Index theorem for inhomogeneous hypoelliptic differential operators

On the index of maximally hypoelliptic differential operators

We give an index formula for the class of all *-maximally hypoelliptic differential operators on any closed manifold with vector bundle coefficients, generalising previous index formulas by Atiyah-Singer and van Erp. Using this formula, we give new explicit index computations for Hormander's sum of squares operators of arbitrary rank.

An equivariant index theorem for hypoelliptic operators

Let M be a foliated manifold and G a discrete group acting on M by diffeomorphisms mapping leaves to leaves. Then G naturally acts by automorphisms on the algebra of Heisenberg pseudodifferential operators on the foliation. Our main result is an index theorem for hypoelliptic-type operators which belong to the crossed product of the Heisenberg pseudodifferential operators with the group G. As a corollary, we get a solution to Connes-Moscovici's transverse problem in arbitrary codimensions, by exhibiting an explicit formula in terms of characteristic classes of equivariant vector bundles over M…

Explicit construction of Atiyah-Singer indices for maximally hypoelliptic operators on contact manifolds

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. Enlightening from Alain Connes' tangent groupoid proof of the index theorem and van Erp's research for the Heisenberg index theory on contact manifolds, we give an explicit construction of a series of maps, whose induced map in K-theory is the Heisenberg Atiyah-Singer index map on contact manifolds. Our methods derive from Higson's construction for symbol class in K-theory.

Carnot流形上亚椭圆算子的指标理论

Index theory of hypoelliptic operators on Carnot manifolds

本文研究Carnot流形上Heisenberg伪微分算子的Fredholm指标，利用算子K理论与Baum-Douglas几何K同调建立对偶方法，并给出Toeplitz及Δ_H+γT型算子的显式指标解。

Zeta Functions, Excision in Cyclic Cohomology and Index Problems

The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain a local index formula for "abstract elliptic pseudodifferential operators" associated to spectral triples. This formula is notably well adapted when the zeta function has multiple poles. The second part is devoted to give a concrete realization of this formula by deriving an index theorem on the simple, but significant example of Heisenberg elliptic operators on a trivial foliation, which are in general non-elliptic but hypoelliptic. The l…

Heisenberg椭圆算子和横截Heisenberg椭圆算子的指标理论：$KK$-理论视角

Index theory for Heisenberg elliptic and transversally Heisenberg elliptic operators from $KK$-theoretic viewpoint

本研究从$KK$-理论视角，运用Kasparov方法系统研究了Heisenberg椭圆及横截Heisenberg椭圆算子的指标理论，为亚椭圆微分算子分析提供了稳健框架。