Granular materials are common in everyday experience, but have long-resisted a complete theoretical description. Here, we consider the regime of slow, dense granular flow, for which there is no general model, representing a considerable hurdle to industry, where grains and powders must frequently be manipulated. Much of the complexity of modeling granular materials stems from the discreteness of the constituent particles, and a key theme of this work has been the connection of the microscopic particle motion to a bulk continuum description. This led to development of the "spot model", which provides a microscopic mechanism for particle rearrangement in dense granular flow, by breaking down the motion into correlated group displacements on a mesoscopic length scale. The spot model can be used as the basis of a multiscale simulation technique which can accurately reproduce the flow in a large-scale discrete element simulation of granular drainage, at a fraction of the computational cost. In addition, the simulation can also successfully track microscopic packing signatures, making it one of the first models of a flowing random packing. To extend to situations other than drainage ultimately requires a treatment of material properties, such as stress and strain-rate, but these quantities are difficult to define in a granular packing, due to strong heterogeneities at the level of a single particle. However, they can be successfully interpreted at the mesoscopic spot scale, and this information can be used to directly test some commonly-used hypotheses in modeling granular materials, providing insight into formulating a general theory.

Epidemiology, the study of disease risk factors in populations, emerged between the 16th and 19th centuries in response to terrifying epidemics of infectious diseases such as yellow fever, cholera and bubonic plague. Traditional epidemiological studies have led to modifications in hygiene, diet, sex, and many other practices that have profoundly altered the dynamic between humans and diseases.

In this thesis, we develop mathematical techniques to address modern challenges, including emerging diseases such as SARS and West Nile virus, the threat of bioterrorism, and stringent legislation protecting patient privacy. For real-time disease surveillance, in which the goal is early detection of outbreaks based on time-series data, we introduce a generalized additive model that maintains constant specificity on various time scales. Within spatial epidemiology, one problem is to map the risk of disease across space (i.e., disease mapping), and another is to analyze the data for clustering. We propose a general technique, cartograms created from exact patient location data, which can address both of these problems. We also develop a graph-theoretical method to detect spatial clusters of any shape based on Euclidean minimum spanning trees. For mapping applications, we present an optimal strategy for mapping patient locations that preserves both privacy and spatial patterns within the data.

We present a general theory for the dynamics of thin viscous sheets. Employing concepts from differential geometry and tensor calculus, we derive the governing equations in terms of a coordinate system that moves with the film. Special attention is given to incorporating inertia and the curvature forces that arise from the thickness variations along the film. Exploiting the slenderness of the film, we assume that the transverse fluid velocity is small compared to the longitudinal one and perform a perturbation expansion to obtain the leading order equations when the center-surface that defines the coordinate system is parametrized by lines of curvature. We then focus on the dynamics of flat film rupture, in an attempt to gain some insights into the sheet breakup and its fragmentation into droplets. By combining analytical and numerical methods, we extend the prior work on the subject and compare our numerical simulations with experimental work reported in the literature.

We prove an analogue of the Hodge cohomology theorem for a certain class of non-compact manifolds. Specifically, let M be a compact manifold with boundary ∂M, and let g be a metric on Int(M ). Assume that there exists a collar neighborhood of the boundary, U≅ (0,1)_x × ∂M, in which g has the form

This thesis consists of two parts. In the first part of the thesis, we prove an index theorem for Dirac operators of conic singularities with co-dimension $2$. As a result, a positive mass theorem for some special even dimensional non-spin manifolds is derived. And an obstruction for a minimal embedded sphere to be characteristic in a four manifold with positive scalar curvature also follows from the index theorem.

In the second part, we study the moduli space of monopoles with singularities along an embedded surface. We prove that when the base manifold is Kahler, there is a holomorphic description of the singular monopoles. The compactness for this is proved.

In this thesis, we study ad-nilpotent ideals and its relations with nilpotent orbits, affine Weyl groups, sign types and hyperplane arrangements.

This thesis is divided into three parts. The first and second parts deal with ad-nilpotent ideals for complex reductive Lie groups. In the first part, we study the left equivalence relation of ad-nilpotent ideals and relate it to some equivalence relation of affine Weyl groups and sign types. In the second part, we prove that for classical groups there always exist ideals of minimal dimension as conjectured by Sommers.

In the third part, we define an analogous object for connected real reductive Lie groups, which is called $\theta$-nilpotent subspaces. We relate $\theta$-nilpotent subspaces to dominant regions of some real hyperplane arrangement and get the characteristic polynomials of the real hyperplane arrangement in the case of $U(m, n)$ and $Sp(m, n)$. We conjecture a general formula for other types.