere) and objects (like the surface of a cube) which contain non-smooth parts, referred to as singularities. The emphasis in this project, which has three distinct sections, is on the study of the singular parts of various classes of objects. The first section is concerned with a class of objects which may have no smooth parts whatsoever ,and yet, they can be studied by methods of calculus. Surprisingly, such objects arise``naturally'' in various mathematical contexts. An even bigger surprise is that their study has applications to theoretical computer science. The second section of the project considers objects which are smoothly curved and whose curvature is constrained in a certain way. (They satisfy the so-called Einstein equation.) One wants to understand what are the``worst" examples of such objects. This leads in limiting cases to objects with singularities and the goal is to understand precisely what kinds of singularities can arise in this way. For instance, some of the examples in the first section of the project can arise in this way, but many cannot. The third section is concerned with objects which have smooth versions and also "piecewise flat" versions (with singularities). For instance, from the standpoint of topology, the surface of a sphere and the surface of a cube are equivalent i.e. one can be deformed continuously into the other. The sphere is smooth, while the surface of a cube is "piecewise flat", in the sense that it can be assembled by appropriately joining together 6 (flat) squares along their edges. We study certain topological measurements of such piecewise flat objects and show that they can be computed by adding up certain geometrical quantities whichare measured only at the most singular parts.

建议中有三个主要的调查领域。 第一部分讨论了测度加倍且Poincare不等式成立的度量测度空间的解析结构和几何结构，以及从度量测度空间到Banach空间L_1 $的Lipschitz映射的性质。这项研究与数学和理论计算机科学的一些领域有联系和应用。该建议的第二个方面涉及爱因斯坦度量的退化，特别是在4维，特别强调崩溃的情况。 最终的目标是完全理解所有可能的简并。第三个领域涉及的谱几何方法获得的Pontrjagin类的三角流形的$\eta$-不变量的链接的组合公式。提出了一个修改该公式的方案，从而得到一个在所有迭代链路上都是局部的公式。这是以在链接上选择某些附加数据为代价的;如果想要得到这种类型的公式，一些这样的选择是不可避免的。数学中的一个基本区别是处处光滑的对象（如球体的表面）和包含非光滑部分的对象（如立方体的表面），称为奇点。这个项目有三个不同的部分，重点是研究各类对象的奇异部分。 第一部分是关于一类可能没有任何光滑部分的物体，然而，它们可以用微积分的方法来研究。令人惊讶的是，这样的对象在各种数学环境中“自然”出现。更令人惊讶的是，他们的研究可以应用于理论计算机科学。该项目的第二部分考虑的是平滑弯曲的物体，其曲率以某种方式受到约束。 (They这就是所谓的爱因斯坦方程（Einstein Equation）。人们想知道这些物体的“最坏”例子是什么。这导致了对具有奇点的物体的限制，目标是精确地理解以这种方式可以产生什么样的奇点。例如，项目第一部分中的一些示例可以以这种方式出现，但许多示例不能。 第三部分涉及的对象，有光滑的版本，也“分段平坦”的版本（与奇点）。例如，从拓扑学的角度来看，球体的表面和立方体的表面是等效的，即一个可以连续变形为另一个。球体是光滑的，而立方体的表面是“分段平坦的”，在这个意义上，它可以通过适当地将6个（平坦的）正方形沿着它们的边缘连接在一起来组装。我们研究了这种分段平面物体的某些拓扑测量，并证明了它们可以通过将仅在最奇异部分测量的某些几何量相加来计算。