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We study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or “Dirac points”. We then show that the introduction of an “edge”, via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized “edge states”. These bound states are associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. Our model captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene. The states we construct can be realized as highly robust TM-electromagnetic modes for a class of photonic waveguides with a phase-defect.

We prove that the two-dimensional Schrödinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (Dirac points) at the vertices of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb structures is associated with many novel electronic and optical properties of materials such as graphene.

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric inn+2dimensions that encodes a conformal class of metrics inndimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric inn+1dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics. The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.

We study a class of periodic Schrödinger operators on ℝ that have Dirac points. The introduction of an “edge” via adiabatic modulation of a periodic potential by a domain wall results in the bifurcation of spatially localized “edge states,” associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. The bound states we construct can be realized as highly robust transverse-magnetic electromagnetic modes for a class of photonic waveguides with a phase defect. Our model captures many aspects of the phenomenon of topologically protected edge states for 2D bulk structures such as the honeycomb structure of graphene.

In this paper, we study the quotient space of equivalence classes of ‐tuples in a metric space , equipped with the metric induced by the minimal total pairing distance. Given a continuous path , we prove that there exist continuous functions such that for every , the multiset represents . That is, lifts to a continuous path in . Furthermore, we prove that if is Hölder continuous of order , the lift is Hölder continuous of the same order. The proof proceeds by induction on , with the fundamental step of showing that local lifts exist around all configurations, including those with point collisions.

We exhibit optimal control strategies for a simple toy problem in which the underlying dynamics depend on a parameter that is initially unknown and must be learned. We consider a cost function posed over a finite time interval, in contrast to much previous work that considers asymptotics as the time horizon tends to infinity. We study several different versions of the problem, including Bayesian control, in which we assume a prior distribution on the unknown parameter; and “agnostic” control, in which we assume nothing about the unknown parameter. For the agnostic problems, we compare our performance with that of an opponent who knows the value of the parameter. This comparison gives rise to several notions of “regret”, and we obtain strategies that minimize the “worst-case regret” arising from the most unfavorable choice of the unknown parameter. In every case, the optimal strategy turns out to be a Bayesian strategy or a limit of Bayesian strategies.

For any h∈(1,2], we give an explicit construction of a compactly supported, uniformly continuous, and (weakly) divergence-free velocity field in R2 that weakly advects a measure whose support is initially the origin but for positive times has Hausdorff dimension h. These velocities are uniformly continuous in space-time and compactly supported, locally Lipschitz except at one point and satisfy the conditions for the existence and uniqueness of a Regular Lagrangian Flow in the sense of Di Perna and Lions theory. We then construct active scalar systems in R2 and R3 with measure-valued solutions whose initial support has co-dimension 2 but such that at positive times it only has co-dimension 1. The associated velocities are divergence free, compactly supported, continuous, and sufficiently regular to admit unique Regular Lagrangian Flows. This is in part motivated by the investigation of dimension conservation for the support of measure-valued solutions to active scalar systems. This question occurs in the study of vortex filaments in the three-dimensional Euler equations.

The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for testing the existence of a manifold that fits a probability distribution supported in a separable Hilbert space, only using i.i.d. samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution P\\mathcal {P} supported on the unit ball of a separable Hilbert space H\\mathcal {H}. Let G(d,V,τ)\\mathcal {G}(d, V, \\tau ) be the set of submanifolds of the unit ball of H\\mathcal {H} whose volume is at most VV and reach (which is the supremum of all rr such that any point at a distance less than rr has a unique nearest point on the manifold) is at least τ\\tau. Let L(M,P)\\mathcal {L}(\\mathcal {M}, \\mathcal {P}) denote the mean-squared distance of a random point from the probability distribution P\\mathcal {P} to M\\mathcal {M}. We obtain an algorithm that tests the manifold hypothesis in the following sense. The algorithm takes i.i.d. random samples from P\\mathcal {P} as input and determines which of the following two is true (at least one must be): There exists M∈G(d,CV,τC)\\mathcal {M} \\in \\mathcal {G}(d, CV, \\frac {\\tau }{C}) such that L(M,P)≤Cϵ.\\mathcal {L}(\\mathcal {M}, \\mathcal {P}) \\leq C {\\epsilon }. There exists no M∈G(d,V/C,Cτ)\\mathcal {M} \\in \\mathcal {G}(d, V/C, C\\tau ) such that L(M,P)≤ϵC.\\mathcal {L}(\\mathcal {M}, \\mathcal {P}) \\leq \\frac {\\epsilon }{C}. The answer is correct with probability at least 1−δ1-\\delta.