The Spectral Geometry of Operators of Dirac and Laplace Type
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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) . -
Heat Kernels and Function Theory on Metric Measure Spaces
Heat kernels and function theory on metric measure spaces Alexander Grigor’yan Imperial College London SW7 2AZ United Kingdom and Institute of Control Sciences of RAS Moscow, Russia email: [email protected] Contents 1 Introduction 2 2 Definition of a heat kernel and examples 5 3 Volume of balls 8 4 Energy form 10 5 Besov spaces and energy domain 13 5.1 Besov spaces in Rn ............................ 13 5.2 Besov spaces in a metric measure space . 14 5.3 Identification of energy domains and Besov spaces . 15 5.4 Subordinated heat kernel ......................... 19 6 Intrinsic characterization of walk dimension 22 7 Inequalities for walk dimension 23 8 Embedding of Besov spaces into H¨olderspaces 29 9 Bessel potential spaces 30 1 1 Introduction The classical heat kernel in Rn is the fundamental solution to the heat equation, which is given by the following formula 1 x y 2 pt (x, y) = exp | − | . (1.1) n/2 t (4πt) − 4 ! x y 2 It is worth observing that the Gaussian term exp | − | does not depend in n, − 4t n/2 whereas the other term (4πt)− reflects the dependence of the heat kernel on the underlying space via its dimension. The notion of heat kernel extends to any Riemannian manifold M. In this case, the heat kernel pt (x, y) is the minimal positive fundamental solution to the heat ∂u equation ∂t = ∆u where ∆ is the Laplace-Beltrami operator on M, and it always exists (see [11], [13], [16]). Under certain assumptions about M, the heat kernel can be estimated similarly to (1.1). -
Heat Kernel Analysis on Graphs
Heat Kernel Analysis On Graphs Xiao Bai Submitted for the degree of Doctor of Philosophy Department of Computer Science June 7, 2007 Abstract In this thesis we aim to develop a framework for graph characterization by com- bining the methods from spectral graph theory and manifold learning theory. The algorithms are applied to graph clustering, graph matching and object recogni- tion. Spectral graph theory has been widely applied in areas such as image recog- nition, image segmentation, motion tracking, image matching and etc. The heat kernel is an important component of spectral graph theory since it can be viewed as describing the flow of information across the edges of the graph with time. Our first contribution is to investigate how to extract useful and stable invari- ants from the graph heat kernel as a means of clustering graphs. The best set of invariants are the heat kernel trace, the zeta function and its derivative at the origin. We also study heat content invariants. The polynomial co-efficients can be computed from the Laplacian eigensystem. Graph clustering is performed by applying principal components analysis to vectors constructed from the invari- ants or simply based on the unitary features extracted from the graph heat kernel. We experiment with the algorithms on the COIL and Oxford-Caltech databases. We further investigate the heat kernel as a means of graph embedding. The second contribution of the thesis is the introduction of two graph embedding methods. The first of these uses the Euclidean distance between graph nodes. To do this we equate the spectral and parametric forms of the heat kernel to com- i pute an approximate Euclidean distance between nodes. -
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. -
Inverse Spectral Problems in Riemannian Geometry
