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In these talks I shall give examples of some applications of elliptic operators on manifolds centering around the fixed point formula and the index theorem. In his lectures at this session, R. T. Seeley will review pseudo-differential operators on manifolds and the elementary properties of the index. This will allow me to assume much of the analytical facts concerning elliptic operators. Since the primary discipline of the majority of the participants is analysis, I will spend more time with the geometry and topology but motivate the proofs of the main theorems from the analytic viewpoint. By and large analysts find the geometrical and topological background needed here formidable. It is my hope that these expository talks will help orient the listener and allow him to read the relevant papers listed at the end more easily.

If a master field exists, for large N gauge theories, it is an extreme point of the set of positive linear central functionals on the group algebra of a group of loops. In two dimensions master fields exist. We exhibit it explicitly for the plane.

We study the orbit space of vector potentials acted upon by gauge transformations. We describe its topology, its natural functional connection, and its curvature under a natural metric. A zeta function regularization is given for the functional Laplacian. When Sobolev metrics are used as an ultraviolet cutoff, the orbit space has a Gauss-Wiener measure.

In Parts I and II of this paper ((4), (5)) we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator A on a compact manifold we definedwhere λ runs over the eigenvalues of A. For the particular operators of interest in Riemannian geometry we showed that ηA(s) had an analytic continuation to the whole complex s-plane, with simple poles, and that s = 0 was not a pole. The real number ηA(0), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.

We shall accept as primitive (undefined) the concepts of a set (collection, family) of objects and the concept of an object belonging to a set.

Definition. Let M be an oriented Riemannian manifold of dimension 2. Let T(M) denote the tangent bundle of M. Let \(S(M) = [(m,v) \in T(M); = 1]\).

In Chapter 1 we dealt with topological concepts where it was not particularly important whether open sets were plentiful or scarce. However, the topological spaces usually encountered in applications have plenty of open sets, almost always enough to “separate” points, and often enough to “separate” closed sets.

We have defined the De Rham cohomology groups Hl(X, d) for a smooth manifold X. These groups came from a sequence of maps \({C^\infty }(X,{\Lambda ^{l - 1}}(X))\xrightarrow{d}{C^\infty }(X,{\Lambda ^l}(X))\xrightarrow{d}{C^\infty }(X,{\Lambda ^{l + 1}}(X))\) and Hl(X, d) = Ker d/Im d. We saw that the dimension of H0(X, d) measured the number of connected components of X, and we saw, at least for the circle X = S1, that the dimension of H1(X, d) measured the number of “holes” in X. We shall now develop similar groups for simplicial complexes. We shall study a sequence of maps \({C_{l - 1}}\xleftarrow{\partial }{C_l}\xleftarrow{\partial }{C_{l + 1}},\) where each Ck is an abelian group and where \({\partial ^2} = 0\).

In the previous chapter we studied the intrinsic geometry of a surface. Now we look at the properties of a surface as it lies in R3.

Almost all theories of fundamental interactions are nowadays based on the gauge concept. Starting with the historical example of quantum electrodynamics, we have been led to the successful unified gauge theory of weak and electromagnetic interactions, and finally to a non abelian gauge theory of strong interactions with the notion of permanently confined quarks. The. early theoretical work on gauge theories was devoted to proofs of renormalizability, investigation of short distance behaviour, the discovery of asymptotic freedom, etc . ., aspects which were accessible to tools extrapolated from renormalised perturbation theory. The second phase of the subject is concerned with the problem of quark confinement which necessitates a non-perturbative understanding of gauge theories. This phase has so far been marked by the introduc tion of ideas from geometry, topology and statistical mechanics in particular the theory of phase transitions. The 1979 Cargese Institute on "Recent Developments on Gauge Theories" was devoted to a thorough discussion of these non-perturbative, global aspects of non-abelian gauge theories. In the lectures and seminars reproduced in this volume the reader wilf find detailed reports on most of the important developments of recent times on non perturbative gauge fields by some of the leading experts and innovators in this field. Aside from lectures on gauge fields proper, there were lectures on gauge field concepts in condensed matter physics and lectures by mathematicians on global aspects of the calculus of variations, its relation to geometry and topology, and related topics."