Pure Mathematics

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Browsing Pure Mathematics by Author "Bailey, Toby N."

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    Complex analysis and the Funk transform
    (Korean Mathematical Society, 2003) Bailey, Toby N.; Eastwood, Michael George; Gover, A. Rod; Mason, L. J.; School of Mathematical Sciences : Pure Mathematics
    The Funk transform is defined by integrating a function on the two-sphere over its great circles. We use complex analysis to invert this transform.
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    Exceptional invariants in the parabolic invariant theory of conformal geometry
    (American Mathematical Society, 1995) Bailey, Toby N.; Gover, A. Rod
    We provide a construction for the exceptional invariants of certain modules for a parabolic subgroup of a pseudo-orthogonal group. The invariant theory of these modules has applications in conformal geometry.
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    The Funk transform as a Penrose transform
    (Cambridge University Press, 1999) Bailey, Toby N.; Eastwood, Michael George; Gover, A. Rod; Mason, Lionel J.
    The Funk transform is the integral transform from the space of smooth even functions on the unit sphere S²[subset or is implied by][open face R]³ to itself defined by integration over great circles. One can regard this transform as a limit in a certain sense of the Penrose transform from [open face C][open face P]₂ to [open face C][open face P]*ast;₂. We exploit this viewpoint by developing a new proof of the bijectivity of the Funk transform which proceeds by considering the cohomology of a certain involutive (or formally integrable) structure on an intermediate space. This is the simplest example of what we hope will prove to be a general method of obtaining results in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.
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    Smoothly parameterized Cech cohomology of complex manifolds
    (Mathematica Josephina Inc, 2005) Bailey, Toby N.; Eastwood, Michael George; Gindikin, Simon G.; School of Mathematical Sciences : Pure Mathematics
    A Stein covering of a complex manifold may be used to realize its analytic cohomology in accordance with the Cˇech theory. If, however, the Stein covering is parameterized by a smooth manifold rather than just a discrete set, then we construct a cohomology theory in which an exterior derivative replaces the usual combinatorial Cˇech differential. Our construction is motivated by integral geometry and the representation theory of Lie groups.
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    Twistor results for integral transforms
    (American Mathematical Society, 2001) Bailey, Toby N.; Eastwood, Michael George; AMS-IMS-SIAM Joint Summer Research Conference on Radon Transforms and Tomography (2000 : Mount Holyoke College); School of Mathematical Sciences : Pure Mathematics
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    Zero-energy fields on real projective space
    (Springer, 1997) Bailey, Toby N.; Eastwood, Michael George
    A smooth 1-form on real projective space with vanishing integral along all geodesics is said to have zero energy. Such a 1-form is necessarily the exterior derivative of a smooth function. We formulate a general version of this theorem for tensor fields on real projective space and prove it using methods of complex analysis. A key ingredient is the cohomology of involutive structures.