The Chung-Yau graph invariants were originated from Chung-Yau’s work on discrete Green’s function. They are useful to derive explicit formulas and estimates for hitting times of random walks on discrete graphs. In this thesis, we study properties of Chung-Yau invariants and apply them to study some questions:

(1) The relationship of Chung-Yau invariants to classical graph invariants; (2) The change of hitting times under natural graph operations;

(3) Properties of graphs with symmetric hitting times;

(4) Random walks on weighted graphs with different weight schemes.

When a pathogen invades the body, an acute inflammatory response is activated to eliminate the intruder. In some patients, runaway activation of the immune system may lead to collateral tissue damage and, in the extreme, organ failure and death.

Experimental studies have found an association between severe infections and depletion in levels of adenosine triphosphate (ATP), increase in nitric oxide production, and accumulation of lactate, suggesting that tissue energetics is compromised.

We present a computational model consisting of ordinary differential equations to explore the dynamics of the acute inflammatory response against infections caused when a pathogen makes its way into a host. This model incorporates energy production along with the energy requirements that arise when fighting such an infection. In particular, we investigate the role of energetics during infection and explore the relation between overproduction of nitric oxide (NO), lactate, altered adenosine triphosphate (ATP) levels, and sepsis.

Finally, a data-driven approach is used to extend our model as an effort to better understand the role of energy in sepsis. This extended model is calibrated by fitting animal data from a study done in thirty-two baboons that were induced into sepsis after infusing E. coli intravenously. Using Bayesian analysis, we quantify uncertainty in model parameters and with them we investigate differences across different populations, including survivors and non-survivors.

A toric degeneration is a flat family over $\mathbb{A}^1$ that is trivial away from the special fiber (fiber over zero) and whose special fiber is a variety acted linearly by a torus with a dense orbit; i.e., the special fiber is a non-normal = not-necessarily-normal toric variety. We introduce a systematic method to construct toric degenerations of a projective variety (embedded up to Veronese embeddings). Part 1 develops the general theory of non-normal toric varieties by generalizing the more conventional theory of toric varieties. A new characterization of non-normal toric varieties as a complex of toric varieties is given. Given a projective variety $X$ of dimension $d$, the main result of the thesis (Part 2) constructs a finite sequence of flat degenerations with irreducible and reduced special fibers such that the last one is a non-normal toric variety. The degeneration sequence depends on the choice of a full flag of closed subvarieties $X = Y_0 \supset Y_1 \supset \cdots \supset Y_d$ such that each $Y_i$ is a \emph{good divisor} in $Y_{i-1}$. The notion of a good divisor comes from the asymptotic ideal theory in commutative algebra and the goodness ensures the finite generation of the defining graded ring of the special fiber in each step. This is a generalization of degeneration (or deformation) to normal cone in intersection theory and can be regarded as geometric reinterpretation of the construction of a valuation in \cite{Oko}, the key step in the construction of a Newton--Okounkov body. Part 3 reformulates the main result of \cite{Oko} in terms of an equivariant Hilbert function; this reformulation may be thought of as a special case of the equivariant Riemann--Roch theorem.

On the sequence space $\ell^{\infty}$, we construct Banach limits that are invariant under the Ces\`aro averaging operator. On the function space $L^{\infty}(0,\infty)$, we start by defining a new operator $J^{\alpha}$, for each $\alpha >0$. This new operator extends the definition of $J^{n}$, with $n \in \mathbb{N}$, which is the operator obtained by composing the Ces\`aro averaging operator with itself $n$ times. We show that the family of operators $\left(J^{\alpha} \right)_{\alpha >0}$ has the semigroup property. We also construct Banach limits on $L^{\infty}(0,\infty)$ that are invariant under the members of this family of operators. Finally, on the operator space $\mathcal{B}(\ell^2(\mathbb{N}_0))$, we define a Ces\`aro averaging operator from this space to itself. We also discuss known results about vector-valued Banach limits on $\ell^{\infty}(\ell^2(\mathbb{Z}))$ that preserve Ces\`aro convergence, and use them to construct a continuous linear functional on $\mathcal{B}(\ell^{2}(\mathbb{N}_0))$ with Ces\`aro-invariance-like properties.

