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 Vasy, András, author.
 Providence, Rhode Island : American Mathematical Society, [2015]
 Description
 Book — x, 281 pages : illustrations ; 27 cm.
 Summary

 * Introduction* Where do PDE come from* First order scalar semilinear equations* First order scalar quasilinear equations* Distributions and weak derivatives* Second order constant coefficient PDE: Types and d'Alembert's solution of the wave equation* Properties of solutions of second order PDE: Propagation, energy estimates and the maximum principle* The Fourier transform: Basic properties, the inversion formula and the heat equation* The Fourier transform: Tempered distributions, the wave equation and Laplace's equation* PDE and boundaries* Duhamel's principle* Separation of variables* Inner product spaces, symmetric operators, orthogonality* Convergence of the Fourier series and the Poisson formula on disks* Bessel functions* The method of stationary phase* Solvability via duality* Variational problems* Bibliography* Index.
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MATH17301
 Course
 MATH17301  Theory of Partial Differential Equations
 Instructor(s)
 Fredrickson, Laura Joy
 Vasy, András.
 Paris : Société mathématique de France, 2000.
 Description
 Book — iv, 151 p : ill. ; 24 cm.
 Online
SAL1&2 (oncampus shelving)
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Shelved by Series title V.262  Unknown 
 Melrose, Richard B.
 Paris : Société mathématique de France, 2013.
 Description
 Book — vi, 135 pages : illustrations ; 24 cm.
 Online
SAL1&2 (oncampus shelving)
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Shelved by Series title V.351  Unknown 
 Norte, Richard A.
 [2007].
 Description
 Book — 14 leaves.
 Online
Special Collections
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4471 2007 N  Inlibrary use 
 Basel ; New York : Birkhäuser, ©2013.
 Description
 Book — 1 online resource.
 Summary

 Semiclassical and adiabatic limits
 Local Smoothing with a Prescribed Loss for the Schrödinger Equation / Hans Christianson, Jared Wunsch
 Propagation Through Trapped Sets and Semiclassical Resolvent Estimates / Kiril Datchev, András Vasy
 SpaceAdiabatic Theory for RandomLandau Hamiltonian: Results and Prospects / Giuseppe De Nittis
 Microlocal Analysis of FIOs with Singularities / Raluca Felea
 A Nonlinear Adiabatic Theorem for Coherent States / Clotilde Fermanian Kammerer, Rémi Carles
 Adiabatic Limits and Related Lattice Point Problems / Yuri A. Kordyukov, Andrey A. Yakovlev
 The Effective Hamiltonian in Curved Quantum Waveguides and When It Does Not Work / David Krejčiřík, Helena Šediváková
 The Adiabatic Limit of the Laplacian on Thin Fibre Bundles / Jonas Lampart, Stefan Teufel
 Adiabatic Limit with Isolated Degenerate Fibres / Richard B. Melrose
 Microlocal Analysis and Adiabatic Problems: The Case of Perturbed Periodic Schrödinger Operators / Gianluca Panati
 Recent Results in Semiclassical Approximation with Rough Potentials / T. Paul.
 Singular spaces
 On the Closure of Elliptic Wedge Operators / Juan B. Gil, Thomas Krainer, Gerardo A. Mendoza
 Generalized BlowUp of Corners and Fiber Products / Chris Kottke, Richard Melrose
 Trace Expansions for Elliptic Cone Operators / Thomas Krainer, Juan B. Gil, Gerardo A. Mendoza
 Spectral Geometry for the Riemann Moduli Space / Rafe Mazzeo
 Invariant Integral Operators on the Oshima Compactification of a Riemannian Symmetric Space: Kernel Asymptotics and Regularized Traces / Pablo Ramacher, Aprameyan Parthasarathy
 Pseudodifferential Operators on Manifolds with Foliated Boundaries / Frédéric Rochon
 The Determinant of the Laplacian on a Conically Degenerating Family of Metrics / David A. Sher
 Spectral and scattering theory
 Relatively Isospectral Noncompact Surfaces / Pierre Albin, Clara Aldana, Frédéric Rochon
 Microlocal Analysis of Scattering Data for Nested Conormal Potentials / Suresh Eswarathasan
 Equidistribution of Eisenstein Series for Convex Cocompact Hyperbolic Manifolds / Colin Guillarmou, Frédéric Naud.
 Lower Bounds for the Counting Function of an Integral Operator / Yuri Safarov
 The Identification Problems in SPECT: Uniqueness, Nonuniqueness and Stability / Plamen Stefanov
 Eigenvalues and Spectral Determinants on Compact Hyperbolic Surfaces / Alexander Strohmaier, Ville Uski
 Wave propagation and topological applications
 A Support Theorem for a Nonlinear Radiation Field / Dean Baskin, António Sá Barreto
 Propagation of Singularities Around a Lagrangian Submanifold of Radial Points / Nick Haber, András Vasy
 Local Energy Decay for Several Evolution Equations on Asymptotically Euclidean Manifolds / Dietrich Häfner, JeanFrançois Bony
 Rayleigh Surface Waves and Geometric Pseudodifferential Calculus / Sönke Hansen
 Topological Implications of Global Hypoellipticity / Gerardo A. Mendoza
 ChernSimons Line Bundle on Teichmüller Space / Sergiu Moroianu, Colin Guillarmou
 A Simple Diffractive Boundary Value Problem on an Asymptotically Antide Sitter Space / Ha Pham
 Quantization in a Magnetic Field / Radu Purice, Viorel Iftimie, Marius Măntoiu
 Price's Law on Black Hole SpaceTimes / Daniel Tataru.
Online 6. Boundary fibration structures and quasihomogeneous geometries [electronic resource] [2017]
 Thorvaldsson, Sverrir.
 2017.
 Description
 Book — 1 online resource.
 Summary

