Cyclic cohomology and algebra extensions

We construct a spectral sequence to study cyclic cohomology for an extension A = R/I of algebras. When R is a free algebra we describe the cyclic cohomology of A in terms of traces defined on R or powers of I. Explicit cyclic cocycles representing the cyclic cohomology class belonging to the trace are constructed as analogues of Chern character and Chern-Simons forms.Dedicated to Alexander Grothendieck     

Eigenfunctions of transfer operators and cohomology

The eigenfunctions with eigenvalues 1 or −1 of the transfer operator of Mayer are in bijective correspondence with the eigenfunctions with eigenvalue 1 of a transfer operator connected to the nearest integer continued fraction algorithm. This is shown by relating these eigenspaces of these operators to cohomology groups for the modular group with coefficients in certain principal series representations.     

Cyclic operators with finite support

We begin a systematic study of cyclic operators with finite support, a notion which was introduced by P. Enflo in the paper where he solved the invariant subspace problem. We give spectral properties of these operators as well as surprising examples.     

Cyclic cohomology of etale groupoids

We extend Connes's computation of the cyclic cohomology groups of smooth algebras arising from foliations with separated graphs. We find that the characteristic classes of foliations factor through these groups. Our results also explain some results of Atiyah and Segal on orbifold Euler characteristic in the setting of cyclic homology.Partially supported by NSF grants DMS 92-03517 and DMS 89-03248.Partially supported by NSF grant DMS 92-05548.     

Cyclic cohomology of Hopf algebras

We give a construction of the Connes–Moscovici cyclic cohomology for any Hopf algebra equipped with a character (solving the technical problem raised in Connes and Moscovici (Comm. Math. Phys. 198 (1998) 199–246)). Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the Weil complex arises naturally when looking at Hopf algebra actions and invariant higher traces, to give a non-commutative version of the usual Chern–Weil theory.     

Cyclic cohomology and the family Lefschetz theorem

Any closed current on the base of a compact fibration gives rise to a cyclic cocycle on the smooth convolution algebra. We prove that such cocycle furnishes additive maps from the vertically equivariant K-theory to the scalars. This enables to associate to any closed current on the base of the fibration, a Lefschetz formula for fiber-preserving isometries. Using geometric operators on the base, we deduce the integrality of some characteristic numbers. Received: 28 June 2001 / Published online: 1 February 2002     

Trace formulas for almost commuting operators,cyclic cohomology and subnormal operators

Some formulas related to cyclic cohomology for the trace of the product of commutators are established. A simple complete unitary invariant for some subnormal operator with simply connected spectrum is found.This work is supported in part by NSF grant.     

Generalized cyclic cohomology associated with deformed commutators

The generalized cyclic cohomology is introduced which is associated with -deformed commutators . Some formulas related to the trace of the product of -deformed commutators are established. The Chern character of odd dimension associated with -deformed commutators is studied.

     

Integral operators and integral cohomology classes of Hilbert schemes

The methods of integral operators on the cohomology of Hilbert schemes of points on surfaces are developed. They are used to establish integral bases for the cohomology groups of Hilbert schemes of points on a class of surfaces (and conjecturally, for all simply connected surfaces).Mathematics Subject Classification (2000): 14C05, 14F43, 17B69Partially supported by an NSF grant.     

Macdonald difference operators and Harish-Chandra series

We analyse the centralizer of the Macdonald difference operatorin an appropriate algebra of Weyl group invariant differenceoperators. We show that it coincides with Cherednik's commutingalgebra of difference operators via an analog of the Harish-Chandraisomorphism. Analogs of Harish-Chandra series are defined andrealized as solutions to the system of basic hypergeometricdifference equations associated to the centralizer algebra.These Harish-Chandra series are then related to both Macdonaldpolynomials and Chalykh's Baker–Akhiezer functions.