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Using the Witten deformation and localization algebra techniques, we compute the $G$-equivariant $K$-homology class of the de Rham operator on a proper cocompact $G$-spin manifold, where $G$ is an almost connected Lie group. By applying a…

Let $D$ be a (generalized) Dirac operator on a non-compact complete Riemannian manifold $M$ acted on by a compact Lie group $G$. Let $v:M --> Lie(G)$ be an equivariant map, such that the corresponding vector field on $M$ does not vanish…

We consider a generalized APS boundary problem for a G-invariant Dirac-type operator, which is not of product type near the boundary. We establish a delocalized version (a so-called Kirillov formula) of the equivariant index theorem for…

This paper is devoted to mathematical and physical properties of the Dirac operator and spectral geometry. Spin-structures in Lorentzian and Riemannian manifolds, and the global theory of the Dirac operator, are first analyzed. Elliptic…

We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients.…

Let $f\colon M \to M$ be a uniformly quasiregular self-mapping of a compact, connected, and oriented Riemannian $n$-manifold $M$ without boundary, $n\ge 2$. We show that, for $k \in \{0,\ldots, n\}$, the induced homomorphism $f^* \colon…

Index theory for Lorentzian Dirac operators is a young subject with significant differences to elliptic index theory. It is based on microlocal analysis instead of standard elliptic theory and one of the main features is that a nontrivial…

This paper studies the K-homology of a crossed product of a discrete group acting smoothly on a manifold, with a better understanding of the noncommutative geometry of the crossed-product as the primary goal, and the Baum-Connes apparatus…

In this paper, we study $l^1$-higher index theory and its pairing with cyclic cohomology for both closed manifolds and compact manifolds with boundary. We first give a sufficient geometric condition for the vanishing of the $l^1$-higher…

We develop two connections between the quantitative framework of operator $K$-theory for geometric $C^*$-algebras and the problem of positive scalar curvature. First, we introduce a quantitative notion of higher index and use it to give a…

We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator $D_m$ on the maximal Connes-Skandalis Hilbert module and…

In this paper, we first establish an $S^1$-equivariant index theorem for Spin$^c$ Dirac operators on $\mathbb{Z}/k$ manifolds, then combining with the methods developed by Taubes \cite{MR998662} and Liu-Ma-Zhang \cite{MR1870666,MR2016198},…

In this article, we explore the boundedness properties of pseudo-differential operators on radial sections of line bundles over the Poincar\'e upper half plane, even when dealing with symbols of limited regularity. We first prove the…

In this paper we discuss the index problem for geometric differential operators (Spin-Dirac operator, Gau{\ss}-Bonnet operator, Signature operator) on manifolds with metric horns. On singular manifolds these operators in general do not have…

We describe a topological predual to differential forms constructed as an inductive limit of a sequence of Banach spaces. This subspace of currents has nice properties, in that Dirac chains and polyhedral chains are dense, and its operator…

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as Andr\'e-Oort conjecture. Among the many tools developed in…

In this lecture we review apprearance of the Riemann-Roch Theorem in classical function theory, Algebraic topology, in theory of pseudo-differential operators and finally in noncommutative geometry. We show also it usefulness in many…

Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C^*_rG)\to \mathbb{C}$ defined by orbital integrals to recover group theoretic information about $G$, including information contained in $K$-theory classes not…

We define two types of pseudo-differential perturbations of the Dirac operator within the framework of the noncommutative geometry. And we obtain the noncommutative residue of the inverse square of these perturbations on 4-dimensional…

In view of the Segal construction each category with a coherent operation gives rise to a cohomology theory. Similarly each open stable differential relation $R$ imposed on smooth maps of manifolds determines cohomology theories $k^*$ and…