Starting from the Amann-Conley-Zehnder finite reduction framework in the non-compact Viterbo's version we discuss the existence of global generating function with a finite number of auxiliary parameters describing the two-points Characteristic Relation related to the geodesic problem in the Hamiltonian formalism. This applies both to Analytical Mechanics and to General Relativity - we construct a global object generalizing the World Function introduced by Synge, which is well-defined only locally. Whenever the auxiliary parameters can be fully removed, Synge's World Function is restored.

The submanifold quantum mechanics was opened by Jensen and Koppe and has been studied for more than three decades. This article gives its more algebraic definition and show the essential aspects of the submanifold quantum mechanics from an algebraic viewpoint.

The profound, beautiful and, at times, rather mysterious symbiosis between mathematics and physics has always been a source of wonder, but, in the past twenty years, the intensity of the mutual interaction between these two has become nothing short of startling. Our objective here is to provide an introduction, in terms as elementary as possible, to one small aspect of this relationship. Toward this end we shall tell a story. Although we make no attempt to relate it chronologically, the story can be said to begin with the efforts of Yang and Mills to construct a nonabelian generalization of classical electromagnetic theory, and to culminate in a remarkable conjecture of Witten concerning the Donaldson invariants of a smooth four-manifold.