METRIC MEASURE SPACES, EINSTEIN METRICS, SPECTRAL GEOMETRY

提案中有三个主要的调查领域。第一部分讨论了度量度量空间的解析结构和几何结构，其中度量是加倍的，并且庞加莱不等式成立，以及从这些空间到巴拿赫空间L_1$的Lipschitz映射的行为。这项研究与数学和理论计算机科学的许多领域都有联系和应用。该提议的第二个领域涉及爱因斯坦度量的退化，特别是在4维，特别强调坍缩的情况。最终的目标是完全理解所有可能的退化。第三个领域涉及光谱几何方法，以获得三角化流形的Pontrjagin类的组合公式，其中包含连杆的$\eta$不变量。提出了一种修改该公式的方案，使该公式在所有迭代链路上都是局部的。这是以在链接上选择某些额外数据为代价的；如果想要这种类型的公式，某些这样的选择是无法避免的。数学中一个基本的区别是处处光滑的物体（如球体表面）和包含非光滑部分的物体（如立方体表面），这些部分被称为奇点。这个项目有三个不同的部分，重点是研究不同类别物体的单个部分。第一部分讨论的是一类物体，它们可能没有光滑的部分，但是它们可以用微积分的方法来研究。令人惊讶的是，这些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。项目的第二部分考虑了光滑弯曲的物体，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。这就导致了对具有奇点的物体的限制，我们的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例子可以以这种方式出现，但许多例子不能。第三部分涉及具有光滑版本和“分段平坦”版本（具有奇点）的对象。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，即一个可以连续地变形成另一个。球体是光滑的，而立方体的表面是“分段平坦的”，也就是说，它可以通过沿边缘将6个（平坦的）正方形适当地连接在一起而组装起来。我们研究了这种分段平面物体的某些拓扑测量，并表明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.