Masoud Khalkhali works in an area of mathematics called
"Noncommutative Geometry". Noncommutative geometry, pioneered
by Alain Connes since 1980's, has its roots in operator algebras and
problems in analysis, geometry, topology, algebra and mathematical
physics that can be most naturally and effectively tackled through the
use of special classes of noncommutative algebras. A perfect
example of this is the advent of quantum mechanics by Heisenberg
and Schroedinger in 1920's, where one replaces the commutative
algebra of classical observables by a noncommutative algebra of
operators on a Hilbert space. The high level of sophistication and
precision obtained by passing from classical commutative spaces to
quantum noncommutative spaces is very well reflected in this
example. One of the main tools of noncommutative geometry is
cyclic cohomology theory. This theory replaces de Rham (co)homology in
noncommutative settings and through a Chern character map is mapped to
K-theory. An analytic variant of cyclic cohomology is the entire cyclic
cohomology of Banach algebras defined by Connes in late 1980's. In his
thesis, Masoud Khalkhali proved, for the first time, some
of the most fundamental properties of the entire cyclic
cohomology like homotopy
invariance, Morita invariance, and isomorphism theorems between various
formulations of the theory. Using results of Cuntz and Quillen,
he was then able to show that for Banach algebras of finite
cohomological dimension the entire and continuous cyclic
cohomology are isomorphic. This result opened the way for future
developments in asymptotic and analytic cyclic cohomology theories.
The idea of symmetry, encoded by the action of a group or Lie algebra
in classical geometry and physics, finds its most natural extension
in noncommutative geometry in the theory of quantum groups and Hopf
algebras. In recent years Masoud Khalkhali, together with his students
R. Akbarpour and B. Rangipour, has studied a new cyclic
cohomology theory for Hopf algebras and quantum groups. This
theory, initiated by Connes and Moscovici in their work on index
theory of transversally elliptic operators, is a fully noncommutative
analogue of cohomology theories for Lie algebras and (Lie)groups and
the theory of characteristic classes. In one direction, Khalkhali and
Akbarpour have developed Hopf algebra equivariant cyclic theories and
K-theories and have established spectral sequences relating equivariant
theories to Hopf crossed products. Together with Rangipour,
khalkhali has developed a new cyclic theory for H-algebras called
invariant cyclic (co)homology. It is shown that all known examples of
Hopf-cyclic theories as well as the ordinary cyclic (co)homology theory
are examples of invariant cyclic (co)homology. In joint work P.
Hajac, Y. Sommerhaeuser, and B. Rangipour, the
most general class of coefficient systems for invariant cyclic homology
are found. These systems of coefficients strongly resemble local
systems
for de Rham theory.
