Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Author:
Melas Dimitrios-Chrysovalantis
Supervisors info:
Ιακωβος Ανδρουλιδάκης, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Διονύσιος Λάππας, Αφυπ. Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Δημήτριος Χελιώτης, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Original Title:
Index theorems via groupoids, KK theory and cyclic cohomology
Translated title:
Index theorems via groupoids, KK theory and cyclic cohomology
Summary:
In the 1960s, Atiyah and Singer gave a vast generalisation of a series of
results connecting Topology with Analysis. The Atiyah-Singer theorem
generalises the classical Gauss-Bonnet theorem, Chern-Weil theory and the
Riemann-Roch theorem. It gives a formula for the calculation of the
analytic index of an elliptic (pseudo)differential operator using
characteristic classes.
This dissertation presents the recent proof of the Atiyah-Singer theorem
using Lie groupoids and K-theory. This proof arises from the observation
that the analytic index depends only from the class of the principal
symbol in K-theory. Starting from this, Alain Connes used a deformation
groupoid and its associated extension of C* algebras to describe the
relation of the elliptic operator with its principal symbol. The
connecting map in K-theory is the analytic index.
Claire Debord showed that the topological index can also be expressed in
K-theory using deformation groupoids and the Thom isomorphism. The proof
of the equality of the two indices is a Poincare duality type theorem,
expressed through Kasparov's KK-theory.
In this framework, the calculation of the index can be possible by the
pairing of K-theory with cyclic cohomology. Partial results in this
direction have been given by Pflaum-Posthuma-Tang.
Main subject category:
Science
Keywords:
Index theory, noncommutative geometry, K theory -K homology, KK theory, cyclic cohomology,
