Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

Funder: Comunidad de Madrid; Id: http://dx.doi.org/10.13039/100012818Funder: Consejería de Educación e Investigación; Id: http://dx.doi.org/10.13039/501100010774Funder: Engineering and Physical Sciences Research Council; Id: http://dx.doi.org/10.13039/501100000266Funder: Cantab Capital Institute for the Mathematics of InformationFunder: Cambridge Centre for AnalysisAbstract: We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, exp ( n s z ) , over the interval [ − 1 , 1 ] , where s ∈ C is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, s ∈ i R , due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as n → ∞ have recently been studied for s ∈ i R , and our main goal is to extend these results to all s in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so‐called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter s is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter s approaches a breaking curve, by considering double scaling limits as s approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points s = ± 2 or some other points on the breaking curve

Inflation:Generic predictions and nilpotent superfields

Met de observaties van de Planck satelliet begint de kosmologie aan een nieuw tijdperk waarin het universum bestudeerd kan worden met hoge precisie. Dit levert interessante informatie op over het zeer vroege universum, inclusief de inflatieperiode. Deze kosmologische inflatie werd in dit doctoraat op twee manieren theoretisch bestudeerd. Met de eerste methode wordt een groot aantal inflatiemodellen met de data van de CMB, zoals gemeten door de Planck satelliet, met elkaar vergeleken. Hieruit kunnen generieke voorspellingen worden afgeleid voor de verschillende parametrisaties waarmee deze modellen zijn verkregen. Door verschillende parametrisaties van de potentiaal te vergelijken, wordt geconcludeerd dat modellen behorende tot de groep van de plateau inflatiemodellen beter overeenkomen met de CMB dan de zogeheten polynomische modellen. De tweede benadering bestudeert inflatiemodellen in supergravitatie, hetgeen een extensie is van zowel het standaardmodel van de deeltjesfysica als van de algemene relativiteitstheorie door middel van een nieuwe symmetrie genaamd supersymmetrie. In dit onderzoek bestuderen we een bepaalde inbedding van inflatie in supergravitatie en bestuderen we de theoretische consistentie. Daarnaast zijn we in staat om donkere materie, een ander probleem in de kosmologie, te bestuderen. In een deel van de parameterruimte van ons model wordt deze donkere materie verklaard door een deeltje in de supersymmetrische sector. Hierdoor zijn wij in staat om in ons model zowel supersymmetriebreking, inflatie als donkere materie te beschrijven

A Riemann-Hilbert Approach to the Kissing Polynomials

Motivated by the numerical treatment of highly oscillatory integrals, this thesis studies a family of polynomials known as the Kissing Polynomials through Riemann-Hilbert techniques. The Kissing Polynomials are a family of non-Hermitian orthogonal polynomials, which are orthogonal with respect to the complex weight function $\exp(i\omega z)$ over the interval $[-1,1]$, where $\omega>0$. Although they have already been used to derive complex quadrature rules, there remain two main questions which this thesis addresses. The first is the existence of such polynomials; the second is the behavior of these polynomials throughout the complex plane. The first two chapters of this thesis provide the necessary background needed for the main results presented in the later chapters. In the first chapter, the connection between the numerical integration of highly oscillatory integrals and the Kissing Polynomials is established. Furthermore, we present the theory of non-Hermitian orthogonal polynomials and provide a more detailed description of the results in this thesis. The second chapter is a review on the formulation of the Kissing Polynomials as a solution to a matrix valued Riemann-Hilbert problem. This formulation is crucial to establishing both the existence of the Kissing Polynomials and its properties throughout the complex plane. Moreover, we also provide an overview of the powerful non-commutative steepest descent technique developed by Deift and Zhou in the mid 1990s used to compute the asymptotics for oscillatory Riemann-Hilbert problems, which will be used extensively in Chapters 4 and 5. In Chapter 3, we utilize the Riemann-Hilbert approach of Fokas, Its, and Kitaev to establish our first main result: the existence of the even degree Kissing polynomials for all values of $\omega>0$. First, we use the Riemann-Hilbert problem to show that the Kissing Polynomials satisfy a certain linear ordinary differential equation. Then, using standard results on differential equations, along with previous results on the Kissing Polynomials found in the literature, we are able to provide the desired result. In Chapter 4, we turn our attention to the behavior of the Kissing Polynomials as both the degree $n$ and parameter $\omega$ become large. To achieve this, we formulate this problem in terms of varying-weight Kissing polynomials, whose asymptotics can be handled with the Deift-Zhou steepest descent analysis. Now, the weight function depends now on $n$, the degree of the underlying polynomial. We are able to provide uniform asymptotics of the Kissing Polynomials as both $n$ and $\omega$ go to infinity at a linear rate such that the ratio $\omega/n >t_c$, where $t_c$ is a to be specified critical value. In Chapter 5, we generalize the results of Chapter 4 and study polynomials which are orthogonal with respect to the varying, complex weight, $\exp(n s z)$, over the interval $[-1,1]$, where now $s\in\mathbb{C}$. We will see that there are curves in the $s$-plane, called breaking curves, which separate regions corresponding to differing asymptotic behavior of the polynomials. In this chapter, we provide the large $n$ behavior of the recurrence coefficients associated to these polynomials. Finally, we also study the behavior of these recurrence coefficients as the parameter $s$ approaches a breaking curve in a specified double scaling limit.PhD Studentship: Cantab Capital Institute for the Mathematics of Informatio

