ANALYTIC APPROACHES IN LAGRANGIAN GEOMETRY

The focus of this thesis is two equations that arise in special Lagrangian geometry: the degenerate special Lagrangian equation (DSL) and the Lagrangian mean curvature flow (LMCF). A significant part of this focus centers on Dirichlet duality, subequations, and viscosity solutions, the analytic framework which we use to formulate and study both equations. Given a Calabi--Yau manifold $(X, \omega, J, \Omega)$ and a model manifold $M$, one can construct a kind of moduli space of Lagrangians in $X$ called the space of positive Lagrangians. A Lagrangian $L\subset X$ belongs to this infinite-dimensional space if $L$ is diffeomorphic to $M$ and Re$(\Omega|_L) >0$. % i.e., the set of Lagrangians that are Hamiltonian deformations of a fixed Lagrangian, A Hamiltonian deformation class of the space of positive Lagrangians admits an $L^2$-type Riemannian metric which allows one to study this space from a geometric point of view. Geodesics in this space play a crucial role in a program initiated by Solomon \cite{S1, S2} to understand the existence and uniqueness of special Lagrangian submanifolds in Calabi–Yau manifolds. They also play a key role in a new approach to the Arnold conjecture put forth by Rubinstein–Solomon and in the development of a pluripotential theory for Lagrangian graphs \cite{RS, DR}. The DSL arises as the geodesic equation in the space of positive graph Lagrangians when $X= \mathbb{C}^n$ and $\omega$ and $\Omega$ are associated to the Euclidean structure \cite{RS}. Building on the results of Rubinstein--Solomon \cite{RS}, we show that the DSL induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth semi-flat Calabi--Yau torus fibration are governed by the Riemannian DSL on the product of the base manifold and an interval. The geodesic endpoint problem in this setting thus corresponds to solving the Dirichlet problem for the DSL. However, the DSL is a degenerate-elliptic, fully non-linear, second-order equation, and so the standard elliptic theory does not furnish solutions. Moreover, for Lagrangians with boundary the natural domains on which one would like to solve the Dirichlet problem are cylindrical and thus not smooth. These issues are resolved by Rubinstein--Solomon in the Euclidean setting by adapting the Dirichlet duality framework of Harvey--Lawson to domains with corners \cite{RS}. We further develop these analytic techniques, specifically modifications of the Dirichlet duality theory in the Riemannian setting to obtain continuous solutions to the Dirichlet problem for the Riemannian DSL and hence, in certain settings, continuous geodesics in the space of positive Lagrangians. The uniqueness of solutions to the Dirichlet problem in the Euclidean formulation of Dirichlet duality theory relies on an important convex-analytic theorem of Slodkowski \cite{Slod}. Motivated by the significance of this result and the technical, geometric nature of its proof, we provide a detailed exposition of the proof. We then study some of the quantities involved using the Legendre transform, offering a dual perspective on this theorem. Given a Lagrangian submanifold in a Calabi--Yau, a fundamental and still open question is whether or not there is a special Lagrangian representative in its homology or Hamiltonian isotopy class. A natural approach to this problem is the Lagrangian mean curvature flow, which preserves not only the Lagrangian condition but also the homology and isotopy class. Assuming the flow exists for all time and converges, it will converge to a minimal (i.e., zero mean curvature) Lagrangian. In the Calabi--Yau setting these are precisely the special Lagrangian submanifolds. A major conjecture in this area is the Thomas--Yau conjecture \cite{TY}, which posits certain stability conditions on the initial Lagrangian under which the LMCF will exist for all time and converge to the unique special Lagrangian in that isotopy class. Thomas--Yau stated a variant of their conjecture for a related, more tractable flow, called the almost Lagrangian mean curvature flow (ALMCF). In the setting of highly symmetric Lagrangian spheres in Milnor fibers, and under some additional technical assumptions, they make significant progress towards a proof of this variant of the conjecture \cite{TY}. We study the flow of $2$-spheres from a slightly different perspective and provide a relatively short proof of the longtime existence of viscosity solutions under certain stability conditions, and their convergence to a special Lagrangian sphere

Imunossensor eletroquímico label-free baseado em ácido polilático/grafeno e impresso 3D para a detecção de microcistina-LR

