Let M#k(64) be the space of weakly holomorphic modular forms in level 64 and weight k which can have poles only at infinity, and let S#k(64) be the subspace of M#k(64) consisting of forms which vanish at all cusps other than infinity. We explicitly construct canonical bases for these spaces and show that the coefficients of these basis elements satisfy Zagier duality. We also compute the generating function for the canonical basis.

Let N ∈ {8,9,16,25} and let M#0(N) be the space of level N weakly holomorphic modular functions with poles only at the cusp at infinity. We explicitly construct a canonical basis for M#0(N) indexed by the order of the pole at infinity and show that many of the coefficients of the elements of these bases are divisible by high powers of the prime dividing the level N. Additionally, we show that these basis elements satisfy an interesting duality property. We also give an argument that extends level 1 results on congruences from Griffin to levels 2, 3, 4, 5, 7, 8, 9, 16, and 25.

Let M#k(52) be the space of weight k level 52 weakly holomorphic modular forms with poles only at infinity, and S#k(52) the subspace of forms which vanish at all cusps other than infinity. For these spaces we construct canonical bases, indexed by the order of vanishing at infinity. We prove that the coefficients of the canonical basis elements satisfy a duality property. Further, we give closed forms for the generating functions of these basis elements.

Thesis (Ph. D.)--University of Hong Kong, 2000. / Includes bibliographical references (leaves 50-53).

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 72-74).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 1999. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph.D.)--University of Hong Kong, 2000. / Includes bibliographical references (leaves 50-53) Also available in print.

Orientador: Jorge Tulio Mujica Ascui / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-07T16:44:31Z (GMT). No. of bitstreams: 1 Takatsuka_Paula_D.pdf: 3540981 bytes, checksum: 643e40ac81900cf042cfe1cfb3737b0d (MD5) Previous issue date: 2006 / Resumo: Este trabalho estende teoremas clássicos da teoria de funções holomorfas de uma variável complexa para espaços localmente convexos de dimensão infinita. Serão dadas várias caracterizações de famílias normais, n¿ao apenas com relação à topologia compacto-aberta, mas também para outras topologias naturais no espaço de aplicações holomorfas. Teoremas de tipo Montel e de tipo Schottky, bem como outros resultados correlatos, ser¿ao estabelecidos em dimensão infinita para as diferentes topologias. Teoremas de limita¸c¿ao universal sobre famílias de funções holomorfas que omitem dois valores distintos ser¿ao formulados para espaços de Banach / Abstract: The present work extends some classical theorems from the theory of holomorphic functions of one complex variable to infinite dimensional locally convex spaces. Several characterizations of normal families are given, not only for the compact-open topology, but also for other natural topologies on spaces of holomorphic mappings. Montel-type and Schottky-type theorems and various related results are established in infinite dimension for these different topologies. Universal boundedness theorems concerning families of holomorphic functions which omit two distinct values are formulated for Banach spaces / Doutorado / Mestre em Matemática

In this thesis, we study compactifiations of ten-dimensional heterotic supergravity at O(α'), focusing on the moduli of such compactifications. We begin by studying supersymmetric compactifications to four-dimensional maximally symmetric space, commonly referred to as the Strominger system. The compactifications are of the form M₁₀ = M₄ x X, where M₄ is four-dimensional Minkowski space, and X is a six-dimensional manifold of what we refer to as heterotic SU(3)-structure. We show that this system can be put in terms of a holomorphic operator D on a bundle Q = T* X ⊕ End(TX) ⊕ End(V ) ⊕ TX, defined by a series of extensions. Here V is the E₈ x E₈ gauge-bundle, and TX is the tangent bundle of the compact space X. We proceed to compute the infinitesimal deformation space of this structure, given by TM = H^(0,1)(Q), which constitutes the infinitesimal spectrum of the lower energy four-dimensional theory. In doing so, we find an over counting of moduli by H^(0,1)(End(TX)), which can be reinterpreted as O(α') field redefinitions. In the next part of the thesis, we consider non-maximally symmetric compactifications of the form M₁₀ = M₃ x Y , where M₃ is three-dimensional Minkowski space, and Y is a seven-dimensional non-compact manifold with a G₂-structure. We write X → Y → ℝ, where X is a six dimensional compact space of half- at SU(3)-structure, non-trivially fibered over ℝ. These compactifications are known as domain wall compactifications. By focusing on coset compactifications, we show that the compact space X can be endowed with non-trivial torsion, which can be used in a combination with %α'-effects to stabilise all geometric moduli. The domain wall can further be lifted to a maximally symmetric AdS vacuum by inclusion of non-perturbative effects in a heterotic KKLT scenario. Finally, we consider domain wall compactifications where X is a Calabi-Yau. We show that by considering such compactifications, one can evade the usual no-go theorems for flux in Calabi-Yau compactifications, allowing flux to be used as a tool in such compactifications, even when X is Kähler. The ultimate success of these compactifications depends on the possibility of lifting such vacua to maximally symmetric ones by means of e.g. non-perturbative effects.

Since the 1980s, much progress has been done in completely determining which functions share a Julia set. The polynomial case was completely solved in 1995, and it was shown that the symmetries of the Julia set play a central role in answering this question. The rational case remains open, but it was already shown to be much more complex than the polynomial one. In this thesis, we review existing results on rational maps sharing a Julia set, and offer results of our own on the symmetry group of such maps. / Desde a década de oitenta, um enorme progresso foi feito no problema de determinar quais funções têm o mesmo conjunto de Julia. O caso polinomial foi completamente respondido em 1995, e mostrou-se que as simetrias do conjunto de Julia têm um papel central nessa questão. O caso racional permanece aberto, mas já se sabe que ele é muito mais complexo do que o polinomial. Nesta dissertação, nós revisamos resultados existentes sobre aplicações racionais com o mesmo conjunto de Julia e apresentamos nossos próprios resultados sobre o grupo de simetrias de tais aplicações.