Spectral Theory and Mathematical Physics
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Spectral Geometry Spring 2016
Spectral Geometry Spring 2016 Michael Magee Lecture 5: The resolvent. Last time we proved that the Laplacian has a closure whose domain is the Sobolev space H2(M)= u L2(M): u L2(M), ∆u L2(M) . { 2 kr k2 2 } We view the Laplacian as an unbounded densely defined self-adjoint operator on L2 with domain H2. We also stated the spectral theorem. We’ll now ask what is the nature of the pro- jection valued measure associated to the Laplacian (its ‘spectral decomposition’). Until otherwise specified, suppose M is a complete Riemannian manifold. Definition 1. The resolvent set of the Laplacian is the set of λ C such that λI ∆isa bijection of H2 onto L2 with a boundedR inverse (with respect to L2 norms)2 − 1 2 2 R(λ) (λI ∆)− : L (M) H (M). ⌘ − ! The operator R(λ) is called the resolvent of the Laplacian at λ. We write spec(∆) = C −R and call it the spectrum. Note that the projection valued measure of ∆is supported in R 0. Otherwise one can find a function in H2(M)where ∆ , is negative, but since can be≥ approximated in Sobolev norm by smooth functions, thish will contradicti ∆ , = g( , )Vol 0. (1) h i ˆ r r ≥ Note then the spectral theorem implies that the resolvent set contains C R 0 (use the Borel functional calculus). − ≥ Theorem 2 (Stone’s formula [1, VII.13]). Let P be the projection valued measure on R associated to the Laplacan by the spectral theorem. The following formula relates the resolvent to P b 1 1 lim(2⇡i)− [R(λ + i✏) R(λ i✏)] dλ = P[a,b] + P(a,b) . -
Arxiv:1903.00779V1 [Math.SP] 2 Mar 2019 O Clrsh¨Dne Prtr ( Schr¨Odinger Operators Scalar for Rn O P29177
THE INVERSE APPROACH TO DIRAC-TYPE SYSTEMS BASED ON THE A-FUNCTION CONCEPT FRITZ GESZTESY AND ALEXANDER SAKHNOVICH Abstract. The principal objective in this paper is a new inverse approach to general Dirac-type systems of the form ′ y (x,z)= i(zJ + JV (x))y(x,z) (x ≥ 0), ⊤ where y = (y1,...,ym) and (for m1, m2 ∈ N) Im1 0m1×m2 0m1 v J = , V = ∗ , m1 + m2 =: m, 0m2×m1 −Im2 v 0m2 m ×m for v ∈ C1([0, ∞)) 1 2 , modeled after B. Simon’s 1999 inverse approach to half-line Schr¨odinger operators. In particular, we derive the A-equation associated to this Dirac-type system in the (z-independent) form x ∂ ∂ ∗ A(x,ℓ)= A(x,ℓ)+ A(x − t, ℓ)A(0,ℓ) A(t, ℓ) dt (x ≥ 0,ℓ ≥ 0). ∂ℓ ∂x ˆ0 Given the fundamental positivity condition ST > 0 in (1.14) (cf. (1.13) for details), we prove that this integro-differential equation for A( · , · ) is uniquely solvable for initial conditions 1 m ×m A( · , 0) = A( · ) ∈ C ([0, ∞)) 2 1 , m ×m and the corresponding potential coefficient v ∈ C1([0, ∞)) 1 2 can be recovered from A( · , · ) via ∗ v(ℓ) = −iA(0,ℓ) (ℓ ≥ 0). Contents 1. Introduction 1 2. Preliminaries 6 3. The A-Function for Dirac-Type Systems: Part I 7 4. The A-Function for Dirac-Type Systems: Part II 9 5. The A-Equation for Dirac-Type Systems 13 6. The Inverse Approach 15 arXiv:1903.00779v1 [math.SP] 2 Mar 2019 Appendix A. Various Results in Support of Step 2 in the Proof of Theorem 5.1 19 References 29 1. -
Spectral Geometry for Structural Pattern Recognition
Spectral Geometry for Structural Pattern Recognition HEWAYDA EL GHAWALBY Ph.D. Thesis This thesis is submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy. Department of Computer Science United Kingdom May 2011 Abstract Graphs are used pervasively in computer science as representations of data with a network or relational structure, where the graph structure provides a flexible representation such that there is no fixed dimensionality for objects. However, the analysis of data in this form has proved an elusive problem; for instance, it suffers from the robustness to structural noise. One way to circumvent this problem is to embed the nodes of a graph in a vector space and to study the properties of the point distribution that results from the embedding. This is a problem that arises in a number of areas including manifold learning theory and