Related papers: Some questions about the index of quantized contac…

In this work we define operator-valued Fourier transforms for suitable integrable elements with respect to the Plancherel weight of a (not necessarily Abelian) locally compact group. Our main result is a generalized version of the Fourier…

The notion of pseudo-differential operators with coefficients in a continuous trace algebra over a manifold are introduced and their index theory is studied. The algebra of principal symbols in this calculus provides an abstract Poincar\'e…

We consider manifolds endowed with a contact pair structure. To such a structure are naturally associated two almost complex structures. If they are both integrable, we call the structure a normal contact pair. We generalize the Morimoto's…

A linear or multi-linear valuation on a finite abstract simplicial complex can be expressed as an analytic index dim(ker(D)) -dim(ker(D^*)) of a differential complex D:E -> F. In the discrete, a complex D can be called elliptic if a…

A geometric model for twisted $K$-homology is introduced. It is modeled after the Mathai-Melrose-Singer fractional analytic index theorem in the same way as the Baum-Douglas model of $K$-homology was modeled after the Atiyah-Singer index…

This paper establishes the orderability of contact manifolds which are quotients of fillable contact manifolds under finite group actions compatible with the filling, the prototypical example being $\mathbb{R}P^{2n-1}$ as the quotient of…

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of…

Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral…

We define Maslov--type indices associated to contact and symplectic transformation groups. There are two such families of indices. The first class of indices are maps from the homotopy groups of the contactomorphism or symplectomorphism…

We describe how Lie groupoids are used in singular analysis, index theory and non-commutative geometry and give a brief overview of the theory. We also expose groupoid proofs of the Atiyah-Singer index theorem and discuss the Baum-Connes…

We present an index theorem for certain hypoelliptic differential operators on foliated manifolds. Our proof is a development of Alain Connes tangent groupoid proof of the Atiyah-Singer index theorem. The paper is largely self-contained.

Given two closed contact three-manifolds, one can form their contact connected sum via the Weinstein one-handle attachment. We study how pseudo-holomorphic curves in the symplectization behave under this operation. As a result, we give a…

We give an integral formula for the total $Q^\prime$-curvature of a three-dimensional CR manifold with positive CR Yamabe constant and nonnegative Paneitz operator. Our derivation includes a relationship between the Green's functions of the…

We consider the index of a Dirac operator on a compact even dimensional manifold with a domain wall. The latter is defined as a co-dimension one submanifold where the connection jumps. We formulate and prove an analog of the…

In this note we present a formula for the equivariant index of the cohomological complex obtained from localization of $\mathcal{N}=2$ SYM on simply-connected compact four-manifolds with a $T^2$-action. Knowledge of said index is essential…

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category…

We formulate and prove a generalization of the Atiyah-Singer family index theorem in the context of the theory of spaces of manifolds \`a la Madsen, Tillmann, Weiss, Galatius and Randal-Williams. Our results are for Dirac-type operators…

This paper gives a survey of the index theory of tangentially elliptic and transversally elliptic operators on foliated manifolds as well as of related notions and results in non-commutative geometry.

Let M be a compact manifold with a fixed spin structure \chi. The Atiyah-Singer index theorem implies that for any metric g on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending…

The $L^2$-Index Theorem of Atiyah \cite{atiyah} expresses the index of an elliptic operator on a closed manifold $M$ in terms of the $G$-equivariant index of some regular covering $\widetilde{M}$ of $M$, with $G$ the group of covering…