Mathematics, in large part, can be said to have originated from the study of equations in integers. Fermat, Euler, Gauss, Hilbert and many great mathematicians have contributed to the field, and numerous results have been obtained through the centuries to resolve particular instances. It is only in recent times, however, that a general method, based on transcendence theory, has been successfully developed in this context. The method has deep connections with the theory of elliptic curves, with class number questions, with p-adic L-functions and with many aspects of Diophantine geometry. The lecture will survey the current state of the subject.
This is a report on joint research by myself and Professor Bent Fuglede of the University of Copenhagen. It is an essay on harmonic maps (i. e., energy extremals) between certain singular spaces (especially, Riemannian polyhedra). The guiding principle has been T. Ishihara's characterization  for maps between smooth Riemannian manifolds: A continuous map is harmonic iff it pulls germs of convex functions back to germs of subharmonic functions. Gromov  has shown that geodesic spaces (in the sense of Alexandrov and Busemann) are natural targets; and our starting point was the realization that—in view of the above characterization—harmonic spaces (in the sense of Brelot) are natural domains. We have not been able to work in that generality; but have discovered that admissible Riemannian polyhedra (described in the course of this lecture) are both geodesic and harmonic spaces. Thus harmonic maps (especially, when presented in their variational context, via the Dirichlet integral) between Riemannian polyhedra are our main object of study. Now we describe the broad lines of development—first for functions, and then for maps.