Thesis (Th. M.)--Westminster Theological Seminary, Philadelphia, 1993. / Includes bibliographical references (leaves 82-94).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, December 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Th. M.)--Westminster Theological Seminary, Philadelphia, 1993. / Includes bibliographical references (leaves 82-94).

Univ., Diss., 2007--Duisburg-Essen. / Literaturverz. S. 223 - 225.

Thesis (M.F.A.)--Georgia State University, 2008. / Title from title page (Digital Archive@GSU, viewed July 6, 2010) Constance Thalken, committee chair; Sheldon Schiffer, Nancy Floyd, committee members. Includes bibliographical references (p. 35).

Este trabalho apresenta os conceitos envolvidos na definição de um complexo elíptico sobre uma variedade compacta M e desenvolve a teoria de Hodge neste complexo. O principal resultado em questão é o teorema de Hodge. No caso mais simples, dados dois fibrados vetoriais E → M , F → M e um operador diferencial elíptico L : Γ(E) → Γ(F ), agindo nas seções destes fibrados, o teorema de Hodge garante que a dimensão de seu núcleo N(L) é finita e que podemos decompor Γ(E) = N(L) ⊕ Im(L∗), onde Im(L∗) é a imagem da adjunta de L. Para a demonstração apresentada aqui, são empregadas as propriedades dos espaços de Sobolev Hm(E) das seções de E. Certa ênfase é dada na obtenção de propriedades globais a partir de resultados locais. / This work presents concepts involved in the definition of an elliptic complex on a compact manifold M and develops the Hodge theory over this complex. The main result at hand is the Hodge theorem. In the simplest case, given two vector bundles E → M , F → M and an elliptic differential operator L : Γ(E) → Γ(F ), acting on sections of these bundles, the Hodge theorem ensures that the dimension of its kernel N(L) is finite and that we can decompose Γ(E) = N(L) ⊕ Im(L∗), where Im(L∗) is the range of the adjoint of L. In the proof presented here, we employ properties of the Sobolev spaces Hm(E) of sections of E. We give an emphasis to obtaining global properties from local results.

Dans cette thèse, nous généralisons les inégalités universelles de Yang etde Levitin et Parnovski, concernant les valeurs propres du laplacien de Dirichlet sur undomaine euclidien borné, au cas du laplacien de Hodge-de Rham sur une sous-variétéeuclidienne fermée. Cela permet une extension de l’inégalité de Reilly et de l’inégalitéd’Asada, concernant respectivement la première valeur propre du laplacien et celle dulaplacien de Hodge-de Rham, à toutes les valeurs propres de ces deux opérateurs. Ensuite,nous obtenons une nouvelle inégalité algébrique qui relie les valeurs propres d’un opérateurauto-adjoint sur un espace d’Hilbert à deux familles d’opérateurs symétriques et antisymétriqueset à leurs commutateurs. Cette inégalité permet d’unifier et d’améliorer denombreux résultats connus concernant le laplacien, le laplacien de Hodge-de Rham, lecarré de l’opérateur de Dirac et plus généralement le laplacien agissant sur les sections d’unfibré vectoriel riemannien au-dessus d’une sous-variété euclidienne, le laplacien de Kohn,les puissances du laplacien... Dans une dernière partie, nous montrons une majoration dela première valeur propre du problème de Steklov sur un domaine Ω d’une sous-variétéeuclidienne ou sphérique, en fonction des r-courbures moyennes de son bord ∂Ω. / In this thesis, we generalize the Yang and the Levitin and Parnovski universalinequalities, concerning the eigenvalues of the Dirichlet Laplacian on a Euclideanbounded domain, to the case of the Hodge-de Rham Laplacian on a Euclidean closed submanifold.This gives an extension of Reilly’s inequality and Asada’s inequality, concerningthe first eigenvalues of the Laplacian and the Hodge-de Rham Laplacian respectively, toall eigenvalues of these operators. We also obtain a new abstract inequality relating theeigenvalues of a self-adjoint operator on a Hilbert space to two families of symmetric andskew-symmetric operators and their commutators. This inequality is proved useful both forunifying and for improving numerous known results concerning the Laplacian, the Hodgede Rham Laplacian, the square of the Dirac operator and more generally the Laplacianacting on sections of a Riemannian vector bundle on a Euclidean submanifold, the KohnLaplacian, a power of the Laplacian...In the last part, we obtain an upper bound for thefirst eigenvalue of Steklov problem on a domain Ω of a Euclidean or a spherical submanifoldin terms of the r-th mean curvatures of ∂Ω

