We study analysis and partial differential equations on metric measure spaces by investigating the property of Sobolev functions or Sobolev mappings and studying the viscosity solutions to some partial differential equations.

This manuscript consists of two parts. The first part is focused on the theory of Sobolev spaces on metric measure spaces. We investigate the continuity of Sobolev functions in the critical case in some general metric spaces including compact connected one-dimensional spaces and fractals. We also constructe a large class of pathological $n$-dimensional spheres in $\mathbb{R}^{n+1}$ by showing that for any Cantor set $C\subset\mathbb{R}^{n+1}$ there is a topological embedding

$f:\mathbb{S}^n\to\mathbb{R}^{n+1}$ of the Sobolev class $W^{1,n}$ whose image contains the Cantor set $C$.

The second part is focused on the theory of viscosity solutions for nonlinear partial differential equations in metric spaces, including the Heisenberg group as an important special case. We study Hamilton-Jacobi equations on the Heisenberg group $\mathbb{H}$ and show uniqueness of viscosity solutions with exponential growth at infinity. Lipschitz and horizontal convexity preserving properties of these equations under appropriate assumptions are also investigated. In this part, we also study a recent game-theoretic approach to the viscosity solutions of various equations, including deterministic and stochastic games. Based on this interpretation, we give new proofs of convexity preserving properties of the mean curvature flow equations and normalized $p$-Laplace equations in the Euclidean space.

The work presented in this thesis is motivated by questions arising about pathogen dynamics. The effects of pathogens can be observed on a variety of spatial scales, from within-host interactions with the immune system on the microscopic level, to the spread of communicable disease on the population level. We present analyses of three common pathogens on three different scales. In Chapter 2, we derive and study a system of ordinary differential equations modeling the competition for space and resources between a mammalian host's native intestinal microbiota and an invasive species of Salmonella Typhimurium. We use our model to discuss optimal invasion strategies that maximize the salmonella's likelihood of successfully displacing the microbiota for a spot on the intestinal wall. In Chapter 3, we analyze an anomalous behavior observed in which two interacting pulses of E. coli in a one-dimensional nutrient gradient will turn around move away from one another rather than combine. To this end, we derive a novel system of ordinary differential equations approximating the dynamics of the classic Keller-Segel partial differential equations model for bacterial chemotaxis, and use this approximation to make testable predictions about mechanisms driving the turn around behavior. Finally, in Chapter 4, we use a two-strain SIR-type model of rotavirus transmission to study the effects of vaccination on a population exposed to multiple endemic strains.

In this thesis, we study two topics related to defaults. First, we provide a Probability of Default (PD) calculation method for privately-held U.S. regional banks, using free and transparent data from the Federal Deposit Insurance Corporation (FDIC). Our method is efficient and useful for both investors and regulators. We have improved Moody's proprietary RiskCalc PD model [17] by creating a new cautionary index, which is able to capture default behaviors very well and has a very high predictive power over both one-year and six-month time horizons as shown by our numerical results. We also find that this performance is robust over different historical periods. We describe the factors we chose, the modeling methodology, and the model's accuracy in detail.

Second, we propose two strategies to reduce the frequency of defaults in home mortgages (foreclosures). The first is a new mortgage insurance contract (American put option with the house as the underlying asset). Our analysis differs from that for the standard put option in equity markets in that our strike (the remaining value of the mortgage) is time dependent, and the drift and volatility in the Geometric Brownian Motion are time dependent (step functions) due to a regime switch from declining to increasing house prices. Both theoretical derivations and numerical results will be obtained. We will also analyze the Adjustable Balance Mortgage (ABM) in continuous time as a second alternative to avoiding foreclosures. Here the mortgage payments are reduced if the house price falls below the remaining value of the mortgage.

In the following thesis, we explore the notion of rational Fivebrane structures. This is done through a combination of obstruction theory and rational homotopy theory. We show that these structures can be classified to some degree by the underlying Spin bundle. From there we turn our focus to the differential setting. Using this relation to the Spin bundle, we apply the classical machinery of Cheeger and Simons to understand differential rational Fivebrane classes. Finally we use these classes to obtain information for differential trivializations in the integral case. In doing this we introduce the exact braid diagram.