In this thesis we extend work by Mazzeo on conformally compact manifolds to a class of manifolds with quasihomogeneous geometries, which we call kappamanifolds. Our results show that there are complete noncompact manifolds of negative curvature, that have 0 in the essential spectrum for the Hodge Laplacian on forms, and this applies in a range of degrees centered at the middle degree. As is typical for boundary fibration structures our methods give much more, namely we provide a general framework to study elliptic partial differential operators on kappamanifolds based on microlocal methods. We construct a calculus of pseudodifferential operators on the manifold, and give precise conditions for the existence of a parametrix for elliptic differential operators in this calculus. This work takes up the bulk of the thesis. We then apply this to the spectral theory of the Hodge Laplacian on a kappamanifold. This step requires detailed analysis of the Hodge Laplacian on a simpler model space, which in turn requires detailed study of a system of ordinary differential equations.
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Online 7. Diffraction of elastic waves by edges [electronic resource] [2015]
 Katsnelson, Vitaly.
 2015.
 Description
 Book — 1 online resource.
 Summary

The purpose of this thesis is to investigate the diffraction of singularities of solutions to the linear elastic equation on manifolds with edge singularities. Such manifolds are modeled on the product of a smooth manifold and a cone over a compact fiber. For the fundamental solution, the initial pole generates a pressure wave (pwave), and a secondary, slower shear wave (swave). If the initial pole is appropriately situated near the edge, we show that when a pwave strikes the edge, the diffracted pwaves and swaves (i.e. loosely speaking, are not limits of prays which just miss the edge) generated from such an interaction are weaker in a Sobolev sense than the incident pwave. More generally, we show that subject to a "coinvolutivity" hypothesis, if a psingularity (or ssingularity) of any solution strikes the edge, the diffracted p and s wavefronts are smoother that the incident singularity.
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Online 8. Global analysis of linear and nonlinear wave equations on cosmological spacetimes [electronic resource] [2015]
 Hintz, Peter.
 2015.
 Description
 Book — 1 online resource.
 Summary

We develop a general framework for the global analysis of linear and nonlinear wave equations on geometric classes of Lorentzian manifolds, based on microlocal analysis on compactified spaces. The main examples of manifolds that fit into this framework are cosmological spacetimes such as de Sitter and Kerrde Sitter spaces, as well as Minkowski space, and perturbations of these spacetimes. In particular, we establish the global solvability of quasilinear wave equations on cosmological black hole spacetimes and obtain the asymptotic behavior of solutions using a novel approach to the global study of nonlinear hyperbolic equations. The framework directly applies to nonscalar problems as well, and we present linear and nonlinear results both for scalar equations and for equations on natural vector bundles. To a large extent, our work was motivated by the black hole stability problem for cosmological spacetimes, and we expect the resolution of this problem to be within reach with the methods presented here.
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Online 9. The vacuum Einstein constraint equations on manifolds with ends of cylindrical type [electronic resource] [2015]
 Leach, Jeremy.
 2015.
 Description
 Book — 1 online resource.
 Summary

