Fix a positive integer N. A manifold X of dimension N (also called an N-manifold) is a geometric object that locally looks like N-dimensional Euclidean space. Informally, people like to say that if you were a myopic "bug" living on X then you would think you were living in R^N. Europeans in the fifteenth century were like this bug. They looked at the world around them and thought it was flat---a copy of R². They did not realize until Columbus and other intrepid explorers circumnavigated the globe that in fact the world is a sphere.

So a sphere is a prime example of a 2-manifold. See Figure 1. Another good example is the torus---see Figure 2.

Figure 1

Figure 2

It is a remarkable fact from the mid-nineteenth century that any closed 2-manifold in 3-space is topolically equivalent to a sphere with handles attached. The torus, for example, is a sphere with one handle attached. See Figure 3. Other examples are shown in Figure 4.

Figure 3

Figure 4

Higher-dimensional manifolds are much more difficult to understand. Various differential geometric tools have been developed as devices for understanding the "shape" of a manifold. One of the chief among these devices is the concept of curvature.

Classically, we understand curvature by way of the Gauss map. First, associate to each point x ∈ X the outward unit normal ν_x to X at x. Then, to each tangent vector τ to X at x, assign the derivative τ ν_x, where the derivative is taken in the x variable. Since

< ν_x, ν_x > ≡ 1 ,

we see that

0 = τ < ν_x, ν_x > = 2 < τ ν_x, ν_x > .

Thus τ ν_x is itself a tangent vector.

So we have constructed a map from the tangent space to itself. The determinant of this mapping called the curvature of the manifold X at the point x. Instances of positive and negative curvature are illustrated in Figure 5.

Figure 5

For higher-dimensional manifolds, it is often convenient to consider sectional curvature---which means that we consider the curvature of two-dimensional slices. This information is harnessed using various tensors.

The philosophy of this workshop was that curvature properties of a manifold will reveal structural and topological characteristics. Particular interest was given to manifolds of nonnegative curvature. In this context there is a particularly rich structure. Typical results are

Hopf: A simply-connected manifold of constant positive curvature is a sphere.

Bonnnet-Meyers: A manifold having a metric with positive curvature has finite dimensional first homotopy group. This puts severe restrictions on the number of "holes" that the manifold can have.

Hopf: It is an important conjecture that an even-dimensional manifold with positive curvature has positive Euler characteristic. Again, this puts measurable limitations on the number of "holes" in the manifold.

Other topics of interest at this workshop were submersions--differentiable mappings of manifolds having "large" derivatives. Also the Ricci flow--an idea of Richard Hamilton which was used by G. Perelman to prove the Poincaré conjecture--- is turning out to be useful in this classical geometric context.

This workshop proved to be a meeting ground for techniques from geometry, partial differential equations, Lie groups, and other fields as well. The afternoon working groups were vigorous and productive, and indications are that many new directions were launched.