Karen Uhlenbeck

"Most explicit information on the eigenfunctions of a Laplace operator on a compact manifold comes from computations where a high degree of symmetry is present. In these cases, eigenspaces may be of large dimension, the zeros of the eigenfunctions are often critical points, and the eigenfunctions usually have degenerate critical points. However, these properties are all unstable under small perturbations of the metric, and are therefore rather misleading to one's intuition."

"In the last several years, the study of gauge theories in quantum field theory has led to some interesting problems in nonlinear elliptic differential equations. One such problem is the local behavior of Yang-Mills fields ... over Euclidean 4-space. Our main result is a local regularity theorem: A Yang-Mills field with finite energy over a 4-manifold cannot have isolated singularities. Apparent point singularities (including singularities in the bundle) can be removed by a gauge transformation. In particular, a Yang-Mills field for a bundle over R4 which has finite energy may be extended to a smooth field over R4 \cup {∞} = S4."

"How did gauge theory appear and become successful in mathematics in the space of a few years? The fundamental mathematical ingredients were in place. The basics of fibre and vector bundles and their connections were in daily use by geometers. Chern-Weil theory (and even Chern-Simons invariants) were studied in most graduate courses in differential geometry. De Rham cohomology and its realization via the Hodge theory of harmonic forms were standard items in differential topology. In hindsight, the Yang-Mills equations were waiting to be discovered. Yet mathematicians were in themselves unable to create them. Gauge field theory is an adopted child."