Séance Séminaire

Colloquium de Mathématiques

Atiyah-Singer index theory, fractional variant and applications

I will begin with a review of the renowned Atiyah-Singer index theorem. The laws of nature are often expressed by differential equations, involving their rates of change. If elliptic, they have an index, which is the number of solutions minus the number of constraints imposed. The Atiyah-Singer index theorem gives a striking calculation of this index in terms of the geometry and topology of the space on which the elliptic equations are defined. I will then sketch the concept of a projective differential operator, introduced in fairly recent joint work with Melrose and Singer, whose index is defined and is a fraction in general and the index theorem this context will be discussed. In particular this solved the open problem of defining a Dirac operator (as a projective differential operator) on an oriented manifold (even in the absence of a spin structure). Applications to a model of the fractional quantum Hall effect will be sketched.