Heat kernel

Abstract: 1 Background on Differential Geometry.- 1.1 Fibre Bundles and Connections.- 1.2 Riemannian Manifolds.- 1.3 Superspaces.- 1.4 Superconnections.- 1.5 Characteristic Classes.- 1.6 The Euler and Thorn Classes.- 2 Asymptotic Expansion of the Heat Kernel.- 2.1 Differential Operators.- 2.2 The Heat Kernel on Euclidean Space.- 2.3 Heat Kernels.- 2.4 Construction of the Heat Kernel.- 2.5 The Formal Solution.- 2.6 The Trace of the Heat Kernel.- 2.7 Heat Kernels Depending on a Parameter.- 3 Clifford Modules and Dirac Operators.- 3.1 The Clifford Algebra.- 3.2 Spinors.- 3.3 Dirac Operators.- 3.4 Index of Dirac Operators.- 3.5 The Lichnerowicz Formula.- 3.6 Some Examples of Clifford Modules.- 4 Index Density of Dirac Operators.- 4.1 The Local Index Theorem.- 4.2 Mehler's Formula.- 4.3 Calculation of the Index Density.- 5 The Exponential Map and the Index Density.- 5.1 Jacobian of the Exponential Map on Principal Bundles.- 5.2 The Heat Kernel of a Principal Bundle.- 5.3 Calculus with Grassmann and Clifford Variables.- 5.4 The Index of Dirac Operators.- 6 The Equivariant Index Theorem.- 6.1 The Equivariant Index of Dirac Operators.- 6.2 The Atiyah-Bott Fixed Point Formula.- 6.3 Asymptotic Expansion of the Equivariant Heat Kernel.- 6.4 The Local Equivariant Index Theorem.- 6.5 Geodesic Distance on a Principal Bundle.- 6.6 The heat kernel of an equivariant vector bundle.- 6.7 Proof of Proposition 6.13.- 7 Equivariant Differential Forms.- 7.1 Equivariant Characteristic Classes.- 7.2 The Localization Formula.- 7.3 Bott's Formulas for Characteristic Numbers.- 7.4 Exact Stationary Phase Approximation.- 7.5 The Fourier Transform of Coadjoint Orbits.- 7.6 Equivariant Cohomology and Families.- 7.7 The Bott Class.- 8 The Kirillov Formula for the Equivariant Index.- 8.1 The Kirillov Formula.- 8.2 The Weyl and Kirillov Character Formulas.- 8.3 The Heat Kernel Proof of the Kirillov Formula.- 9 The Index Bundle.- 9.1 The Index Bundle in Finite Dimensions.- 9.2 The Index Bundle of a Family of Dirac Operators.- 9.3 The Chern Character of the Index Bundle.- 9.4 The Equivariant Index and the Index Bundle.- 9.5 The Case of Varying Dimension.- 9.6 The Zeta-Function of a Laplacian.- 9.7 The Determinant Line Bundle.- 10 The Family Index Theorem.- 10.1 Riemannian Fibre Bundles.- 10.2 Clifford Modules on Fibre Bundles.- 10.3 The Bismut Superconnection.- 10.4 The Family Index Density.- 10.5 The Transgression Formula.- 10.6 The Curvature of the Determinant Line Bundle.- 10.7 The Kirillov Formula and Bismut's Index Theorem.- References.- List of Notation.

Abstract: Preface. The Laplacian. The Basic Examples. Curvature. Isoperimetric Inequalities. Eigenvalues and Kinematic Measure. The Heat Kernel for Compact Manifolds. The Dirichlet Heat Kernel for Regular Domains. The Heat Kernel for Noncompact Manifolds. Topological Perturbations with Negligible Effect. Surfaces of Constant Negative Curvature. The Selberg Trace Formula. Miscellanea. Laplacian on Forms. Bibliography. Index.

Abstract: We propose a novel point signature based on the properties of the heat diffusion process on a shape. Our signature, called the Heat Kernel Signature (or HKS), is obtained by restricting the well-known heat kernel to the temporal domain. Remarkably we show that under certain mild assumptions, HKS captures all of the information contained in the heat kernel, and characterizes the shape up to isometry. This means that the restriction to the temporal domain, on the one hand, makes HKS much more concise and easily commensurable, while on the other hand, it preserves all of the information about the intrinsic geometry of the shape. In addition, HKS inherits many useful properties from the heat kernel, which means, in particular, that it is stable under perturbations of the shape. Our signature also provides a natural and efficiently computable multi-scale way to capture information about neighborhoods of a given point, which can be extremely useful in many applications. To demonstrate the practical relevance of our signature, we present several methods for non-rigid multi-scale matching based on the HKS and use it to detect repeated structure within the same shape and across a collection of shapes.

Abstract: This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.