Inverse Spectral Problems in Riemannian Geometry Peter A. Perry Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027, U.S.A. 1 Introduction Over twenty years ago, Marc Kac posed what is arguably one of the simplest inverse problems in pure mathematics: "Can one hear the shape of a drum?" [19]. Mathematically, the question is formulated as follows. Let /2 be a simply connected, plane domain (the drumhead) bounded by a smooth curve 7, and consider the wave equation on /2 with Dirichlet boundary condition on 7 (the drumhead is clamped at the boundary): Au(z,t) = ~utt(x,t) in/2, u(z, t) = 0 on 7. The function u(z,t) is the displacement of the drumhead, as it vibrates, at position z and time t. Looking for solutions of the form u(z, t) = Re ei~tv(z) (normal modes) leads to an eigenvalue problem for the Dirichlet Laplacian on B: v(x) = 0 on 7 (1) where A = ~2/c2. We write the infinite sequence of Dirichlet eigenvalues for this problem as {A,(/2)}n=l,c¢ or simply { A-},~=1 co it" the choice of domain/2 is clear in context. Kac's question means the following: is it possible to distinguish "drums" /21 and/22 with distinct (modulo isometrics) bounding curves 71 and 72, simply by "hearing" all of the eigenvalues of the Dirichlet Laplacian? Another way of asking the question is this. What is the geometric content of the eigenvalues of the Laplacian? Is there sufficient geometric information to determine the bounding curve 7 uniquely? In what follows we will call two domains isospectral if all of their Dirichlet eigenvalues are the same. -
Three Examples of Quantum Dynamics on the Half-Line with Smooth Bouncing
Three examples of quantum dynamics on the half-line with smooth bouncing C. R. Almeida Universidade Federal do Esp´ıritoSanto, Vit´oria,29075-910, ES, Brazil H. Bergeron ISMO, UMR 8214 CNRS, Univ Paris-Sud, France J.P. Gazeau Laboratoire APC, UMR 7164, Univ Paris Diderot, Sorbonne Paris-Cit´e75205 Paris, France A.C. Scardua Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180 - Rio de Janeiro, RJ, Brazil Abstract This article is an introductory presentation of the quantization of the half-plane based on affine coherent states (ACS). The half-plane is viewed as the phase space for the dynamics of a positive physical quantity evolving with time, and its affine symmetry is preserved due to the covariance of this type of quantization. We promote the interest of such a procedure for transforming a classical model into a quantum one, since the singularity at the origin is systematically removed, and the arbitrari- ness of boundary conditions can be easily overcome. We explain some important mathematical aspects of the method. Three elementary examples of applications are presented, the quantum breathing of a massive sphere, the quantum smooth bounc- ing of a charged sphere, and a smooth bouncing of \dust" sphere as a simple model of quantum Newtonian cosmology. Keywords: Integral quantization, Half-plane, Affine symmetry, Coherent states, arXiv:1708.06422v2 [quant-ph] 31 Aug 2017 Quantum smooth bouncing Email addresses: [email protected] (C. R. Almeida), [email protected] (H. Bergeron), [email protected] (J.P. -
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. -
Introduction
Traceglobfin June 3, 2011 Copyrighted Material Introduction The purpose of this book is to use the hypoelliptic Laplacian to evaluate semisimple orbital integrals, in a formalism that unifies index theory and the trace formula. 0.1 The trace formula as a Lefschetz formula Let us explain how to think formally of such a unified treatment, while allowing ourselves a temporarily unbridled use of mathematical analogies. Let X be a compact Riemannian manifold, and let ΔX be the corresponding Laplace-Beltrami operator. For t > 0, consider the trace of the heat kernel Tr pexp h ΔX 2m . If X is the Hilbert space of square-integrable functions t = L2 on X, Tr pexp htΔX =2 m is the trace of the ‘group element’ exp htΔX =2 m X acting on L2 . X X Suppose that L2 is the cohomology of an acyclic complex R on which Δ acts. Then Tr pexp htΔX =2 m can be viewed as the evaluation of a Lefschetz trace, so that cohomological methods can be applied to the evaluation of this trace. In our case, R will be the fibrewise de Rham complex of the total space Xb of a flat vector bundle over X, which contains TX as a subbundle. The Lefschetz fixed point formulas of Atiyah-Bott [ABo67, ABo68] provide a model for the evaluation of such cohomological traces. R The McKean-Singer formula [McKS67] indicates that if D is a Hodge like Laplacian operator acting on R and commuting with ΔX , for any b > 0, X L2 p hX m R p h X R 2m Tr exp tΔ =2 = Trs exp tΔ =2 − tD =2b : (0.1) In (0.1), Trs is our notation for the supertrace. -