This thesis introduces two notions of motive associated to a log scheme. We introduce a category of log motives à la Voevodsky, and prove that the embedding of Voevodsky motives is an equivalence, in particular proving that any homotopy-invariant cohomology theory of schemes extends uniquely to log schemes. In the case of a log smooth degeneration, we give an explicit construction of the motivic Albanese of the degeneration, and show that the Hodge realization of this construction gives the Albanese of the limit Hodge structure.

Este trabalho apresenta os conceitos envolvidos na definição de um complexo elíptico sobre uma variedade compacta M e desenvolve a teoria de Hodge neste complexo. O principal resultado em questão é o teorema de Hodge. No caso mais simples, dados dois fibrados vetoriais E → M , F → M e um operador diferencial elíptico L : Γ(E) → Γ(F ), agindo nas seções destes fibrados, o teorema de Hodge garante que a dimensão de seu núcleo N(L) é finita e que podemos decompor Γ(E) = N(L) ⊕ Im(L∗), onde Im(L∗) é a imagem da adjunta de L. Para a demonstração apresentada aqui, são empregadas as propriedades dos espaços de Sobolev Hm(E) das seções de E. Certa ênfase é dada na obtenção de propriedades globais a partir de resultados locais. / This work presents concepts involved in the definition of an elliptic complex on a compact manifold M and develops the Hodge theory over this complex. The main result at hand is the Hodge theorem. In the simplest case, given two vector bundles E → M , F → M and an elliptic differential operator L : Γ(E) → Γ(F ), acting on sections of these bundles, the Hodge theorem ensures that the dimension of its kernel N(L) is finite and that we can decompose Γ(E) = N(L) ⊕ Im(L∗), where Im(L∗) is the range of the adjoint of L. In the proof presented here, we employ properties of the Sobolev spaces Hm(E) of sections of E. We give an emphasis to obtaining global properties from local results.

Este trabalho apresenta os conceitos envolvidos na definição de um complexo elíptico sobre uma variedade compacta M e desenvolve a teoria de Hodge neste complexo. O principal resultado em questão é o teorema de Hodge. No caso mais simples, dados dois fibrados vetoriais E → M , F → M e um operador diferencial elíptico L : Γ(E) → Γ(F ), agindo nas seções destes fibrados, o teorema de Hodge garante que a dimensão de seu núcleo N(L) é finita e que podemos decompor Γ(E) = N(L) ⊕ Im(L∗), onde Im(L∗) é a imagem da adjunta de L. Para a demonstração apresentada aqui, são empregadas as propriedades dos espaços de Sobolev Hm(E) das seções de E. Certa ênfase é dada na obtenção de propriedades globais a partir de resultados locais. / This work presents concepts involved in the definition of an elliptic complex on a compact manifold M and develops the Hodge theory over this complex. The main result at hand is the Hodge theorem. In the simplest case, given two vector bundles E → M , F → M and an elliptic differential operator L : Γ(E) → Γ(F ), acting on sections of these bundles, the Hodge theorem ensures that the dimension of its kernel N(L) is finite and that we can decompose Γ(E) = N(L) ⊕ Im(L∗), where Im(L∗) is the range of the adjoint of L. In the proof presented here, we employ properties of the Sobolev spaces Hm(E) of sections of E. We give an emphasis to obtaining global properties from local results.