graph-drawing. In this thesis, our first contribution is to investigate the heat kernel embed- ding as a route to computing geometric characterisations of graphs. The reason for turning to the heat kernel is that it encapsulates information concerning the distribution of path lengths and hence node affinities on the graph. The heat ker- nel of the graph is found by exponentiating the Laplacian eigensystem over time. The matrix of embedding co-ordinates for the nodes of the graph is obtained by performing a Young-Householder decomposition on the heat kernel. Once the embedding of its nodes is to hand we proceed to characterise a graph in a geometric manner. With the embeddings to hand, we establish a graph character- ization based on differential geometry by computing sets of curvatures associated ii Abstract iii with the graph nodes, edges and triangular faces. -
Scientific Report for 2012
Scientific Report for 2012 Impressum: Eigent¨umer,Verleger, Herausgeber: The Erwin Schr¨odingerInternational Institute for Mathematical Physics - University of Vienna (DVR 0065528), Boltzmanngasse 9, A-1090 Vienna. Redaktion: Joachim Schwermer, Jakob Yngvason. Supported by the Austrian Federal Ministry of Science and Research (BMWF) via the University of Vienna. Contents Preface 3 The Institute and its Mission . 3 Scientific activities in 2012 . 4 The ESI in 2012 . 7 Scientific Reports 9 Main Research Programmes . 9 Automorphic Forms: Arithmetic and Geometry . 9 K-theory and Quantum Fields . 14 The Interaction of Geometry and Representation Theory. Exploring new frontiers. 18 Modern Methods of Time-Frequency Analysis II . 22 Workshops Organized Outside the Main Programmes . 32 Operator Related Function Theory . 32 Higher Spin Gravity . 34 Computational Inverse Problems . 35 Periodic Orbits in Dynamical Systems . 37 EMS-IAMP Summer School on Quantum Chaos . 39 Golod-Shafarevich Groups and Algebras, and the Rank Gradient . 41 Recent Developments in the Mathematical Analysis of Large Systems . 44 9th Vienna Central European Seminar on Particle Physics and Quantum Field Theory: Dark Matter, Dark Energy, Black Holes and Quantum Aspects of the Universe . 46 Dynamics of General Relativity: Black Holes and Asymptotics . 47 Research in Teams . 49 Bruno Nachtergaele et al: Disordered Oscillator Systems . 49 Alexander Fel'shtyn et al: Twisted Conjugacy Classes in Discrete Groups . 50 Erez Lapid et al: Whittaker Periods of Automorphic Forms . 53 Dale Cutkosky et al: Resolution of Surface Singularities in Positive Characteristic . 55 Senior Research Fellows Programme . 57 James Cogdell: L-functions and Functoriality . 57 Detlev Buchholz: Fundamentals and Highlights of Algebraic Quantum Field Theory . -
A New Approach to Inverse Spectral Theory, II. General Real Potentials
Annals of Mathematics, 152 (2000), 593–643 A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure∗ By Fritz Gesztesy and Barry Simon Abstract We continue the study of the A-amplitude associated to a half-line d2 2 Schr¨odinger operator, 2 + q in L ((0, b)), b . A is related to the Weyl- − dx ≤∞ Titchmarsh m-function via m( κ2)= κ a A(α)e 2ακ dα+O(e (2a ε)κ) for − − − 0 − − − all ε> 0. We discuss five issues here. First,R we extend the theory to general q in L1((0, a)) for all a, including q’s which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure 1 ρ: A(α) = 2 ∞ λ− 2 sin(2α√λ) dρ(λ) (since the integral is divergent, this − −∞ formula has toR be properly interpreted). Third, we provide a Laplace trans- form representation for m without error term in the case b < . Fourth, we ∞ discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly. 1. Introduction In this paper we will consider Schr¨odinger operators d2 (1.1) + q −dx2 in L2((0, b)) for 0 <b< or b = and real-valued locally integrable q. arXiv:math/9809182v2 [math.SP] 1 Sep 2000 ∞ ∞ There are essentially four distinct cases. ∗This material is based upon work supported by the National Science Foundation under Grant No. -
Noncommutative Geometry, the Spectral Standpoint Arxiv
