Narrow Results

This research comprehensively describes the basic theory of transversally Heisenberg elliptic operators, and investigates the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators from the perspective of $KK$-theory, applying Kasparov’s methodology. Moreover, the analysis methodically examines specific conditions, with a focus on the Fourier transform of the nilpotent group $C^{\ast }$-algebra. We demonstrate enhanced methods for analyzing the hypoellipticity of operators, presenting a robust framework for defining and understanding transversal Heisenberg ellipticity in a $KK$-theoretic context. This work provides a solid foundation for future research into the properties of hypoelliptic differential operators in complicated manifolds.

We study the classification of D-branes and Ramond–Ramond fields in Type I string theory by developing a geometric description of KO-homology. We define an analytic version of KO-homology using KK-theory of real C*-algebras, and construct explicitly the isomorphism between geometric and analytic KO-homology. The construction involves recasting the Cℓ_n-index theorem and a certain geometric invariant into a homological framework which is used, along with a definition of the real Chern character in KO-homology, to derive cohomological index formulas. We show that this invariant also naturally assigns torsion charges to non-BPS states in Type I string theory, in the construction of classes of D-branes in terms of topological KO-cycles. The formalism naturally captures the coupling of Ramond–Ramond fields to background D-branes which cancel global anomalies in the string theory path integral. We show that this is related to a physical interpretation of bivariant KK-theory in terms of decay processes on spacetime-filling branes. We also provide a construction of the holonomies of Ramond–Ramond fields in Type II string theory in terms of topological K-chains.

This paper is a survey of the ${\mathbb{Z}}_2$-valued invariant of topological insulators used in condensed matter physics. The $\mathbb{Z}$-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The ${\mathbb{Z}}_2$ invariant is more mysterious; we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the classification of topological insulators with different symmetries in which K-theory plays an important role. Moreover, we establish that both invariants are realizations of index theorems which can also be understood in terms of condensed matter physics.

We define the secondary invariants L²-eta and -rho forms for families of generalized Dirac operators on normal coverings of fiber bundles. On the covering family we assume transversally smooth spectral projections and Novikov–Shubin invariants bigger than 3(dim B + 1) to treat the large time asymptotic for the heat operator. In the case of a bundle of spin manifolds, we study the L²-rho class in relation to the space of positive scalar curvature vertical metrics.

In 1978, Rabinowitz proved the existence of a non-constant $T$-periodic solution for nonlinear Hamiltonian systems on ${\mathbf{\text{R}}}^{2n}$ with Hamiltonian function being super-quadratic at the infinity and zero for any given $T>0$. Since the minimal period of this solution may be $T/k$ for some positive integer $k$, he proposed the question whether there exists a solution with $T$ as its minimal period for such a Hamiltonian system. This is the so-called Rabinowitz minimal periodic solution conjecture. In the last more than 40 years, this conjecture has been deeply studied by many mathematicians. But under the original structural conditions of Rabinowitz, the conjecture is still open when $n\ge 2$. In this paper, I give a brief survey on the studies of this conjecture and hope to lead to more interests on it.

Several known results, by Rivin, Calegari-Maher and Sisto, show that an element φ_n ∈ Out(F_r), obtained after n steps of a simple random walk on Out(F_r), is fully irreducible with probability tending to 1 as n → ∞. In this paper, we construct a natural "train track directed" random walk 𝒲 on Out(F_r) (where r ≥ 3). We show that, for the element φ_n ∈ Out(F_r), obtained after n steps of this random walk, with asymptotically positive probability the element φ_n has the following properties: φ_n is an ageometric fully irreducible, which admits a train track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that φ_n has "rotationless index" (so that the geometric index of the attracting tree T_φn of φ_n is 2r - 3), has index list and the ideal Whitehead graph being the complete graph on 2r - 1 vertices, and that the axis bundle of φ_n in the Outer space CV_r consists of a single axis.