TCC (graduação) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Química.A contaminação ambiental das águas naturais, ocasionada pelo gerenciamento inadequado de resíduos industriais, agrícolas e urbanos, causa um grande impacto à saúde pública. Como consequência, ocorre a eutrofização da água, favorecendo a proliferação excessiva de cianobactérias. Além de causarem sabor e odor desagradáveis na água, as cianobactérias podem produzir algumas toxinas chamadas de microcistinas, que podem causar câncer hepático ou levar à óbito. Tais cianobactérias também são caracterizadas pela propriedade de bioacumulação, podendo ser encontrada no fígado e músculo de peixes, intoxicando os que deles se alimentam. Assim, a OMS estabelece um limite de 1,0 μg L-1 ou 1,0 ng mL-1 de microcistina-LR na água potável. Dentre as formas de detecção de microcistinas, a utilização de sensores eletroquímicos oferece vantagens como elevada sensibilidade, simplicidade instrumental, relativo baixo custo e possibilidade de miniaturização. Nos últimos anos, com o desenvolvimento da tecnologia de impressão 3D, uma nova classe de eletrodos plásticos com aplicações em eletroanálise tem sido estudada. Nesse tipo de eletrodo, materiais como grafeno são adicionados ao polímero ácido polilático, tornando-o condutor. Entretanto, devido à baixa concentração do material condutor, pré-tratamentos como a ativação eletroquímica em NaOH são aplicados para melhorar o desempenho dos eletrodos. Neste projeto, o eletrodo 3D foi ativado eletroquimicamente em NaOH 1,0 mol L-1 pela aplicação de +1,6 V vs. Ag/AgCl, KCl(saturado) por 1800 s. Após ativado, um imunossensor 3D para a detecção de microcistina-LR foi construído através da imobilização covalente do anticorpo anti-microcistina-LR via EDC/NHS. Com a realização da otimização dos parâmetros da técnica eletroanalítica de voltametria de onda quadrada, uma curva de calibração foi construída. Os limites de detecção e de quantificação foram de 1,28 e 3,88 ng mL-1, respectivamente.The environmental contamination of natural waters, caused by inadequate management of industrial, agricultural and urban waste, has a major impact on public health. As a consequence, water eutrophication occurs, favoring the excessive proliferation of cyanobacteria. In addition to causing unpleasant taste and odor in water, cyanobacteria can produce some toxins called microcystins, which can cause liver cancer or lead to death. Such cyanobacteria are also characterized by the property of bioaccumulation, and can be found in the liver and muscle of fish, intoxicating those who feed on them. Thus, the WHO establishes a limit of 1.0 μg L-1 or 1.0 ng mL-1 of microcystin-LR in drinking water. Among the ways of detecting microcystins, the use of electrochemical sensors offers advantages such as high sensitivity, instrumental simplicity, relatively low cost and the possibility of miniaturization. In recent years, with the development of 3D printing technology, a new class of plastic electrodes with applications in electroanalysis has been studied. In this type of electrode, materials such as graphene are added to the polylactic acid polymer, making it conductive. However, due to the low additive concentration, pre-treatments such as electrochemical activation in NaOH are applied to improve the performance of the electrodes. In this project, the 3D electrode was electrochemically activated in 1.0 mol L-1 NaOH by applying +1.6 V vs. Ag/AgCl, KCl(saturated) for 1800 s. After activation, a 3D immunosensor for the detection of microcystin-LR was constructed through covalent immobilization of the anti-microcystin-LR antibody via EDC/NHS. With the optimization of the parameters of the electroanalytical technique of square wave voltammetry, a calibration curve was constructed. The limits of detection and quantification were 1.28 and 3.88 ng mL-1, respectively

Embryology of the spider crabs Leurocyclus tuberculosus (H. Milne-Edwards & Lucas 1842) and Libinia spinosa (H. Milne-Edwards 1834) (Brachyura, Majoidea)

