Related papers: A Callias-type index theorem with degenerate poten…
We consider a complete Riemannian manifold M whose boundary is a disjoint union of finitely many complete connected Riemannian manifolds. We compute the index of a local boundary value problem for a strongly Callias-type operator on M. Our…
We provide Fredholm conditions for compatible differential operators on certain Lie manifolds (that is, on certain possibly non-compact manifolds with nice ends). We discuss in more detail the case of manifolds with cylindrical, hyperbolic,…
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.
We examine the noncommutative index theory associated to the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials, and…
We consider a complete Riemannian manifold, which consists of a compact interior and one or more $\varphi$-cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here $\varphi$ is a function of the radial…
A semiclassical argument is used to show that the low-lying spectrum of a selfadjoint operator, the so-called spectral localizer, determines the number of Dirac or Weyl points of an ideal semimetal. Apart from the IMS localization…
We show that for an associative algebra A and its ideal I such that the I-adic topology on A coincides with the p-adic topology, the relative continuous K-theory pro-spectrum "lim"K(A_i, IA_i), where A_i :=A/p^i A, is naturally isogenous to…
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…
In this paper and its sequel we consider locally-free $\mathscr{O}_X$-modules together with a connection over a quasi-smooth Berkovich curve $X$. We obtain necessary and sufficient conditions for the finite dimensionality of their de Rham…
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.
We establish an $L^2$-Gamma index theorem for the Dirac operator on a globally hyperbolic manifold $M$ with Cauchy hypersurface $\Sigma$ being a Galois covering of a compact smooth manifold with Galois group $\Gamma$. Our argument rewrites…
Motivated by the study of layer potentials on manifolds with straight conical or cylindrical ends, we introduce and study two classes (or calculi) of pseudodifferential operators defined on manifolds with cylindrical ends: the class of…
It is well known that the integral identity conjecture is of prime importance in Kontsevich-Soibelman's theory of motivic Donaldson-Thomas invariants for non-commutative Calabi-Yau threfolds. In this article we consider its numerical…
In a previous work, the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. This index is refined to obtain a Conley index for families of…
We introduce a new fixed point theorem of Krasnoselskii type for discontinuous operators. As an application we use it to study the existence of positive solutions of a second-order differential problem with separated boundary conditions and…
We give a proof of the cobordism invariance of the index of elliptic pseudodifferential operators on sigma-compact manifolds, where, in the non-compact case, the operators are assumed to be multiplication outside a compact set. We show…
We study quantitative unique continuation at infinity for Dirac equations with bounded matrix-valued potentials. For the massless Dirac operator $\mathcal{D}_n$ in $\mathbb{R}^n$, we establish a Landis-type estimate showing that the…
New formulas on the inverse problem for the continuous skew-self-adjoint Dirac type system are obtained. For the discrete skew-self-adjoint Dirac type system the solution of a general type inverse spectral problem is also derived in terms…
In this paper we solve the general case of the cohomological relative index problem for foliations of non-compact manifolds. In particular, we significantly generalize the groundbreaking results of Gromov and Lawson, [GL83], to Dirac…
Let $G$ be a connected, real semisimple Lie group. Let $K<G$ be maximal compact, and let $\Gamma < G$ be discrete and such that $\Gamma \backslash G$ has finite volume. If the real rank of $G$ is $1$ and $\Gamma$ is torsion-free, then…