This dissertation concerns the vacuum Einstein constraint equations on manifolds possessing end regions which are asymptotically periodic, including the special case where the ends are conformally asymptotically cylindrical. We will first apply the conformal method to construct a large class of vacuum initial data on any such manifold with positive Yamabe invariant, and then extend these existence results to manifolds which may also have asymptotically Euclidean ends. We will also show that, in the conformally asymptotically cylindrical case, any solution to the constraints one obtains via the conformal method which preserves the end geometry must have a unique asymptotic limit. Finally, we describe two different approaches to gluing generic asymptotically periodic initial data sets endtoend, thereby allowing us to construct a large family of initial data sets with ``long neck'' regions.
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Online 10. Microlocal analysis of lagrangian submanifolds of radial points [electronic resource] [2013]
 Haber, Nick.
 2013.
 Description
 Book — 1 online resource.
 Summary

Microlocal analysis relies on correspondences between quantum physics and classical physics to give information about certain PDEs  for instance, linear variablecoefficient PDEs on manifolds. PDEs are interpreted as quantum systems. The corresponding classical systems tell us, for example, function spaces on which problems are solvable or almost solvable, existence and uniqueness results, and the structure of solution operators. Landmark papers of Hörmander and Duistermaat and Hörmander establish key results for the standard calculus of microlocal analysis, which gives a broad framework for dealing with variablecoefficient PDEs on manifolds. Their work is wellsuited for dealing with PDEs which, in a generalized sense, are hyperbolic, with corresponding classical dynamics looking like wave propagation of geometric optics. In this thesis, we aim to extend many of their results to situations in which the corresponding classical dynamics are less wellbehaved: those with a Lagrangian submanifold of radial points.
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3781 2013 H  Inlibrary use 
 Sher, David Alexander.
 2012.
 Description
 Book — 1 online resource.
 Summary

We consider a family of smooth Riemannian manifolds which degenerate to a manifold with a conical singularity. Such families arise in various settings in spectral theory, including the study of the isospectral problem. We investigate the behavior of the determinant of the Laplacian under the degeneration. Our main result is an approximation formula for the determinant, including all terms which do not vanish in the limit. The key idea is a uniform parametrix construction for the heat kernel on the degenerating family of manifolds, which enables us to analyze the determinant via the heat trace. It becomes clear in the construction that we need to understand both the shorttime and longtime behavior of the heat kernel on an asymptotically conic manifold. Using techniques of Melrose and building on previous work of Guillarmou and Hassell, we give a complete description of the asymptotic structure of this heat kernel in all spatial and temporal regimes.
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Online 12. A model diffractive boundary value problem on an asymptotically antide Sitter space [electronic resource] [2012]
 Pham, Ha Ngoc.
 2012.
 Description
 Book — 1 online resource.
 Summary

We study the propagation of singularities (in the sense of smooth wave front set) of the solution of a model case initialboundary value problem with glancing rays for a concave domain on an asymptotically antide Sitter manifold. The main result addresses the diffractive problem and establishes that there is no propagation of singularities into the shadow for the solution, i.e. the diffractive result for codimension1 smooth boundary holds in this setting. The approach adopted is motivated by the work done for a conformally related diffractive model problem by Friedlander, in which an explicit solution was constructed using the Airy function. This work was later generalized by Melrose and by Taylor, via the method of parametrix construction. Our setting is a simple case of asympotically antide Sitter spaces, which are Lorentzian manifolds modeled on antide Sitter space at infinity but whose boundary are not totally geodesic (unlike the exact antide Sitter space). Most technical difficulties of the problem reduce to studying and constructing a global resolvent for a semiclassical ODE on the real half line, which at one end is a boperator (in the sense of Melrose) while having a scattering behavior at infinity. We use different techniques near zero and infinity to analyze the local problems: near infinity we use local resolvent bounds and near zero we build a local semiclassical parametrix. After this step, the `gluing' method by DatchevVasy serves to combine these local estimates to get the norm bound for the global resolvent.
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Online 13. On harmonic maps into conic surfaces [electronic resource] [2011]
 Description
 Book — 1 online resource.
 Summary

