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 which are measured only at the most singular parts.
Our research has had two distinct but closely related themes. The first one has been to study spaces which, although they can be quite wild (for example fractal in nature) are still nice enough to be studied by the methods of calculus. In past research, we indentified two conditions on spaces which guaranteed that they were nice in the above sense. In our recent research, we and our collaborators have constructed broad classes of examples of spaces which satisfy our conditions and identified additional hidden aspects of their infinitesimal structure in the general case. This area of study has surprising connections with theoretical computer science which have figured prominently in our previous work. The second theme has concerned nonlinear equations (partial differential equations) whcih typically arise from geometric problems. A familiar example is provided by soap films which minimize area among all surfaces with the same boundary. Since solutions to such nonlinear problems need not be smooth, an important classical issue has been to describe in each case, the size of their singular sets. We and our collaborators have developed a new general methodology which gives more precise results of this type than could be previously obtained by classical methods. To date, we have applied this methodology in seven distinct cases, including the minimal surface problem alluded to above. A key advantage of our technique is that it gives control not just on the size set of the where the solution is singular, but also, in a precise quantitative sense, on the larger set of points at which the solution is "almost singular" i.e. although smooth, nonetheless very badly behaved. We know that the class of problems to which our methods apply has yet to be exhausted.
