Room 517, Meng Wah Complex, HKU
Professor Paolo
Piazza
Università di Roma
"La Sapienza", Italy
Doing geometry with Dirac
operators
Abstract
There is a very interesting class of
first order elliptic differential operators attached to a Riemannian manifold (M, g), the so-called
operators of Dirac type. Examples include the Dolbeault operator on an almost complex manifold, the
Gauss-Bonnet operator on an orientable manifold, the Atiyah-Singer operator on a
spin manifold. In this talk I will explain how it is possible to use analytic
invariants attached to these operators in order to prove purely geometric
theorems. I will begin by talking about the index of these operators, as in the
seminal work of Atiyah and Singer; I will then move
to more sophisticated invariants, the rho-invariants, and explain recent
results about their use in Riemannian geometry and differential topology.
Our main tools will be elliptic theory both on closed manifolds and on
manifolds with boundary, and two generalized homology theories:
K-homology and bordism.