We prove the existence and uniqueness of harmonic maps in degree one homotopy classes of closed, orientable surfaces of positive genus, where the target has nonpositive gauss curvature and conic points with cone angles less than $2\pi$. For a homeomorphism $w$ of such a surface, we prove existence and uniqueness of minimizers in the homotopy class of $w$ relative to the inverse images of the cone points with cone angles less than or equal to $\pi$. We show that such maps are homeomorphisms and that they depend smoothly on the target metric. For fixed geometric data, the space of minimizers in relative degree one homotopy classes is a complex manifold of (complex) dimension equal to the number of cone points with cone angles less than or equal to $\pi$. When the genus is zero, we prove the same relative minimization provided there are at least three cone points of cone angle less than or equal to $\pi$.
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Online 14. Topology of spaces of microimages, and an application to texture discrimination. [electronic resource] [2011]
 Description
 Book — 1 online resource.
 Summary

Using the fact that most 3by3 pixel highcontrast patches from natural images accumulate around a space with the topology of a Klein bottle, we present in this thesis a novel method for texture representation and discrimination. Given a texture image, most of its highcontrast patches can be projected onto the aforementioned Klein bottle. We analyze this sample in terms of its underlying probability density function, which we show can be represented via Fourierlike coefficients. These coefficients in turn, can be estimated with high confidence from the sample. Dissimilarity measures are defined on the set of estimated coefficients, and performance of the method is tested on a large collection of texture images.
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Online 15. Wave equations on asymptotically de Sitter spaces [electronic resource] [2010]
 Baskin, Dean Russell.
 2010.
 Description
 Book — 1 online resource.
 Summary

Asymptotically de Sitter spaces are Lorentzian manifolds modeled on the de Sitter space of general relativity. In this dissertation, we construct the forward fundamental solution for the wave and KleinGordon equations on asymptotically de Sitter spaces. We adapt classes of conormal and paired Lagrangian distributions to this setting and show that the lift of the kernel of the forward fundamental solution to a blownup space is a sum of distributions in these classes. We use the structure of the kernel of the fundamental solution to study its mapping properties. We show that Strichartz estimates with loss hold for the positive mass KleinGordon equation on asymptotically de Sitter spaces. When the mass parameter is the conformal value, Strichartz estimates hold without loss. As an application of these estimates, we prove a smalldata global existence result for a defocusing KleinGordon equation.
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 Conference on Inverse Problems (2012 : Irvine, Calif.)
 Providence, Rhode Island : American Mathematical Society, [2014]
 Description
 Book — vii, 309 pages : illustrations ; 26 cm.
 Summary

 Spectral theory of a NeumannPoincaretype operator and analysis of cloaking by anomalous localized resonance II by H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton Hybrid inverse problems and redundant systems of partial differential equations by G. Bal A direct imaging method for inverse scattering using the generalized FoldyLax formulation by G. Bao, K. Huang, P. Li, and H. Zhao The inverse scattering problem for a penetrable cavity with internal measurements by F. Cakoni, D. Colton, and X. Meng A Neumann series based method for photoacoustic tomography on irregular domains by E. Chung, C. Y. Lam, and J. Qian Nonlinear inversion from partial EIT data: Computational experiments by S. J. Hamilton and S. Siltanen Increasing stability of the inverse boundary value problem for the Schrodinger equation by V. Isakov, S. Nagayasu, G. Uhlmann, and J.N. Wang Recent progress of inverse scattering theory on noncompact manifolds by H. Isozaki, Y. Kurylev, and M. Lassas On an inverse problem for the Steklov spectrum of a Riemannian surface by A. Jollivet and V. Sharafutdinov Recent progress in the Calderon problem with partial data by C. Kenig and M. Salo Local reconstruction of a Riemannian manifold from a restriction of the hyperbolic DirichlettoNeumann operator by M. Lassas and L. Oksanen Damping mechanisms for regularized transformationacoustics cloaking by J. Li, H. Liu, and H. Sun Hybrid inverse problem for porous media by S. Moskow and J. C. Schotland Efficient algorithms for ptychographic phase retrieval by J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini Matrix elements of Fourier integral operators by S. Zelditch.
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