Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 127-131).

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2005. / Includes bibliographical references (leaves 75-79). Also available in electronic version.

Thesis (Ph. D.)--University of Chicago, 1922. / Biography. "A Dissertation, submitted to the Faculty of the Ogden Graduate School of Science in candidacy for the Degree of Doctor of Philosophy, Department of Mathematics. "Private edition, distributed by the University of Chicago Libraries, Chicago, Illinois." "Reprinted from Transactions American mathematical society, vol. XXIV, no. 2, September, 1922." Includes bibliographical references.

Following an examination of the properties of the conformal group in 4-space, a review is made of the procedure by which conformally covariant massless field equations are written in manifestly covariant form. By writing the Minkowski coordinates in terms of coordinates on the null hyperquadric of a 6-dimensional flat space with two timelike directions, the action of the group is linearized and field equations are written in rotationally covariant form in 6-dimensional space. It is then shown that extending the 6-coordinates off the null surface generalizes Minkowski space to a 5-dimensional space. Such a generalization necessitates employing a method of descent to 4-dimensional space from six dimensions which differs from the usual procedure, and allows one to encompass massive field theories in the manifest formalism. It is demonstrated that these massive fields can be understood as manifestations in Minkowski space of massless fields in 5-dimsnsional space. For the case of spinors, the field equation can accomodate precisely two species of particle having two different masses. An action principle is developed in the 6-space, and a method of field quantization is devised. As examples of the method, the special cases of spin-0, spin-1/2, and spin-1 fields are examined in detail, and minimal coupling of the spinor field equation is carried out. The formalism presented in this investigation provides a means by which one can apprehend a massive compensating field within the confines of a gauge invariant theory. The interactions which are obtained in Minkowski space include not only the usual couplings with massive vector or pseudovector fields, but as well the pseudoscalar coupling occurs automatically within this gauge invariant formulation. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

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Kuranishi atlases were introduced by McDuff and Wehrheim as a means to build a virtual fundamental cycle on moduli spaces of J-holomorphic curves and resolve some of the challenges in this field. This thesis considers genus zero Gromov-Witten invariants on a general closed symplectic manifold. We complete the construction of these invariants using Kuranishi atlases. To do so, we show that Gromov-Witten moduli spaces admit a smooth enough Kuranishi atlas to define a virtual fundamental class in any virtual dimension. In the process, we prove a stronger gluing theorem. Once we have defined genus zero Gromov-Witten invariants, we show that they satisfy the Gromov-Witten axioms of Kontsevich and Manin, a series of main properties that these invariants are expected to satisfy. A key component of this is the introduction of the notion of a transverse subatlas, a useful tool for working with Kuranishi atlases.