Degree Theory beyond Continuous Maps

I am very honoured and pleased to participate in the commemoration of the 50 th anniversary of the founding of the CWI of the Dutch Foundation Mathematical Centre. The subject of my talk is concerned not with diierential equations but with something that is used all the time in various nonlinear analytic problems , namely degree theory. It seems particularly appropriate to speak on this subject here since degree theory was developed by L.E.J. Brouwer. The innnite dimensional extension of degree theory, the Leray{Schauder degree, is a basic tool in attacking nonlinear diierential equations. This talk describes some joint work with H. Brezis 5]]6] and is concerned with nite dimensional degree. I recall the notion of degree (and its properties) of a map u from one smooth n-dimensional compact oriented manifold X without boundary to a connected one Y of the same dimension. The degree measures, in a suitable sense, the number of times Y is covered. It may be deened using homology theory but we describe it in more analytic terms: Suppose u is in C 1 and that y 2 Y is a regular value of the map, i.e., the preimage of y; u ?1 (y), consists of a nite number of points, x 1 ; : : : ; x k in X, and the Jacobian matrix J u , in terms of local coordinates near x j and near y, is nonsingular at each x j. If we choose local coordinates compatible with the given orientations on X and Y , then degree of u at y, denoted deg(u; X; y), counts the number of points x j in u ?1 (y) algebraically: deg(u; X; y) = k X i=1 sgn det J u (x i) : This number turns out to be independent of y, and is deened as the degree of the map u from X to Y , deg(u; X; Y) = deg u. In case we put Riemannian metrics on X and Y , then degree may be expressed by an integral, 113