The present book contains fifteen contributions on various topics related to
Number Theory, Physics and Geometry. It presents, together with a forthcoming
second volume, most of the courses and seminars delivered at the meeting
entitled “Frontiers in Number Theory, Physics and Geometry”, which took
place at the Centre de Physique des Houches in the french Alps March 9-21,
One of the major advances of science in the 20th century was the discovery of a mathematical
formulation of quantum mechanics by Heisenberg in 1925 .1 From a
mathematical point of view, this transition from classical mechanics to quantum mechanics
amounts to, among other things, passing from the commutative algebra of
classical observables to the noncommutative algebra of quantum mechanical observables.
To understand this better we recall that in classical mechanics an observable of
a system (e.g. energy, position, momentum, etc.
Israel Moiseevich Gelfand is one of the greatest mathematicians of the 20th century.
His insights and ideas have helped to develop new areas in mathematics and to
reshape many classical ones.
The influence of Gelfand can be found everywhere in mathematics and mathematical
physics from functional analysis to geometry, algebra, and number theory. His
seminar (one of the most influential in the history of mathematics) helped to create
a very diverse and productive Gelfand school; indeed, many outstanding mathematicians
proudly call themselves Gelfand disciples....
In this article we study several homology theories of the algebra E ∞ (X) of Whitney functions over a subanalytic set X ⊂ Rn with a view towards noncommutative geometry. Using a localization method going back to Teleman we prove a Hochschild-Kostant-Rosenberg type theorem for E ∞ (X), when X is a regular subset of Rn having regularly situated diagonals. This includes the case of subanalytic X. We also compute the Hochschild cohomology of E ∞ (X) for a regular set with regularly situated diagonals and derive the cyclic and periodic cyclic theories. ...