Noncommutative Geometry, the spectral standpoint Alain Connes October 24, 2019 In memory of John Roe, and in recognition of his pioneering achievements in coarse geometry and index theory. Abstract We update our Year 2000 account of Noncommutative Geometry in [68]. There, we de- scribed the following basic features of the subject: I The natural “time evolution" that makes noncommutative spaces dynamical from their measure theory. I The new calculus which is based on operators in Hilbert space, the Dixmier trace and the Wodzicki residue. I The spectral geometric paradigm which extends the Riemannian paradigm to the non- commutative world and gives a new tool to understand the forces of nature as pure gravity on a more involved geometric structure mixture of continuum with the discrete. I The key examples such as duals of discrete groups, leaf spaces of foliations and de- formations of ordinary spaces, which showed, very early on, that the relevance of this new paradigm was going far beyond the framework of Riemannian geometry. Here, we shall report1 on the following highlights from among the many discoveries made since 2000: 1. The interplay of the geometry with the modular theory for noncommutative tori. 2. Great advances on the Baum-Connes conjecture, on coarse geometry and on higher in- dex theory. 3. The geometrization of the pseudo-differential calculi using smooth groupoids. 4. The development of Hopf cyclic cohomology. 5. The increasing role of topological cyclic homology in number theory, and of the lambda arXiv:1910.10407v1 [math.QA] 23 Oct 2019 operations in archimedean cohomology. 6. The understanding of the renormalization group as a motivic Galois group. -
Spectral Theory of Translation Surfaces : a Short Introduction Volume 28 (2009-2010), P
Institut Fourier — Université de Grenoble I Actes du séminaire de Théorie spectrale et géométrie Luc HILLAIRET Spectral theory of translation surfaces : A short introduction Volume 28 (2009-2010), p. 51-62. <http://tsg.cedram.org/item?id=TSG_2009-2010__28__51_0> © Institut Fourier, 2009-2010, tous droits réservés. L’accès aux articles du Séminaire de théorie spectrale et géométrie (http://tsg.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://tsg.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Séminaire de théorie spectrale et géométrie Grenoble Volume 28 (2009-2010) 51-62 SPECTRAL THEORY OF TRANSLATION SURFACES : A SHORT INTRODUCTION Luc Hillairet Abstract. — We define translation surfaces and, on these, the Laplace opera- tor that is associated with the Euclidean (singular) metric. This Laplace operator is not essentially self-adjoint and we recall how self-adjoint extensions are chosen. There are essentially two geometrical self-adjoint extensions and we show that they actually share the same spectrum Résumé. — On définit les surfaces de translation et le Laplacien associé à la mé- trique euclidienne (avec singularités). Ce laplacien n’est pas essentiellement auto- adjoint et on rappelle la façon dont les extensions auto-adjointes sont caractérisées. Il y a deux choix naturels dont on montre que les spectres coïncident. 1. Introduction Spectral geometry aims at understanding how the geometry influences the spectrum of geometrically related operators such as the Laplace oper- ator. The more interesting the geometry is, the more interesting we expect the relations with the spectrum to be. -
Discrete Spectral Geometry
