# THEMATIC PROGRAMS

May 29, 2016

## Mini-conference on Noncommutative Geometry, the Local Index Formula and Hopf Algebras

### Friday and Saturday, Sept. 24, 25, 2004, held at Fields Institute in Toronto

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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