41 research outputs found
Weighted Traces on Algebras of Pseudo-Differential Operators and Geometry of Loop Groups
Using {\it weighted traces} which are linear functionals of the type defined on the whole
algebra of (classical) pseudo-differential operators (P.D.O.s) and where is
some positive invertible elliptic operator, we investigate the geometry of loop
groups in the light of the cohomology of pseudo-differential operators. We set
up a geometric framework to study a class of infinite dimensional manifolds in
which we recover some results on the geometry of loop groups, using again
weighted traces. Along the way, we investigate properties of extensions of the
Radul and Schwinger cocycles defined with the help of weighted traces.Comment: 36 page
Wodzicki Residue for Operators on Manifolds with Cylindrical Ends
We define the Wodzicki Residue TR(A) for A in a space of operators with
double order (m_1,m_2). Such operators are globally defined initially on R^n
and then, more generally, on a class of non-compact manifolds, namely, the
manifolds with cylindrical ends. The definition is based on the analysis of the
associate zeta function. Using this approach, under suitable ellipticity
assumptions, we also compute a two terms leading part of the Weyl formula for a
positive selfadjoint operator belonging the mentioned class in the case
m_1=m_2.Comment: 24 pages, picture changed, added references, corrected typo
Curvature in Noncommutative Geometry
Our understanding of the notion of curvature in a noncommutative setting has
progressed substantially in the past ten years. This new episode in
noncommutative geometry started when a Gauss-Bonnet theorem was proved by
Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral
geometry and heat kernel asymptotic expansions suggest a general way of
defining local curvature invariants for noncommutative Riemannian type spaces
where the metric structure is encoded by a Dirac type operator. To carry
explicit computations however one needs quite intriguing new ideas. We give an
account of the most recent developments on the notion of curvature in
noncommutative geometry in this paper.Comment: 76 pages, 8 figures, final version, one section on open problems
added, and references expanded. Appears in "Advances in Noncommutative
Geometry - on the occasion of Alain Connes' 70th birthday