Discrete Spectral Geometry Discrete Spectral Geometry Amir Daneshgar Department of Mathematical Sciences Sharif University of Technology Hossein Hajiabolhassan is mathematically dense in this talk!! May 31, 2006 Discrete Spectral Geometry Outline Outline A viewpoint based on mappings A general non-symmetric discrete setup Weighted energy spaces Comparison and variational formulations Summary A couple of comparison theorems Graph homomorphisms A comparison theorem The isoperimetric spectrum ε-Uniformizers Symmetric spaces and representation theory Epilogue Discrete Spectral Geometry A viewpoint based on mappings Basic Objects I H is a geometric objects, we are going to analyze. Discrete Spectral Geometry A viewpoint based on mappings Basic Objects I There are usually some generic (i.e. well-known, typical, close at hand, important, ...) types of these objects. Discrete Spectral Geometry A viewpoint based on mappings Basic Objects (continuous case) I These are models and objects of Euclidean and non-Euclidean geometry. Discrete Spectral Geometry A viewpoint based on mappings Basic Objects (continuous case) I Two important generic objects are the sphere and the upper half plane. Discrete Spectral Geometry A viewpoint based on mappings Basic Objects (discrete case) I Some important generic objects in this case are complete graphs, infinite k-regular tree and fractal-meshes in Rn. Discrete Spectral Geometry A viewpoint based on mappings Basic Objects (discrete case) I These are models and objects in network analysis and design, discrete state-spaces of algorithms and discrete geometry (e.g. in geometric group theory). Discrete Spectral Geometry A viewpoint based on mappings Comparison method I The basic idea here is to try to understand (or classify if we are lucky!) an object by comparing it with the generic ones. -
Pose Analysis Using Spectral Geometry
Vis Comput DOI 10.1007/s00371-013-0850-0 ORIGINAL ARTICLE Pose analysis using spectral geometry Jiaxi Hu · Jing Hua © Springer-Verlag Berlin Heidelberg 2013 Abstract We propose a novel method to analyze a set 1 Introduction of poses of 3D models that are represented with trian- gle meshes and unregistered. Different shapes of poses are Shape deformation is one of the major research topics in transformed from the 3D spatial domain to a geometry spec- computer graphics. Recently there have been extensive lit- trum domain that is defined by Laplace–Beltrami opera- eratures on this topic. There are two major categories of re- tor. During this space-spectrum transform, all near-isometric searches on this topic. One is surface-based and the other deformations, mesh triangulations and Euclidean transfor- one is skeleton-driven method. Shape interpolation [2, 9, 11] mations are filtered away. The different spatial poses from is a surface-based approach. The basic idea is that given two a 3D model are represented with near-isometric deforma- key frames of a shape, the intermediate deformation can be tions; therefore, they have similar behaviors in the spec- generated with interpolation and further deformations can be tral domain. Semantic parts of that model are then deter- predicted with extrapolation. It can also blend among sev- mined based on the computed geometric properties of all the eral shapes. Shape interpolation is convenient for animation mapped vertices in the geometry spectrum domain. Seman- generation but is not flexible enough for shape editing due tic skeleton can be automatically built with joints detected to the lack of local control. -
Notices of the American Mathematical Society
OF THE 1994 AMS Election Special Section page 7 4 7 Fields Medals and Nevanlinna Prize Awarded at ICM-94 page 763 SEPTEMBER 1994, VOLUME 41, NUMBER 7 Providence, Rhode Island, USA ISSN 0002-9920 Calendar of AMS Meetings and Conferences This calendar lists all meetings and conferences approved prior to the date this issue insofar as is possible. Instructions for submission of abstracts can be found in the went to press. The summer and annual meetings are joint meetings with the Mathe· January 1994 issue of the Notices on page 43. Abstracts of papers to be presented at matical Association of America. the meeting must be received at the headquarters of the Society in Providence, Rhode Abstracts of papers presented at a meeting of the Society are published in the Island, on or before the deadline given below for the meeting. Note that the deadline for journal Abstracts of papers presented to the American Mathematical Society in the abstracts for consideration for presentation at special sessions is usually three weeks issue corresponding to that of the Notices which contains the program of the meeting, earlier than that specified below. Meetings Abstract Program Meeting# Date Place Deadline Issue 895 t October 28-29, 1994 Stillwater, Oklahoma Expired October 896 t November 11-13, 1994 Richmond, Virginia Expired October 897 * January 4-7, 1995 (101st Annual Meeting) San Francisco, California October 3 January 898 * March 4-5, 1995 Hartford, Connecticut December 1 March 899 * March 17-18, 1995 Orlando, Florida December 1 March 900 * March 24-25, -
