Proceedings of the Conference Quantum Probability and Infinite Dimensional Analysis : Burg (Spreewald), Germany, 15-20 March, 2001

Description

Fred Almgren created the excess method for proving regularity theorems in the calculus of variations. His techniques yielded Holder continuity except for a small closed singular set. In the sixties and seventies Almgren refined and generalized his methods. Between 1974 and 1984 he wrote a 1,700-page proof that was his most ambitious exposition of his ground-breaking ideas. Originally, this monograph was available only as a three-volume work of limited circulation. The entire text is faithfully reproduced here.This book gives a complete proof of the interior regularity of an area-minimizing rectifiable current up to Hausdorff codimension 2. The argument uses the theory of Q-valued functions, which is developed in detail. For example, this work shows how first variation estimates from squash and squeeze deformations yield a monotonicity theorem for the normalized frequency of oscillation of a Q-valued function that minimizes a generalized Dirichlet integral. The principal features of the book include an extension theorem analogous to Kirszbraun's theorem and theorems on the approximation in mass of nearly flat mass-minimizing rectifiable currents by graphs and images of Lipschitz Q-valued functions Markov property - recent developments on the quantum Markov property, L. Accardi and F. Fidaleo; stationary quantum stochastic processes from the cohomological point-of-view, G.G. Amosov; the Feller property of a class of quantum Markov semigroups II, R. Carbone and F. Fagnola; recognition and teleportation, K.-H. Fichtner et al; prediction errors and completely positive maps, R. Gohm; multiplicative properties of double stochastic product integrals, R.L. Hudson; isometric cocycles related to beam splittings, V. Liebscher; multiplicativity via a hat trick, J.M. Lindsay and S.J. Wills; dilation theory and continuous tensor product systems of Hilbert modules, M. Skeide; quasi-free fermion planar quantum stochastic integrals, W.J. Spring and I.F. Wilde. (Part contents)