Many physical systems feature interacting components that evolve on disparate timescales. Significant insights about the dynamics of such systems have resulted from grouping timescales into two classes and exploiting the timescale separation between classes through the use of geometric singular perturbation theory. It is natural to expect, however, that some dynamic phenomena cannot be captured by a two timescale decomposition. One example is the mixed burst firing mode, observed in both recordings and model pre-B\"{o}tzinger neurons, which appears to involve at least three timescales based on its time course. With this motivation, we construct a model system consisting of a pair of Morris-Lecar systems coupled so that there are three timescales in the full system. We demonstrate that the approach previously developed in the context of geometric singular perturbation theory for the analysis of two timescale systems extends naturally to the three timescale setting. To elucidate which characteristics truly represent three timescale features, we investigate certain reductions to two timescales and the parameter dependence of solution features in the three timescale framework. Furthermore, these analyses and methods are extended and applied to understand multiple timescale bursting dynamics in a realistic single pre-B\"{o}tzinger complex neuron and a heterogeneous population of these neurons, both of which can generate a novel mixed bursting (MB) solution, also observed in pre-B\"{o}tC neuron recordings. Rather surprisingly, we discover that a third timescale is not actually required to generate mixed bursting solution in the single neuron model, whereas at least three timescales should be involved in the latter model to yield a similar mixed bursting pattern. Through our analysis of timescales, we also elucidate how the single pre-B\"{o}tC neuron model can be tuned to improve the robustness of the MB solution.

Parameter estimation is a vital component of model development. Making use of data, one aims to determine the parameters for which the model behaves in the same way as the system observations. In the setting of differential equation models, the available data often consists of time course measurements of the system. We begin by examining the parameter estimation problem in an idealized setting with complete knowledge of an entire single trajectory of data which is free from error. This addresses the question of uniqueness of the parameters, i.e. identifiability. We derive novel, interrelated conditions that are necessary and sufficient for identifiability for linear and linear-in-parameters dynamical systems. One result provides information about identifiability based solely on the geometric structure of an observed trajectory. Then, we look at identifiability from a discrete collection of data points along a trajectory. By considering data that are observed at equally spaced time intervals, we define a matrix whose Jordan structure determines the identifiability. We further extend the investigation to consider the case of uncertainty in the data. Our results establish regions in data space that give inverse problem solutions with particular properties, such as uniqueness or stability, and give bounds on the maximal allowable uncertainty in the data set that can be tolerated while maintaining these characteristics. Finally, the practical problem of parameter estimation from a collection of data for the system is addressed. In the setting of Bayesian parameter inference, we aim to improve the accuracy of the Metropolis-Hastings algorithm by introducing a new informative prior density called the Jacobian prior, which exploits knowledge of the fixed model structure. Two approaches are developed to systematically analyze the accuracy of the posterior density obtained using this prior.

Rhythmic behaviors such as breathing, walking, and scratching are vital to many species. Such behaviors can emerge from groups of neurons, called central pattern generators (CPGs), in the absence of rhythmic inputs. In vertebrates, the identification of the cells that consti- tute the CPG for particular rhythmic behaviors is difficult, and often, its existence has only been inferred. In the second and third chapters of this thesis, we use two reduced mathemat- ical models to investigate the capability of a proposed network to generate multiple scratch rhythms observed in turtles. Under experimental conditions, intact turtles generate sev- eral rhythmic scratch motor patterns corresponding to non-rhythmic stimulation of different body regions. These patterns feature alternating phases of motoneuron activation that occur repeatedly, with different patterns distinguished by the relative timing and duration of activ- ity of hip extensor, hip flexor, and knee extensor motoneurons. We show through simulation that the proposed network can achieve the desired multi-functionality, even though it relies on hip unit generators to recruit appropriately timed knee extensor motoneuron activity. We develop a phase space representation which we use to derive sufficient conditions for the network to realize each rhythm and which illustrates the role of a saddle-node bifurcation in achieving the knee extensor delay. This framework is harnessed to consider bistability and to make predictions about the responses of the scratch rhythms to input changes for future experimental testing. We also consider a stochastic spiking model to reproduce firing rate changes observed in experiment, explore the relative contributions of different parameters in the model to the observed changes, support our collaborators’ hypothesis regarding these changes, and provide our collaborators with predictions for future experiments. In the fourth chapter of this thesis, we present a theoretical study examining whether three mechanisms suggested by deletion experiments can operate in the same CPG for an extensor-flexor pair in the mammalian central nervous system during locomotion. We arrive at unique solution properties produced by each of the three mechanisms for use in future experiments. Our findings propose explanations for the coexistence of the three experimentally suggested yet seemingly contradictory mechanisms for rhythmogenesis.