The embryonic development of the spider crabs Leurocyclus tuberculosus and Libinia spinosa was described based on laboratory reared specimens. Living embryos were removed from female every 48 h and external morphometric traits, dimensions and development time were registered. Total development was divided into five periods based on: I) the cleavage, II) the embryonic primordium, III) the optic lobes, IV) the pigmentation in the optic lobes and V) the chromatophores. Different characteristics like spines, setae, telson morphology and processes, make both species distinguishable from the period III until hatching. Egg volume was greater in Leurocyclus tuberculosus than in Libinia spinosa. The duration of each period was not similar during development; while in L. tuberculosus the period II (morphogenesis) is the longest, in L. spinosa, the period IV (pigmentation) is the longest. Complete embryonic development at 14°C lasted 36 days in Leurocyclus tuberculosus and 57 days in Libinia spinosa. The knowledge about the embryonic development is essential for studies on reproduction and fundamental ecology of marine bottom invertebrates. It provides a useful tool to determine the time elapsed since spawning and the time remaining before hatching in live eggs, in ecological and comparative studies. Fil: Gonzalez Pisani, Ximena. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Nacional Patagónico; ArgentinaFil: Dellatorre, Fernando Gaspar. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Nacional Patagónico; ArgentinaFil: Lopez, Laura Susana. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Biodiversidad y Biologia Experimental y Aplicada; Argentina. Universidad de Buenos Aires. Facultad de Cs.exactas y Naturales. Departamento de Biodiversidad y Biologia Experimental. Laboratorio de Fisiologia de Crustáceos; Argentin

Rapid expansion and potential range of the invasive kelp Undaria pinnatifida in the southwestern Atlantic

Coastal Patagonia, often regarded as a pristine confine of the world ocean, has been invaded by non-indigenous species that are rapidly modifying local ecosystems. One of the most conspicuous invaders is the kelp Undaria pinnatifida. First recorded near the city of Puerto Madryn (Argentina, 42.75°S) in 1992, during the last 15 years its range spread more than five degrees of latitude to the south. By 2007 it was first detected north of Valdes Península (a natural barrier), beginning its northward expansion, and between 2007 and 2011 spread along the coasts of San José and San Matías gulfs at an average rate of ~50 km yr-1. A small population was detected in Mar del Plata (Argentina, 38.04°S) in 2011, more than three degrees of latitude to the north, suggesting a new human-mediated inoculation. Based on experimental information and remotely sensed sea-surface temperature (SST) at the latitudinal range limits of U. pinnatifida populations worldwide, we hypothesize a thermal tolerance window between -0.6 °C and 16.8 °C average SST in the coldest month, and between 20 °C and 28 °C in the warmest month. Using climatologic satellite SST from several locations, we predict a potential latitudinal thermal range extending between Puerto Deseado (Argentina, 47.75°S) and Torres (Brazil; 29.35°S). Salinity and substrate quality however constrain suitable habitat in the southwestern Atlantic to discrete stretches of coastline, suggesting that the northern potential boundary would be around La Coronilla (33.90°S), on the Uruguayan littoral.Fil: Dellatorre, Fernando Gaspar. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Nacional Patagónico; ArgentinaFil: Amoroso, Ricardo Oscar. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Nacional Patagónico; Argentina. University of Washington; Estados UnidosFil: Saravia, José. Provincia del Chubut. Ministerio de Desarrollo Territorial y Sectores Productivos; ArgentinaFil: Orensanz, Jose Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Nacional Patagónico; Argentin

Desarrollo embrionario del cangrejo nadador Ovalipes trimaculatus (Decapoda: Brachyura: Portunoidea)

The embryogenesis of Ovalipes trimaculatus, a member of the highly valued portunid swimming crabs, was studied under nearly constant temperature (13±1°C), salinity (33) and photoperiod (14 h light:10 h dark) conditions. A five-stage scale of embryonic development was defined for the species. Time required to complete development averaged 35.7±2.11 days, showing no significant differences between embryos located in inner, middle and outer portions of the egg mass. The egg chorion was rounded and showed the highest growth in diameter between stages I (morula-blastula-gastrula) and II (primordium of larval structure) and between stages III (appendage formation) and IV (eye formation). Results reported here provide a baseline for optimizing hatchery operations and useful information for management purposes.Se estudió la embriogénesis del cangrejo nadador Ovalipes trimaculatus, miembro de la familia de portúnidos altamente valorados, bajo condiciones constantes de temperatura (13±1°C), salinidad (33) y fotoperíodo (luz 14h:10h oscuridad). Se definió una escala de cinco etapas de desarrollo embrionario para la especie. El tiempo necesario para completar el desarrollo es de 35.7±2.11 días, sin mostrar diferencias significativas entre embriones ubicados en porciones interior, media y exterior de la masa de huevos. Los huevos son esféricos y presentan mayor crecimiento en diámetro entre las etapas I (mórula - blástula - gástrula) y II (primordio de estructuras larvarias) y entre las etapas III (formación de apéndices) y IV (formación de los ojos). Los resultados aquí presentados proporcionan la base para la optimización del cultivo y brindan información útil para el manejo de la especie

