In this paper, we consider non-self-adjoint Dirac operators on a finite interval with complex-valued potentials and quasi-periodic boundary conditions. Necessary and sufficient conditions for a set of complex numbers to be the spectrum of the indicated problem are established.

Let $\mathbb{L}$ be a length function on a group $G$, and let ${{M}_{\mathbb{L}}}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on ${{\ell }^{2}}\left( G \right)$. Following Connes, ${{M}_{\mathbb{L}}}$ can be used as a “Dirac” operator for the reduced group ${{C}^{*}}$-algebra $C_{r}^{*}\left( G \right)$. It defines a Lipschitz seminorm on $C_{r}^{*}\left( G \right)$, which defines a metric on the state space of $C_{r}^{*}\left( G \right)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

We decompose θ(M), the twisted index obstruction to a positive scalar curvature metric for closed oriented manifolds with spin universal cover, into a pairing of a twisted K-homology with a twisted K-theory class and prove that θ(M) does not vanish if M is a closed orientable enlargeable manifold with spin universal cover.

We study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.

We study the spectral stability of solitary wave solutions to the nonlinear Diracequation in one dimension. We focus on the Dirac equation with cubic nonlinearity, knownas the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model.Presented numerical computations of the spectrum of linearization at a solitary wave showthat the solitary waves are spectrally stable. We corroborate our results by findingexplicit expressions for several of the eigenfunctions. Some of the analytic results holdfor the nonlinear Dirac equation with generic nonlinearity.

Twisted K-theory classes over compact Lie groups can be realized as families of Fredholm operators using the representation theory of loop groups. In this paper we show how to deform the Fredholm family in the sense of quantum groups. The family of Dirac-type operators is parametrized by vectors in the adjoint module for a quantum affine algebra and transforms covariantly under a central extension of the algebra.

We explain an array of basic functional analysis puzzles on the way to general spectral flow formulae and indicate a direction of future topological research for dealing with these puzzles.

We prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic (scattering) metric, which have the form D + iΦ, where D is elliptic pseudodifferential with Hermitian symbols, and Φ is a Hermitian bundle endomorphism which is invertible at the boundary and commutes with the symbol of D there. The index of such operators is completely determined by the symbolic data over the boundary. We use the scattering calculus of R. Melrose in order to prove our results using methods of topological K-theory, and we devote special attention to the case in which D is a family of Dirac operators, in which case our theorem specializes to give family versions of the previously known index formulas.

When the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the K—theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreover, the usual spectral assumption on the Novikov-Shubin invariants of the operator is improved.