This thesis concentrates on the inverse problem in classical statistical mechanics and its applications. Let us consider a system of identical particles with the total energy W + U, where W is a fixed scalar function, and V is an additional internal or external potential in the form of a sum of m-particle interactions u. The inverse conjecture states that any positive, integrable function ρ(m) is the equilibrium m-particle density corresponding to some unique potential u. It has been proved for all m ≥ 1 in the grand canonical ensemble by Chayes and Chayes in 1984. Chapter 2 of this thesis contains the proof of the inverse conjecture for m ≥ 1 in the canonical formulation. For m = 1, the inverse problem lies at the foundation of density functional theory for inhomogeneous fluids. More generally, existence and differentiability of the inverse map for m ≥ 1 provides the basis for the variational principle on which generalizations to density functional theory can be formulated. Differentiability of the inverse map in the grand canonical ensemble for m ≥ 1 is proved here in Section 3.2. In particular, this result leads to the existence of a hierarchy of generalized Ornstein-Zernike equations connecting the 2m-,...,m-particle densities and generalized direct correlation functions. This hierarchy is constructed in Section 3.3.

This work seeks to extract topological information from the order-properties of certain pre-ideals and pre-filters associated with topological spaces. In particular, we investigate the neighborhood filter of a subset of a space, the pre-ideal of all compact subsets of a space, and the ideal of all locally finite subcollections of an open cover of a space. The class of directed sets with calibre (omega 1, omega) (i.e. those whose uncountable subsets each contain an infinite subset with an upper bound) play a crucial role throughout our results. For example, we prove two optimal generalizations of Schneider's classic theorem that a compact space with a G_delta diagonal is metrizable. The first of these can be stated as: if X is (countably) compact and the neighborhood filter of the diagonal in X^2 has calibre (omega 1, omega) with respect to reverse set inclusion, then X is metrizable. Tukey quotients are used extensively and provide a unifying language for expressing many of the concepts studied here.

In order to understand how the network structure impacts the underlying dynamics, we seek an assortment of methods for efficiently constructing graphs of interest that resemble their empirically observed counterparts. Since many real world networks obey degree heterogeneity, where different nodes have varying numbers of connections, we consider some challenges in constructing random graphs that emulate the property. Initially we focus on the Uniform Model, where we would like to uniformly sample from all graphs that realize a given bi-degree sequence. We provide easy to implement, sufficient criteria to guarantee that a bi-degree sequence corresponds to a graph. Consequently, we construct novel results regarding asymptotics of the number of graphs that realize a given degree sequence, where knowledge of the aforementioned enumeration result will assist us in constructing realizations from the Uni- form Model. Finally, we consider another random directed graph model that exhibits degree heterogeneity, the Chung-Lu random graph model and prove concentration results regarding the dominating eigenvalue of the corresponding adjacency matrix. We extend our analysis to a more generalized model that allows for intricate community structure and demonstrate the impact of the community structure in networks with Kuramoto and SIS epidemiological dynamics.

The Robert-Asselin (RA) time filter combined with leapfrog scheme is widely used in numerical models of weather and climate. The RA filter suppresses the spurious computational mode associated with the leapfrog method, and successfully stabilizes the numerical solution. However, it also weakly dampens the physical mode and degrades the formal second-order accuracy of the leapfrog scheme to first order. There is a natural intention to reduce the time-stepping error as it has proven to be a substantial part of the total forecast error. Yet a new scheme must be non-intrusive, i.e., easily implementable in legacy codes in order to avoid significant programming undertaking.

