Equivariant Hodge theory and noncommutative geometry

Hodge Theory and Geometry

This expository paper is an expanded version of a talk given at the joint meeting of the Edinburgh and London Mathematical Societies in Edinburgh to celebrate the centenary of the birth of Sir William Hodge. In the talk the emphasis was on the relationship between Hodge theory and geometry, especially the study of algebraic cycles that was of such interest to Hodge. Special attention will be pl...

Equivariant KK-theory and noncommutative index theory

Canonical equivariant extensions using classical Hodge theory

In [4], Lin and Sjamaar show how to use symplectic Hodge theory to obtain canonical equivariant extensions of closed forms in Hamiltonian actions of compact connected Lie groups on closed symplectic manifolds which have the strong Lefschetz property. In this paper, we show how to do the same using classical Hodge theory. This has the advantage of applying far more generally. Our method makes us...

Noncommutative geometry and number theory

Noncommutative geometry is a modern field of mathematics begun by Alain Connes in the early 1980s. It provides powerful tools to treat spaces that are essentially of a quantum nature. Unlike the case of ordinary spaces, their algebraof coordinates is noncommutative, reflecting phenomena like the Heisenberg uncertainty principle in quantum mechanics. What is especially interesting is the fact th...

Noncommutative geometry and number theory

In almost every branch of mathematics we use the ring of rational integers, yet in looking beyond the formal structure of this ring we often encounter great gaps in our understanding. The need to find new insights into the ring of integers is, in particular, brought home to us by our inability to decide the validity of the classical Riemann hypothesis, which can be thought of as a question on t...