Principal Investigator: Jeff Cheeger
The first main goal (joint with G. Tian) is to describe the how Einstein metrics can degenerate. In our previous work we gave a good local description of collapse of Einstein metrics in dimension 4, showing that away from a definite number of points collapse occurs with bounded curvature and in the limit, gives rise to Einstein metrics with continous symmetries. We want to globalize this local description and, in so far as possible, extend it to higher dimensions. A second main goal of the project, joint with B. Kleiner, is to study first order calculus on metric measure spaces for (Lipschitz) functions with values in infinite dimensional Banach spaces. In this context, differentiation theorems can give rise to bi-Lipschitz nonembedding theorems. For applications arising in computer science, the most interesting target is the space "Ell-One". In this case, a Lipschitz function need not be differentiable (anywhere) unless (as Kleiner and I discovered) differentiability is understood in a suitable extended sense. This sense was still strong enough to enable us give a counter example to the Goemans-Linial conjecture. An important aspect of the present project (joint with Kleiner and A. Naor) is to make this nonembedding theorem quantitative (which is what computer scientists really want). Our approach requires the developement of new techniques in geometric measure theory.
The study of certain special (higher dimensional) smoothly curved objects called Einstein manifolds is important in mathematics and in modern physics e.g. in connection with the theory of relativity and in string theory. In particular, we want a theory which tells us what can be the "most distorted" examples of such objects. Our project is concerned with this issue. A second main focus has to do with "metric spaces". By a metric space, one can understand any collection of objects where there is a notion of distance between any two of them. For example, the objects might be finger prints and a suitable notion of distance would enable a computer to detect which finger prints closely matcheda given one. A key issue in both the pure and applied aspects of the subject, is to be able to decide whether a "complicated" metric space which one wants to understand, can be realized (perhaps in a non-obvious way) as a subset of a "simpler" metrric space which one already understands. In our project, sophistocated tools from pure mathematics are developed which can be used in various cases (of pure and applied interest) to decide the feasibility of such an approach.