The object of this work is the development, analysis and validation of novel Robert-Asselin type time filters, addressing both of the above problems. Specifically, we first propose and analyze a higher-order Robert-Asselin (hoRA) type time filter. The analysis reveals that the filtered leapfrog scheme exhibits second- or third-order accuracy depending on the filter parameter. We then investigate its behavior when used in conjunction with the implicit-explicit integration, which is commonly used in weather and climate models to relieve the severe time step restriction induced by unimportant high-frequency waves. Next, we build a framework of constructing a family of hoRA filters with any pre-determined order of accuracy. In particular, we focus on the fourth-order time filter. Finally, we present supplemental analysis for several filters developed by Williams. For each direction, we present comprehensive error and stability analysis, and perform numerical tests to verify theoretical results.

We use various averaging techniques to obtain results in different aspects of functional analysis

and Banach space theory, particularly in fixed point theory.

Specifically, in the second chapter, we discuss the class of so-called mean nonexpansive

maps, introduced in 2007 by Goebel and Japon Pineda, and we prove that mean isometries

must be isometries in the usual sense. We further generalize this class of mappings to

what we call the affine combination maps, give many examples, and study some preliminary

properties of this class.

In the third chapter, we extend Browder's and Opial's famous Demiclosedness Principles

to the class of mean nonexpansive mappings in the setting of uniformly convex spaces and

spaces satisfying Opial's property. Using this new demiclosedness principle, we prove that

the iterates of a mean nonexpansive map converge weakly to a fixed point in the presence of

asymptotic regularity at a point.

In the fourth chapter, we investigate the geometry and fixed point properties of some

equivalent renormings of the classical Banach space c0. In doing so, we prove that all norms

on `1 which have a certain form must fail to contain asymptotically isometric copies of c0.

The purpose of this work is to use motivic integration for the study of reductive groups over p-adic fields (towards applications of the fundamental lemma for groups). We study spherical Hecke algebras from a motivic point of view. We get a field independent description of the spherical Hecke algebra of a reductive group and its structure. We investigate the Satake isomorphism from the motivic point of view. We prove that some data of the Satake isomorphism is motivic.

The aim of this thesis is to compare the efficiency of different algorithms on estimating parameters

that arise in partial differential equations: Kalman Filters (Ensemble Kalman Filter,

Stochastic Collocation Kalman Filter, Karhunen-Lo`eve Ensemble Kalman Filter, Karhunen-

Lo`eve Stochastic Collocation Kalman Filter), Markov-Chain Monte Carlo sampling schemes

and Adjoint variable-based method.

We also present the theoretical results for stochastic optimal control for problems constrained

by partial differential equations with random input data in a mixed finite element form. We

verify experimentally with numerical simulations using Adjoint variable-based method with

various identification objectives that either minimize the expectation of a tracking cost functional

or minimize the difference of desired statistical quantities in the appropriate Lp norm.

In this thesis, we establish a financial credit derivative pricing model for a credit default swap (CDS) contract which is subject to counterparty risks. A credit default swap is an agreement on exchange of cash flows between two parties, the buyer and the seller, about the occurrence of a credit event. The buyer makes a series of payments to the seller before the event and before the expiration date. The seller pays the buyer a fixed compensation at the moment when the event occurs, if it is before the expiry. The model arises a linear partial differential equation problem. We study this model, i.e. differential equation and show that a solution of the PDE problem from structure model can be obtained as the limit of a sequence of PDE problems which comes from intensity model. In addition, we study the infinite horizon problem of the pricing model which leads to a nonlinear ordinary differential equation problem. We obtain a implicit solution of the ODE problem and prove the solution can be converged by the solution of the PDE problem exponentially. Furthermore, the models and theoretical methods in this study get connected between two main risk frameworks: term structure model and intensity model, which greatly extend the area of applicability of structure models in financial problems. Moreover, We obtain the uniqueness, existence, and properties of the solutions of the PDE and ODE problems. Nevertheless, we implement numerical methods to calibrate the parameters of stochastic interest rate model and analyze the numerical solutions of the pricing model.