Alexander Gorokhovsky (Colorado)
Bivariant Chern Character and Connes' Index Theorem.
We show how to compute the action of bivariant Chern character for
the family
of operators equivariant with respect to an etale groupoid. As an
application
we obtain a new superconnection proof of the Connes' index theorem
for etale groupoids.
Nigel Higson (Penn State)
Introduction to the Connes-Moscovici form
I will give a brief introduction to the Connes-Moscovici formula,
including a discussion of the hypotheses for the formula, its relation
with Connes' cyclic Chern character, and a somewhat conceptual view
of the formula using ideas borrowed from Quillen.
X. Hu (Toronto)
Local index theorem for transversally elliptic operators
Transversally elliptic operators relative to a compact Lie groups
(TEOs) were introduced by Atiyah in 1974, in the language of equivariant
$K$-theory. The transversal index generalizes on one hand the Atiyah-Singer
index and on the other Fourier analysis. However not much has been
said on the general index theorem with the classical method until
1997 by Berline and Vergne.
We show that the noncommutative geometric approach gives an index
theorem for TEOs. Specifically we examine the Connes-Moscovici local
index theorem for TEOs. The formula computes the Connes-Chern character
for the smooth crossed-product algebra in terms of its cyclic cohomological
cocycles. The computation of the index amounts to the residue trace-like
functionals as introduced by Connes-Moscovici on the algebra of pseudo-differential
operators over the transformation groupoid. By use of wave front set
argument, resolution of singularities and the asymptotic expansion
of oscillatory integrals we show that the details of Connes Moscovici
local index formulas. For example, in this case we have a discrete
dimension spectrum, we have control over the multiplicities of the
zeta functions. We conclude that the TEO case is finite in nature,
no renormalization is needed.
Jerry Kaminker (IUPUI):
Duality in noncommutative geometr
We will discuss a noncommutative version of Spanier-Whitehead duality
and show how it comes up in a variety of different settings in index
theory and its applications.
Masoud Khalkhali (University of Western
Ontario)
Renormalization and Motivic Galois Theory (after Connes and
Marcolli)
I will try to give a report on a very recent work of Connes and Marcolli
where they construct a ``motivic Galois group'' and show that it acts
on the set of physical theories. I will first explain the work of
Connes and Kreimer where they construct a pro-unipotent Lie group
G via the Hopf algebra of Feynman graphs and show that perturbative
renormalization in quantum field theory can be understood in terms
of Birkhoff decomposition of loops in G. The motivic Galois group
U* is defined through the Tannakian category of flat equisingular
bundles solving a Riemann-Hilbert
correspondence associated to perturbative renormalization. There is
a (non-canonical) isomorphism between U* and the motivic Galois group
of the scheme S_4 of 4-cyclotomic integers. There is also a mysterious
relationship with Connes Moscovici local index formula.
Eckhart Meinrenken (Toronto)
Chern-Weil homomorphism for non-commutative differential algebras
Let $\g$ be a Lie algebra and $A$ a possibly non-commutative differential
algebra, equipped with a $\g$-action in the sense of H. Cartan. Let
$W\g$ denote the Weil algebra. As we will explain in this talk, there
is a canonical equivariant chain map $W\g\to A$, generalizing the
Chern-Weil map for the commutative case. The generalized Chern-Weil
map induces an algebra homomorphism in basic cohomology, even if $A$
is non-commutative. As application we obtain a quick proof of Duflo-type
theorems for quadratic Lie algebras, and a new construction of universal
characteristic forms in the Bott-Shulman complex. Based on joint work
with Anton Alekseev (Geneva).
John Phillips(with A. Carey, A. Rennie
and F. Sukochev)
From Spectal Flow to the Odd Local Index Formula (.pdf
format)
We generalise the odd local index formula of Connes and Moscovici
to the case of unbounded spectral triples (A,N,D) for a -subalgebra
A of a general semifinite von Neumann algebra, N with a fixed faithful,
normal, semifinite trace, . In this setting it gives a cohomological
formula for the pairing of Conne’s Chern Character Ch(u) of an
element in K1(A) (a unitary u 2 A) with a (b,B) cocycle constructed
from the spectral triple.
We start from the spectral flow formula for the index (of the Toeplitz
operator PuP) for finitely summable spectral triples developed by
Carey-Phillips. This spectral flow formula is given by the integral
of a one-form along the straight line path from D to uDu together
with a normalising “constant.” We show how the seemingly
innocuous normalising constant in the formula actually gives a new
approach to the Connes-Moscovici results. Time permitting, we will
indicate how we can prove the even index theorem using similar techniques,
but starting from a semifinite version of the McKean-Singer formula.
Raphael Ponge (Ohio State)
Noncommutative geometry, Heisenberg caclulus and CR geometry
Bahram Rangipour (Victoria)
Cup product in Hopf cyclic cohomology and Connes Moscovici characteristic
map.
We show that the ordinary cup product in cyclic cohomology can be
generalized in Hopf cyclic cohomology and the result of this product
is in cyclic cohomology of a crossed product algebra. In a Dual method
we find that
the Connes-Moscovici characterestric map has a generalization in Hopf
cyclic cohomology and hence the conjecture stated by P.M. Hajac, M.
Khalkhali, B. R., and Y. Sommerhaeuser is true
Andrzej Sitarz (Wroclav)
Local index formula: going beyond spectral triples
Local index formula of Connes-Moscovici is formulated for the spectral
triples. We present two examples (Heisenberg group algebra and quantum
spheres), which are a testing ground for some generalizations of the
notion of spectral geometries and the Dirac operator. Still, for these
generalized object the local index formula holds.
Boris Tsygan (Northwestern):
BV operators in noncommutative geometry.
I will explain how Batalin-Vilkovyski structures arise in noncommutative
geometry, as well as theit relation to the ones that appear in quantum
field theory.
Erik Van Erp (Ohio State)
TBA
Forr more information contact gensci(PUT_AT_SIGN_HERE)fields.utoronto.ca
