Related papers: Coronas and Callias type operators in coarse geome…

A Dirac operator D on quantized irreducible generalized flag manifolds is defined. This yields a Hilbert space realization of the covariant first-order differential calculi constructed by I. Heckenberger and S. Kolb. All differentials…

This monograph explores classification and perturbation problems for integrable systems on a class of Poisson manifolds called $b^m$-Poisson manifolds. Even if the class of $b^m$-Poisson manifolds is not ample enough to represent general…

We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential $K$-theory, or…

In this paper, we introduce several new secondary invariants for Dirac operators on a complete Riemannian manifold with a uniform positive scalar curvature metric outside a compact set and use these secondary invariants to establish a…

Chiral differential operators (CDOs) are closely related to string geometry and the quantum theory of two-dimensional sigma models. This paper investigates two topics about CDOs on smooth manifolds. In the first half, we study how a Lie…

If we are given a smooth differential operator in the variable $x\in {\mathbb R}/2\pi {\mathbb Z},$ its normal form, as is well known, is the simplest form obtainable by means of the $\mbox{Diff}(S^1)$-group action on the space of all such…

We study the index bundle of the Dirac-Ramond operator associated with a family $\pi: Z \to X$ of compact spin manifolds. We view this operator as the formal twisted Dirac operator $\dd \otimes \bigotimes_{n=1}^{\infty}S_{q^n}TM_{\C}$ so…

We define prequantization for Dirac manifolds to generalize known procedures for Poisson and (pre) symplectic manifolds by using characteristic distributions obtained from 2-cocycles associated to Dirac structures. Given a Dirac manifold…

Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the K-theory of the corresponding reduced C*-algebras.…

In this paper, we define a relative $L^2$-$\rho$-invariant for Dirac operators on odd-dimensional spin manifolds with boundary and show that they are invariants of the bordism classes of positive scalar curvature metrics which are collared…

We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of…

This is a survey of the current state of the question "Which closed connected manifolds of dimension $n\ge 5$ admit Riemannian metrics whose scalar curvature function is everywhere positive?" The introduction gives a brief overview of these…

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We…

We give a superconnection proof of the cohomological form of Mathai-Melrose-Singer index theorem for the family of twisted Dirac operators under relaxed conditions.

In this survey article, given a smooth closed manifold M we study the space of Riemannian metrics of positive scalar curvature on M. A long-standing question is: when is this space non-empty (i.e. when does M admit a metric of positive…

The problem of self-adjoint extensions of Dirac-type operators in manifolds with boundaries is analysed. The boundaries might be regular or non-regular. The latter situation includes point-like interactions, also called delta-like…

In this work we characterise the H\"ormander classes $\symbClassOn{m}{\rho}{\delta}{\group,\textnormal{H\"or}}$ on the open manifold $\group = (-1,1)^n$. We show that by endowing the open manifold $\group = (-1,1)^n$ with a group structure,…

The corona problem was motivated by the question of the density of the open unit disk D in the maximal ideal space of the algebra, H1(D), of bounded holomorphic functions on D. In this note we study relationships of the problem with…

We give a detailed microlocal study of X-ray transforms over geodesics-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator.…

On differential manifolds with spinor structure, it is possible to express the Euler and Pontryagin currents in terms of tensors that also appear as source in the Dirac equation. It is hence possible to tie concepts rooted in geometry and…