METRIC MEASURE SPACES, EINSTEIN METRICS, SPECTRAL GEOMETRY

提案中有三个主要的调查领域。第一部分讨论度量度量空间的解析和几何结构，其中度量加倍且Poincare不等式成立，以及从这样的空间到Banach空间$L_1的Lipschitz映射的行为。这项研究与数学和理论计算机科学的许多领域有联系和应用。提案的第二个领域涉及爱因斯坦度规的退化，特别是在4维，特别是在崩溃的情况下。最终目标是对所有可能的退化有一个完整的了解。第三个领域涉及用链环的不变量来获得三角流形的Pontrjagin类的组合公式的谱几何方法。提出了一个修正该公式的方案，从而得到了一个在所有迭代链路上都是局部的公式。这是以选择链接上的某些额外数据为代价的；如果想要这种类型的公式，一些这样的选择是无法避免的。数学上的一个基本区别是到处都是光滑的对象(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，重点是研究各类物体的单数部分。第一部分是关于一类物体，它们可能没有任何光滑的部分，但它们可以用微积分的方法来研究。令人惊讶的是，这样的物体在不同的数学背景下“自然”地出现。更令人惊讶的是，他们的研究适用于理论计算机科学。该项目的第二部分考虑的对象是平滑弯曲的，并且其曲率以某种方式受到约束。(它们满足所谓的爱因斯坦方程。)人们想知道这类物体的“最糟糕”的例子是什么。这导致了将情况限制在具有奇点的物体上，目标是准确地了解以这种方式可以产生什么类型的奇点。例如，项目第一部分中的一些示例可以以这种方式产生，但许多不能。第三部分是关于具有平滑版本和“分段平坦”版本(具有奇点)的对象。例如，从拓扑学的观点来看，球面和立方体的表面是等价的，即一个可以连续变形成另一个。球体是光滑的，而立方体的表面是“分段平坦的”，从这个意义上说，它可以通过将6个(平坦的)正方形沿着它们的边缘适当地连接在一起来组装。我们研究了这种分片扁平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。

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.