Louis de Branges

Research Description

A half-century of research has been invested in fundamental problems which have not been solved, with the notable exception of the Bieberbach conjecture. The Riemann hypothesis is the most important problem as measured by the effort made and the expected impact of a solution. A solution is proposed as an application of Fourier analysis on quaternions, a result which is of interest in theoretical physics because of its relationship to quantum mechanics. The existence of invariant subspaces for continuous linear transformations of a Hilbert space into itself is obtained as an application of analytic function theory in the unit disk in a context of systems theory. The proof of the Riemann hypothesis is a related application of analytic function theory on the upper half-plane. Hilbert spaces whose elements are entire functions then replace Hilbert spaces of functions analytic in the unit disk. Hilbert spaces of entire functions originate in an integration theory for polynomials on a line which is due to Stieltjes. The measure problem is a curious aspect of integration theory which was discovered by Banach and solved under hypothesis in cardinal number theory. The independence of the continuum hypothesis from the axioms of set theory encourages the belief that these hypotheses are also independent. A solution of the measure problem is however proposed in which these hypotheses are shown to be consequences of the axioms of set theory.

Solutions of major problems need not be immediately publishable when the results obtained are contrary to expectations. The support of Purdue University continues to be essential in attaining research objectives for which funding is difficult.

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