FROM METRIC SPACES TO EINSTEIN SPACES

Gromov defined a notion of distance between isometry classes of compact 
metric spaces.  Thus, the collection of all such isometry classes is itself, 
a metric space, M.  Roughly speaking, two metric spaces are close in 
M if, ``to the naked eye'', they look the same (although when examined on a 
microscopic scale, they might look entirely different). 

Gromov showed that a certain important class of smoothly curved spaces, 
RIC_n, contained in M, consisting of n-dimensional Riemannian manifolds 
with a definite bound on diameter and lower bound on Ricci curvature, 
has the property that every sequence, M^n_i in RIC_n, 
has a subsequence which converges to some metric space Y
in the closure of RIC_n. Because of the weak nature of the 
convergence, apriori, Y might not resemble a smoothly curved 
space in any reasonable way. On the other hand, knowledge of 
the extent to which some Y in the closure of RIC_n
could fail to be smoothly curved, would provide information
on the class, RIC_n, itself. How can one get such information?  

Starting with simple examples, we will discuss curvature and  Gromov's
notion of convergence of metric spaces.  Then we will describe 
a theory (developed jointly with T. Colding) which governs the regularity and 
singularity structure of spaces Y in RIC_n.  
If we specialize further to the case of Einstein manifolds, spaces whose
Ricci tensor is a constant multiple of the metric, stronger conclusions
can be obtained.  In this way, we get information on the manner in which 
Einstein metrics can degenerate. 



Lecture 1: Curvature and Gromov-Hausdorff convergence.  


Lecture 2: Spaces with Ricci curvature bounded below. 


Lecture 3: Degeneration of Einstein metrics.