Curriculum Vitae
Isaac B. Michael Curriculum Vitae Education 2019 Ph.D. in Mathematics, Baylor University, Waco, TX. Advisor: Fritz Gesztesy Co-Advisor: Lance Littlejohn Thesis: On Birman–Hardy–Rellich-type Inequalities 2015 M.S. in Mathematics, Baylor University, Waco, TX. 4.0 GPA 2013 B.S. in Mathematics, Tarleton State University, Stephenville, TX. 3.3 GPA; 3.9 Institutional GPA 8-12 Math Teaching Certification Research Interests Primary Analysis (Real, Complex, and Functional), Differential Equations (Ordi- nary and Partial), Spectral Theory, Operator Theory Secondary Deterministic Dynamical Systems, Semigroups, Functional Calculus, Calculus of Variations, Lie Theory Publications and Preprints [1] Net Regular Signed Trees, with M. Sepanski, Australasian Journal of Combinatorics 66(2), 192-204 (2016). [2] On Birman’s Sequence of Hardy–Rellich-Type Inequalities, with F. Gesztesy, L. Littlejohn, and R. Wellman, Journal of Differential Equations 264(4), 2761-2801 (2018). [3] Radial and Logarithmic Refinements of Hardy’s Inequality, with F. Gesztesy, L. Littlejohn, and M. M. H. Pang, Algebra i Analiz, 30(3), 55–65 (2018) (Russian), St. Petersburg Math. J., St. 30, 429–436 (2019) (English). Louisiana State University – Department of Mathematics H (254) 366-6861 • B [email protected] Í math.lsu.edu/∼imichael 1/9 [4] On Weighted Hardy-Type Inequalities, with C. Y. Chuah, F. Gesztesy, L. Littlejohn, T. Mei, and M. M. H. Pang, Math. Ineq. & App. 23(2), 625-646 (2020). [5] A Sequence of Weighted Birman–Hardy–Rellich-type Inequalities with Loga- rithmic Refinements, with F. Gesztesy, L. Littlejohn, and M. M. H. Pang (submitted for publication). [6] Optimality of Constants in Weighted Birman–Hardy–Rellich Inequalities with Logarithmic Refinements, with F. -
New Publications Offered by the AMS
newpubs-dec00.qxp 10/18/00 12:45 PM Page 1438 New Publications Offered by the AMS composable Hopf algebras over Coxeter groups; D. Nikshych, Algebra and Algebraic A duality theorem for quantum groupoids; M. Ronco, Primitive elements in a free dendriform algebra; S. Sachse, On operator Geometry representations of Uq(isl(2, R)); P. Schauenburg, Duals and doubles of quantum groupoids (×R-Hopf algebras); M. Takeuchi, Survey of braided Hopf algebras; S. Westreich, Inner and outer actions of pointed Hopf algebras; J.-H. Lu, CONTEMPORARY New Trends in Hopf MATHEMATICS Algebra Theory M. Yan, and Y. Zhu, Quasi-triangular structures on Hopf alge- 267 bras with positive bases. New Trends in Nicolás Andruskiewitsch, Contemporary Mathematics, Volume 267 Hopf Algebra Theory Universidad Nacional de Nicolás Andruskiewitsch December 2000, 356 pages, Softcover, ISBN 0-8218-2126-1, Walter Ricardo Ferrer Santos Hans-Jürgen Schneider Córdoba, Argentina, Walter LC 00-045366, 2000 Mathematics Subject Classification: 16W30, Editors Ricardo Ferrer Santos, Centro 16W25, 16W35, 16S40; 16L60, Individual member $45, List THEMAT A IC M A L $75, Institutional member $60, Order code CONM/267N N ΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ A S O C I C R I E de Matemática, Montevideo, E ΑΓΕΩΜΕ T M Y A F OU 88 NDED 18 Uruguay, and Hans-Jürgen American Mathematical Society Schneider, Universität Quantum Linear München, Germany, Editors EMOIRS M of the American Mathematical Society Groups and This volume presents the proceedings from the Colloquium on Volume 149 Number 706 Quantum Groups and Hopf Algebras held in Córdoba Quantum Linear Groups Representations of (Argentina) in 1999.