Diferencias geográficas en la forma del caparazón del cangrejo Cyrtograpsus affinis (Decapoda: Varunidae) y sus implicaciones taxonómicas

Cyrtograpsus genus was traditionally considered to be composed of three species: C. angulatus, C. altimanus and C. affinis. However, recent studies have found solid evidence suggesting that C. affinis and C. altimanus belong to a single species and hypothesize that the morphological differences which caused this misclassification could be related to different ecophenotypes or life stages. Here we report a geometric morphometrics study on the carapace shape of Cyrtograpsus specimens from the Río de la Plata estuary (36°S) and the Nuevo Gulf (42.75°S), testing for shape differences between different sizes (allometry) in the two environments. We found that previous morphological descriptions of the two species were associated with different sizes of a continuous, statistically significant allometric shape variation, concluding that C. affinis is a junior synonym of C. altimanus. We also found significant differences in the carapace shape between estuarine and marine environments, suggesting an effect of the environmental variables on carapace shape and a potential adaptive value of this trait.El género Cyrtograpsus fue tradicionalmente considerado como formado por tres especies: C. angulatus, C. altimanus y C. affinis. Trabajos recientes reportan sólida evidencia que sugiere que C. altimanus y C. affinis pertenecen a una única especie y proponen que las formas que originaron esta clasificación errónea pueden corresponderse con diferentes ecofenotipos o estadios sucesivos del ciclo de vida de una misma especie. Este trabajo presenta un estudio de morfometría geométrica sobre la forma del caparazón de especímenes del género Cyrtograpsus provenientes del estuario del Río de la Plata (36°S) y del golfo Nuevo (42.75°S), en el que se prueban las diferencias en la forma del caparazón relativas al tamaño (alometría) y a los distintos ambientes. Nuestros resultados muestran que las descripciones morfológicas previas de ambas especies están asociadas a diferentes regiones de una variación alométrica continua y estadísticamente significativa, y nos permite concluir que C. affinis es un sinónimo juvenil de C. altimanus. Además encontramos diferencias significativas en la forma del caparazón entre especímenes provenientes del ambiente estuárico y el marino, lo cual sugiere un efecto de ciertas variables ambientales sobre la forma del caparazón y un potencial valor adaptativo de esta característica

Prurido do traje de banho: estudo clínico e epidemiológico de 38 casos no Estado de Santa Catarina, Brasil

Seabather's eruption (SBE) is an intensely itchy, papule-erythematous dermatitis that occurs predominantly in regions of the body covered by bathing costumes, after exposure to marine water. The planulae larvae of Linuche unguiculata scyphomedusae (thimble jellyfish) are the etiologic agent of the dermatitis, which is frequent in waters of Caribbean, Gulf of Mexico and Florida. The authors report 38 cases of SBE in the State of Santa Catarina (Southern region of Brazil), with emphasis on their clinical and epidemiological aspects, such as profile of victim, topography of the papules and conditions predisposing to the accident.Os Cnidários são animais marinhos que estão relacionados com uma variedade de acidentes acometendo banhistas em regiões litorâneas. A larva plânula da cifomedusa Linuche unguiculata (uma pequena água-viva) é o agente etiológico de uma dermatite pápulo-eritematosa, intensamente pruriginosa, denominada Prurido do Traje de Banho. Acomete principalmente áreas do corpo cobertas pelo traje de banho, após exposição à água do mar. Sua presença é freqüente nas águas do Caribe, Golfo do México e da Flórida (Estados Unidos), local onde foi descrita pela primeira vez. Os autores relatam uma série de casos de Prurido do Traje de Banho ocorridos no Estado de Santa Catarina (região do sul do Brasil), com ênfase em seus aspectos clínicos e epidemiológicos