The study of periods of automorphic forms using the theta correspondence and the Weil representation was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral weight, periods over tori of modular forms of integral weight and special values of L-functions attached to these modular forms. In this thesis, we show that there are general relations among periods of automorphic forms on groups related by the theta correspondence. For example, if G is a symplectic group and H is an orthogonal group over a number field k, these relations are identities equating Fourier coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and follow from the basic properties of theta functions and the Weil representation; further study is required to show how they compare to the results of Waldspurger. The second part of this thesis shows that, under some restrictions, the identities alluded to above are the result of a comparison of nonstandard relative traces formulas. In this comparison, the relative trace formula for H is standard however the relative trace formula for G is novel in that it involves the trace of an operator built from theta functions. The final part of this thesis explores some preliminary results on local height pairings of special cycles on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods of automorphic forms over orthogonal subgroups).

The study of periods of automorphic forms using the theta correspondence and the Weil representation was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral weight, periods over tori of modular forms of integral weight and special values of L-functions attached to these modular forms. In this thesis, we show that there are general relations among periods of automorphic forms on groups related by the theta correspondence. For example, if G is a symplectic group and H is an orthogonal group over a number field k, these relations are identities equating Fourier coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and follow from the basic properties of theta functions and the Weil representation; further study is required to show how they compare to the results of Waldspurger. The second part of this thesis shows that, under some restrictions, the identities alluded to above are the result of a comparison of nonstandard relative traces formulas. In this comparison, the relative trace formula for H is standard however the relative trace formula for G is novel in that it involves the trace of an operator built from theta functions. The final part of this thesis explores some preliminary results on local height pairings of special cycles on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods of automorphic forms over orthogonal subgroups).

We aim to have a comprehensive understanding of holographic correspondence and to demonstrate how the holographic correspondence (or renormalization) can be applied. Thus this thesis is divided into two parts. The first part is devoted to the former purpose (chapters 1 to 4 including appendix A,B, and C), and the second part is dedicated for the latter purpose (chapter 5 to 7). In Part I, the structure of the AdS/CFT correspondence is analyzed, and the properties of the AdS spacetime is studied in the context of the AdS/CFT correspondence; Here, we investigate the isometry group, the conformal structure, and generation of asymptotic solution near the conformal boundary. This solution yields significant convenience for the process of holographic renormalization. Moreover the properties of the Minkowski spacetime are compared to those of the AdS spacetime. To develop a greater understanding of the Lifshitz/quantum critical theory correspondence, the quantum phase transition is studied. Furthermore The holographic renormalization is briefly reviewed. In part II, the holographic renormalization associated with the Mann-Marolf (MM) counterterm is investigated for the asymptotically Minkowski spacetime in (n+3) dimensions. As a boundary condition, the cylindrical coordinate is considered. The solution of the MM-counterterm is obtained by solving the given algebraic equation, and from the counterterm solution, the boundary stress tensor is calculated. It is proven that the formula for conserved quantities via the boundary stress tensor holds. Next, we investigate deformations of Lifshitz holography with the Gauss-Bonnet term in (n+1) dimensional spacetime. To admit the non-trivial solution of the sub-leading orders, a value of the dynamical critical exponent z is restricted by z= n-1-2(n-2)α̃, where is the (redefined) Gauss-Bonnet coupling constant. Such solution of sub-leading orders correspond to the marginally relevant modes for the massive vector field and are generated by Λ~0, at the asymptotic region. A generic black hole solution, which is characterized by the horizon flux of the vector field and α̃, is considered in the bulk. We explore its thermodynamic properties, which depend on temperature, by varying n and α̃. As a result, the contribution of the marginally relevant mode is found in a function of Λ^z/T, and the relation between the free energy density and the energy density is numerically recovered when the marginally relevant mode is turned off (Λ=0), is obtained.

Thesis (M. F. A.)--Texas Woman's University, 2006. / Includes bibliographical references.

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Gabrielle Roy kept up, throughout her literary life, a regular correspondence with relatives, friends, business relations and readers: to date, more than 2000 letters have been preserved in different archive collections. The 482 letters she wrote to her husband, Doctor Marcel Carbotte, between 1947 and 1979, form the largest subset of the correspondence; a critical edition of these hitherto unpublished letters is presented in the second part of the thesis. / Surprisingly, the correspondence contains few reflexions of an aesthetic nature, and few explicit traces of the novelist's published work, apart from occasional references to the texts she is working on at the moment she is writing to Marcel. Instead, it is the details of her day-to-day life that Roy describes most abundantly. In the first part of the thesis, we study the letters to Marcel in the light of a possible connection with Roy's canonical work: we start by examining the link between the letter and autobiographical genre; we then show that the letters to Marcel can be read as a personal diary; finally, we suggest that the "autobiographical turn" taken by Roy in both her published work and her letters to Marcel indicates that the letters form part of the same system as the autobiographical works.