Heat Kernels
Chapter 3 Heat Kernels In this chapter, we assume that the manifold M is compact and the general- ized Laplacian H is not necessarily symmetric. Note that if M is non-compact and H is symmetric, we can study the heat kernel following the lines of Spectral theorem and the Schwartz kernel theorem. We will not discuss it here. 3.1 heat kernels 3.1.1 What is kernel? Let H be a generalized Laplacian on a vector bundle E over a compact Riemannian oriented manifold M. Let E and F be Hermitian vector bundles over M. Let p1 and p2 be the projections from M × M onto the first and the second factor M respectively. We denote by ⊠ ∗ ⊗ ∗ E F := p1E p2F (3.1.1) over M × M. Definition 3.1.1. A continuous section P (x; y) on F ⊠E∗) is called a kernel. Using P (x; y), we could define an operator P : C 1(M; E) ! C 1(M; F ) by Z (P u)(x) = hP (x; y); u(y)iEdy: (3.1.2) y2M The kernel P (x; y) is also called the kernel of P , which is also denoted by hxjP jyi. 99 100 CHAPTER 3. HEAT KERNELS Proposition 3.1.2. If P has a kernel P (x; y), then the adjoint operator P ∗ has a kernel P ∗(x; y) = P (y; x)∗ 2 C 1(M × M; E∗ ⊠ F )1. Proof. For u 2 L2(M; E), v 2 L2(M; F ∗), we have Z ⟨Z ⟩ (P u; v)L2 = hP (x; y); u(y)iEdy; v(x) dx x2MZ y2⟨M Z F ⟩ ∗ = u(y); hP (x; y) ; v(x)iF dx dy Z ⟨ y2M Z x2M ⟩ E ∗ ∗ = u(x); hP (y; x) ; v(y)iF dy dx = (u; P v)L2 : (3.1.3) x2M y2M E So for any v 2 L2(M; F ∗), Z ∗ ∗ P v = hP (y; x) ; v(y)iF dy: (3.1.4) y2M The proof of Proposition 3.1.2 is completed. -
The Partial View Heat Kernel Descriptor for 3D Object Representation
The Partial View Heat Kernel Descriptor for 3D Object Representation Susana Brandao˜ 1, Joao˜ P. Costeira2 and Manuela Veloso3 Abstract— We introduce the Partial View Heat Kernel sensor cameras, such as the Kinect sensor, that combine (PVHK) descriptor, for the purpose of 3D object representation RGB and depth information, but are also noisy. To address and recognition from partial views, assumed to be partial these multiple challenges and goals, we introduce a new object surfaces under self occlusion. PVHK describes partial views in a geometrically meaningful way, i.e., by establishing representation for partial views, the Partial View Heat Kernel a unique relation between the shape of the view and the (PVHK) descriptor, which is: descriptor. PVHK is also stable with respect to sensor noise and 1) Informative, i.e., a single descriptor robustly describes therefore adequate for sensors, such as the current active 3D each partial view; cameras. Furthermore, PVHK takes full advantage of the dual 3D/RGB nature of current sensors and seamlessly incorporates 2) Stable, i.e., small perturbations on the surface yield appearance information onto the 3D information. We formally small perturbations on the descriptor; define the PVHK descriptor, discuss related work, provide 3) Inclusive, i.e., appearance properties, such as texture, evidence of the PVHK properties and validate them in three can be seamlessly incorporated into the geometry- purposefully diverse datasets, and demonstrate its potential for based descriptor. recognition tasks. The combination of these three characteristics results in a I. INTRODUCTION representation especially suitable for partial views captured We address the 3D representation of objects from multiple from noisy RGB+Depth sensors during robot navigation 3D partial views, where each partial view is the visible or manipulation where the object surfaces are visible with surface of the object as seen from a view angle, with no limited, if any, occlusion. -
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.