Dado um grupo G, a definição dos grupos de cohomologia com coeficientes em um ZG-módulo M podem ser dadas usando as técnicas usuais da Álgebra Homológica, que garantem a existência de resoluções projetivas P de Z como um ZG-módulo trivial, a equivalência entre resoluções distintas etc. Podemos também construir o produto cup em cohomologia, cuja definição depende de uma aproximação da diagonal para a resolução projetiva P. Entretanto, o cálculo explicito de tais resoluções e dos grupos de cohomologia pode ser bastante difícil na prática, e ainda mais difícil a obtenção de uma aproximação da diagonal. Nesta tese, obteremos resoluções livres e aproximações da diagonal para os grupos fundamentais das superfícies que são espaços K(G,1) e também para o grupo fundamental de fibrados do toro com base S^1, bem como a estrutura de anel de cohomologia de tais grupos. Ainda, para certos grupos virtualmente cíclicos G, obteremos o anel de cohomologia calculando diretamente uma resolução livre e uma aproximação da diagonal, ou então usando a sequência espectral de Lyndon-Hochschild-Serre. A motivação para o estudo da primeira família de grupos vem do fato de representarem variedades de dimensão 2 e 3, e da segunda família por ser constituída de grupos que atuam em esferas de homotopia. / Given a group G, a definition for its cohomology groups with coefficients in a given ZG-module M can be given using the standard techniques of Homological Algebra, that ensure the existence of projective resolutions P of Z as a trivial ZG-module, the equivalence between two such resolutions etc . We can also construct the cup product, whose definition depends on a diagonal approximation for a given projective resolution P. However, the explicit computation of such resolutions and of the cohomology groups may be very hard in practice, and even worse may be the task of constructing a diagonal approximation. In this thesis, we obtain free resolutions and diagonal approximations for the fundamental groups of surfaces that are K(G,1) spaces and for the fundamental group of the torus bundle with the circle as the base space, as well as the structure of the cohomology ring of these groups. Also, for some virtually cyclic groups, we obtain the cohomology ring by an explicit computation of a free resolution and a diagonal approximation, or by the Lyndon-Hochschild-Serre spectral sequence. The motivation for the study of the first family of groups comes from the fact that such groups represent manifolds of dimension 2 and 3, and the groups of the second family act on homotopy spheres.

Methyl bromide (MBr) was widely used as a soil fumigant to manage soilborne pests in plasticulture vegetable production; however, it has been banned by the United Nations Environment Programme. Alternatives to MBr must be implemented to sustain fresh market tomato productivity. Possible MBr alternatives include new fumigant compounds, improved plastic mulch, and grafting. Methyl iodide (MeI) and dimethyl disulfide (DMDS) were tested as fumigant alternatives to MBr for the control of yellow nutsedge and soilborne pathogens of tomato. Virtually impermeable film (VIF) and totally impermeable film (TIF) were tested for fumigant retention and yellow nutsedge control in tomato. Grafting onto resistant rootstocks was tested for bacterial wilt and nematode management in tomato. In the absence of a soil fumigant, TIF suppressed yellow nutsedge better than VIF. TIF increased fumigant retention compared to VIF at similar application rates. Reduced fumigant application rates could be used in combination with TIF while maintaining fumigant concentrations, weed control, and crop yields comparable to greater use rates with VIF. Shank applied DMDS rates could be lowered to 281 L/ha under TIF from 468 L/ha under VIF; shank applied MeI application rates could be reduced to 56 L/ha under TIF compared to 93 L/ha under VIF and drip applied DMDS could be reduced from 561 L/ha under VIF film to 374 L/ha under TIF. Grafting susceptible commercial tomato cultivars onto resistant tomato hybrid rootstocks increased yields and plant survival in bacterial wilt infested fields. "Cheong Gang", "BHN 998", and "BHN 1054" were the best performing rootstocks for bacterial wilt resistance and tomato fruit yield in severely infested fields. Grafting increased tomato yield and decreased root galling from root-knot nematodes in an infested field. Hybrid rootstock "RST 106" resulted in the lowest root-knot nematode galling. In conclusion, TIF with reduced rates of DMDS or MeI is a viable MBr alternative for fresh market tomato production to retain effective doses of fumigant, manage yellow nutsedge and maintain yields. Grafting is an effective MBr alternative to manage bacterial wilt and root-knot nematode and maintain tomato yields. / Ph. D.

<p><br></p><p dir="ltr">Over the past decade, optical frequency combs have spurred significant advancements in both classical ultrafast optics and quantum optics. My research contributes to these two fields, catering to applications in precision metrology and optical networking. In the domain of quantum optics, the study delves into biphoton frequency combs with time-energy entanglement, employing novel electro-optic modulation schemes to enhance sensitivity and enable precise measurements of temporal correlations. Additionally, Bell states, a crucial class of entangled quantum bases, are generated in the frequency domain, showcasing their utility in delay metrology and quantum cryptographic protocols. </p><p dir="ltr">In the realm of classical optical frequency combs, this work explores dynamic steering of pulsed optical beams, holding promise for applications in imaging and remote sensing. The concept of time-efficient dynamic beam steering using a spatial array of optical frequency combs is elucidated and experimentally demonstrated through the utilization of a high-resolution spectral disperser, specifically a virtually imaged phased array (VIPA). Furthermore, integrated photonic designs featuring wavelength-selective switches and spectral dispersers are proposed to enable a versatile on-chip implementation of the beam steering approach. In sum, this research leverages the capabilities of classical and quantum optical frequency combs, with implications for emerging applications such as distributed sensing, quantum networking, and light detection and ranging (LIDAR).</p>