To generalize the Hopf index theorem and the Atiyah–Dupont vector fields theory, one is interested in the following problem: for a real vector bundle E over a closed manifold M with rank E = dim M, whether there exist two linearly independent cross sections of E? We provide, among others, a complete answer to this problem when both E and M are orientable. It extends the corresponding results for E = TM of Thomas, Atiyah, and Atiyah–Dupont. Moreover we prove a vanishing result of a certain mod 2 index when the bundle E admits a complex structure. This vanishing result implies many known famous results as consequences. Ideas and methods from obstruction theory, K-theory and index theory are used in getting our results.

In this paper, we consider the existence and multiplicity of nontrivial solutions to a quadratically coupled Schrödinger system

In this note, we resume the geometric quantization approach to the motion of a charged particle on a plane, subject to a constant magnetic field perpendicular to the latter, by showing directly that it gives rise to a completely integrable system to which we may apply holomorphic geometric quantization. In addition, we present a variant employing a suitable vertical polarization and we also make contact with Bott’s quantization, enforcing the property “quantization commutes with reduction”, which is known to hold under quite general conditions. We also provide an interpretation of translational symmetry breaking in terms of coherent states and index theory. Finally, we give a representation theoretic description of the lowest Landau level via the use of an $S^1$-equivariant Dirac operator.

The extended Heisenberg algebra for a contact manifold has a symbolic calculus that accommodates both Heisenberg pseudodifferential operators as well as classical pseudodifferential operators. We derive here a formula for the index of Fredholm operators in this extended calculus. This formula incorporates in a single expression the Atiyah–Singer formula for elliptic operators, as well as Boutet de Monvel's Toeplitz index formula.

We relate the spectral flow to the index for paths of selfadjoint Breuer–Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin–Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold when the metric is varied. We conclude that the tangential signature of a foliated manifold with boundary does not depend on the metric. In the Appendix we reconsider integral formulas for the spectral flow of paths of bounded operators.

The main result of this paper is a new Atiyah–Singer type cohomological formula for the index of Fredholm pseudodifferential operators on a manifold with boundary. The nonlocality of the chosen boundary condition prevents us to apply directly the methods used by Atiyah and Singer in [4, 5]. However, by using the K-theory of C*-algebras associated to some groupoids, which generalizes the classical K-theory of spaces, we are able to understand the computation of the APS index using classic algebraic topology methods (K-theory and cohomology). As in the classic case of Atiyah–Singer ([4, 5]), we use an embedding into a Euclidean space to express the index as the integral of a true form on a true space, the integral being over a C^∞-manifold called the singular normal bundle associated to the embedding. Our formula is based on a K-theoretical Atiyah–Patodi–Singer theorem for manifolds with boundary that is inspired by Connes' tangent groupoid approach, it is not a groupoid interpretation of the famous Atiyah–Patodi–Singer index theorem.

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert $C^{\ast }$-module bundles and prove an index theorem for such operators. As an application, we derive an obstruction to the existence of complete Riemannian metrics of positive scalar curvature on noncompact spin manifolds in terms of closed submanifolds of codimension one. In particular, when $N$ is a closed spin manifold, we show that if the cylinder $N\times ℝ$ carries a complete metric of positive scalar curvature, then the (complex) Rosenberg index on $N$ must vanish.

While indices, index tracking funds and ETFs have grown in popularity during then last ten years, there are many structural problems, tracking errors and biases inherent in index calculation methodologies and the legal/economic structure of ETFs, which raise actionable issues of "suitability" and "fraud" under US securities laws. This article contributes to the existing literature by: (a) introducing and characterizing the errors and biases inherent in "risk-adjusted" index-weighting methods and the associated adverse effects; (b) showing how these biases/effects reduce social welfare, and can facilitate harmful arbitrage activities; (c) introducing new theorems.

We review our recent understandings on the linear stability of periodic orbits of the n-body problem through their variational characterizations. These two aspects are put together by the index theory of Hamiltonian system. Along the way, many challenging problems and further possible extensions are presented.