Opérateurs monopôles dans les transitions hors d'un liquide de spin de Dirac

Dans la description à basse énergie de systèmes fortement corrélés, les champs de jauge peuvent émerger comme excitations collectives couplées à des quasiparticules fractionalisées. En particulier, certains aimants bidimensionnels dits frustrés sont décrits par un liquide de spin de Dirac comportant une symétrie de jauge U(1) compacte. La description infrarouge est donnée par une théorie conforme des champs, soit l'électrodynamique quantique en 2+1 dimensions avec 2N saveurs de fermions sans masse. Dans les aimants typiques, N=2 ou 4. L'aspect compact du champ de jauge implique également l'existence d'excitations topologiques, soit des instantons créés, dans ce contexte, par des opérateurs monopôles. Cette thèse porte sur les transitions de phase quantiques à partir d'un liquide de spin de Dirac et les propriétés des monopôles aux points critiques correspondants. Ces transitions sont induites en activant diverses interactions de type Gross-Neveu. Dans tous les cas à l'étude, la dimension d'échelle des monopôles est obtenue grâce à la correspondance état-opérateur et à un développement en 1/N. L'accent est d'abord mis sur une transition de confinement-déconfinement vers une phase antiferromagnétique décrite par la condensation d'un monopôle. Une levée de dégénérescence est observée au point critique alors que certaines dimensions d'échelle de monopôles sont réduites par rapport à leur valeur dans le liquide de spin de Dirac. Cette hiérarchie est caractérisée quantitativement en comparant les dimensions d'échelle dans des secteurs distincts du spin magnétique à l'ordre dominant en 1/N, puis qualitativement par une analyse en théorie des représentations. Des exposants critiques pour d'autres observables dans la théorie non compacte sont également obtenus. Enfin, deux transitions vers des liquides de spin topologiques, soit le liquide de spin chiral et le liquide de spin Z2, sont considérées. Les dimensions anormales des monopôles sont obtenues à l'ordre sous-dominant en 1/N. Ces résultats permettent de vérifier une dualité conjecturée avec un modèle bosonique et la valeur d'un coefficient universel pour les théories de jauge U(1)In strongly correlated systems, gauge fields can emerge as collective excitations coupled to fractionalized quasiparticles. In particular, certain frustrated two-dimensional quantum magnets are described by a Dirac spin liquid which has a U(1) gauge symmetry. The infrared description is given by a conformal field theory, namely quantum electrodynamics in 2+1 dimensions with 2N flavours of massless fermions. In typical magnets, N=2 or 4. The compact aspect of the gauge field also implies the existence of topological excitations corresponding to instantons, which are created by monopole operators in this context. This thesis focuses on quantum phase transitions out of a Dirac spin liquid and the properties of monopoles at the corresponding critical points. These transitions are driven by activating various types of Gross-Neveu interactions. In all the cases studied, the scaling dimension of monopoles are obtained using the state-operator correspondence and a 1/N expansion. The confinement-deconfinement transition to an antiferromagnetic order produced by a monopole condensate is first studied. A degeneracy lifting is observed at the critical point, as certain monopoles have their scaling dimension reduced in comparison with the value in the Dirac spin liquid. This hierarchy is charactized quantitatively by comparing monopole scaling dimensions in distinct magnetic spin sector at leading-order in 1/N, and qualitatively by an analysis in representation theory. Critical exponents of various other operators are obtained in the non-compact model. Transitions to two topological spin liquids, namely a chiral spin liquid and a Z2 spin liquid, are also considered. Anomalous dimensions of monopoles are obtained at sub-leading order in 1/N. These results allow the verification of a conjectured duality with a bosonic model and the value of a universal coefficient in U(1) gauge theories

Recent Advances in Industrial and Applied Mathematics

This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress

An automatic PML for acoustic finite element simulations in convex domains of general shape

International audienceThis article addresses the efficient finite element solution of exterior acoustic problems with truncated computational domains surrounded by perfectly matched layers (PMLs). The PML is a popular nonreflecting technique that combines accuracy, computational efficiency, and geometric flexibility. Unfortunately, the effective implementation of the PML for convex domains of general shape is tricky because of the geometric parameters that are required to define the PML medium. In this work, a comprehensive implementation strategy is proposed. This approach, which we call the automatically matched layer (AML) implementation, is versatile and fully automatic for the end‐user. With the AML approach, the mesh of the layer is extruded, the required geometric parameters are automatically obtained during the extrusion step, and the practical implementation relies on a simple modification of the Jacobian matrix in the elementwise integrals. The AML implementation is validated and compared with other implementation strategies using numerical benchmarks in two and three dimensions, considering computational domains with regular and nonregular boundaries. A three‐dimensional application with a generally shaped domain generated using a convex hull is proposed to illustrate the interest of the AML approach for realistic industrial cases

Recent Advances in Industrial and Applied Mathematics

This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress