I. Manifolds and Lie Groups.- II. Differential Forms.- III. Bundles and Connections.- IV. Jets and Natural Bundles.- V. Finite Order Theorems.- VI. Methods for Finding Natural Operators.- VII. Further Applications.- VIII. Product Preserving Functors.- IX. Bundle Functors on Manifolds.- X. Prolongation of Vector Fields and Connections.- XI. General Theory of Lie Derivatives.- XII. Gauge Natural Bundles and Operators.- References.- List of symbols.- Author index.

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Natural operations in differential geometry

Advanced Mathematical Methods for Scientists and Engineers

Finite element exterior calculus is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are re- vealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Lapla- cian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

/pdf/finite-element-exterior-calculus-homological-techniques-and-4129s2mxlu.pdf

Finite element exterior calculus, homological techniques, and applications

Introduction to lie algebras and representation theory

Nonlinear equations for symmetric massless higher spin fields in (A)dSd

Thomas's Structure Bundle for Conformal, Projective and Related Structures

Part 1 Lie algebras and flag manifolds: some structure theory borel and parabolic subalgebras generalized flag varieties fibrations of generalized flag varieties. Part 2 Homogeneous vector bundles on G/P: a brief review of representation theory homogeneous bundles on G/P a remark on inverse images. Part 3 The Weyl group, its actions, and Hasse diagrams: the Weyl group the affine Weyl action the Hasse diagram of a parabolic subalgebra relative Hasse diagrams. Part 4 The Bott-Borel-Weil Theorem: a simple proof some examples direct images. Part 5 Realizations of G/P: the projective realization the cell structure of G/P integral cohomology rings Co-Adjoint realizations and moment maps. Part 6 The Penrose transform in principle: pulling-back cohomology pushing-down cohomology a spectral sequence. Part 7 The Bernstein-Gelfand-Gelfand Resolution: a prototype translating BGG resolutions the general case on G/B the story for G/P an algorithm for computation non-standard morphisms relative BGG resolutions. Part 8 The Penrose transform in practice: the homogeneous Penrose transform the real thing the Penrose transform of forms on twistor space other bundles on twistor space the Penrose transform for ambitwistor space higher dimensions - conformal case a Grassmannian example an exceptional example the Ward correspondence. Part 9 Constructing unitary representations: the discrete series of SU(1,1) massless field representations the twistor point of view the twistor transform Hermitian symmetric spaces towards discrete series. Part 10 Module structures on cohomology: verma modules and differential operators invariant differential operators the algebraic Penrose transform K-types, local cohomology, and elementary states homomorphisms of Verma modules.

The Penrose Transform: Its Interaction with Representation Theory

We identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold.

/pdf/higher-symmetries-of-the-laplacian-1gkf379mbq.pdf

Higher symmetries of the laplacian

The geometry of twistors was first introduced in Penrose [28]. Since that time it has played a significant role in solutions of various problems in mathemetical physics of both a linear and nonlinear nature (cf. Penrose [29], Penrose [35], Ward [48], and Atiyah-Hitchin-Drinfeld-Manin [2], see Wells [52] for a recent survey of the topic with a more extensive bibliography). The major role it has played has been in setting up a general correspondence which translates certain important physical field equations in space-time into holomorphic structures on a related complex manifold known as twίstor space. The purpose of this paper is to give a rigorous discussion of this correspondence for the case of the linear massless free-field equations, including Maxwell's source-free equations, the wave equation, the Dirac-Weyl neutrino equations, and the linearized (weakfield limit of) Einstein's vacuum equations. These equations may also be analyzed from this point of view on a background provided by the nonlinear Yang-Mills or Einstein equations in the (anti-) self-dual case. The correspondence is effected by an integral-geometric transform, which transforms complex-analytic data on twistor space to solutions of the linear massless field equations, and is, in fact, a generalization of the classical Radon transform, which is discussed further below. The motivation for finding such a correspondence in general is that it forms an essential part of the \"twistor programme\" according to which one attempts to eliminate the equations of physics by deriving them from the rigidity of complex geometry and holomorphic functions (see, e.g. Penrose [38]). It is, in fact, rather remarkable the extent to which it is possible to achieve this. Success apparently comes about because in twistor-space descriptions the information is \"stored\" nonlocally. The (local) value of a field at a point in space-time depends upon the way that the holomorphic structure in the twistor-space is fitted together in the large. So sheaf cohomology and function theory of several complex variables turn out to be the appropriate tools in the twistor framework. It is hoped that, as part of the general twistor programme, some deeper insights may eventually be gained as to the interrelation between quantum mechanics or quantum field

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103908690

Cohomology and massless fields

Using the AdS/CFT correspondence, we identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold.

Michael Eastwood

Papers

Thomas's Structure Bundle for Conformal, Projective and Related Structures

The Penrose Transform: Its Interaction with Representation Theory

Higher symmetries of the laplacian

Cohomology and massless fields

Higher Symmetries of the Laplacian