Time-accurate simulations of physical phenomena (e.g., ocean dynamics, weather, and combustion) are essential to economic development and the well-being of humanity. For example, the economic toll hurricanes wrought on the United States in 2017 exceeded $\$200$ billon dollars. To mitigate the damage, the accurate and timely forecasting of hurricane paths are essential. Ensemble simulations, used to calculate mean paths via multiple realizations, are an invaluable tool in estimating uncertainty, understanding rare events, and improving forecasting. The main challenge in the simulation of fluid flow is the complexity (runtime, memory requirements, and efficiency) of each realization. This work confronts each of these challenges with several novel ensemble algorithms that allow for the fast, efficient computation of flow problems, all while reducing memory requirements. The schemes in question exploit the saddle-point structure of the incompressible Navier-Stokes (NSE) and Boussinesq equations by relaxing incompressibility appropriately via artificial compressibility (AC), yielding algorithms that require far fewer resources to solve while retaining time-accuracy. Paired with an implicit-explicit (IMEX) ensemble method that employs a shared coefficient matrix, we develop, analyze, and validate novel schemes that reduce runtime and memory requirements. Using these methods as building blocks, we then consider schemes that are time-adaptive, i.e., schemes that utilize varying timestep sizes.

The consideration of time-adaptive artficial compressibility methods, used in the algorithms mentioned above, also leads to the study of a new slightly-compressible fluid flow continuum model. This work demonstrates stability and weak convergence of the model to the incompressible NSE, and examines two associated time-adaptive AC methods. We show that these methods are unconditionally, nonlinearly, long-time stable and demonstrate numerically their accuracy and efficiency.

The methods described above are designed for laminar flow; turbulent flow is addressed with the introduction of a novel one-equation unsteady Reynolds-averaged Navier-Stokes (URANS) model with multiple improvements over the original model of Prandtl. This work demonstrates analytically and numerically the advantages of the model over the original.

Developed by LeBrun, twistor CR manifold is a 5-dimensional CR manifold foliated by Riemann spheres. The CR structure is determined by both the complex structure on the Riemann sphere and the geometric information of the space of leaves, which is a 3-manifold endowed with a conformal class of metrics and a trace-free symmetric (1,1)-tensor.

When the (1,1)-tensor is zero, the twistor CR structure of zero torsion, named as the rival CR structure on LeBrun's paper “Foliated CR Manifolds”, is obtained. These CR structures are embeddable to a complex 3-manifold if and only if the metric tensor is conformal to a real analytic metric.

We try to understand twistor CR structures through the corresponding Fefferman metric defined on the canonical circle bundle of the given CR manifold. The conformal class of the Fefferman metric is preserved over the choice of contact forms of the CR structure, so it makes possible to classify CR structures by the confomal curvature tensor of the Fefferman metric.

Our main results include representing the Weyl tensor in terms of the Cotton tensor on the 3-manifold when the twistor CR structure is of zero torsion. Moreover, we obtain conditions for vanishing Weyl tensor when the space of leaves is under a flat metric.

In this thesis a natural generalization and further extension of Gröbner theory using Kaveh and Manon's Khovanskii basis theory is constructed. Suppose A is a finitely generated domain equipped with a valuation v with

a finite Khovanskii basis. We develop algorithmic processes for computations regarding ideals in the algebra A. We introduce the notion of a Khovanskii-Gröbner basis for an ideal J in A and give an analogue of the Buchberger algorithm for it (accompanied by a Macaulay2 code). We then use Khovanskii-Gröbner bases to suggest an algorithm to solve a system of equations from A. Finally we suggest a notion of relative tropical variety for an ideal in A and

sketch ideas to extend the tropical compactification theorem to this setting.

With the fast-growing pace of advancements in computer science, mathematics, and linguistics, great strides have been made in each field. Here, work regarding the analysis of language families will be presented in an argument for the acceptance of results that are derived from a computational means. Specially, this research leverages machine learning methodologies to gain insight into the relationship between, and classification of, different languages and language families. Further, the higher rate of the availability of data regarding the geospatial aspects of a language spreading allows for the incorporation of this data into an analysis of language spread. This research lays the foundation and establishes a framework in which these two aspects, computational analysis and geospatial data, are intertwined to offer a perspective and glean insight into language.

"Finding integer solutions to polynomial equations, also known as “Diophantine geometry,” is a fundamental topic throughout the history of the study of numbers. In the early 20th century, one of Hilbert’s famous list of mathematical problems was to find an algorithm that would calculate the solutions to any polynomial equation. Around 1970, a group of mathematicians determined that there can be no such algorithm.

Since we lack a general algorithm, it is natural to start with some particular kind of equations. Algebraic curves are a natural candidate: they are distinguished by the complex number solutions to the equation forming a surface (that is, a 2-dimensional space). Remarkably, in the 1980s Faltings proved that when the number of “holes” in this surface is at least 2, then there are finitely many solutions.

However, Faltings’s method gave no way to actually find the solutions. In particular, if you find some collection of solutions, how can you know that you have found them all? Starting around 2010, Minhyong Kim introduced a suite of new ideas to this endeavor, now known as the non-abelian Chabauty method.

The goal of this project is to produce a framework that makes the non-abelian Chabauty method amendable to computation in more cases, thereby making it possible to find all of the solutions on more algebraic curves. "

We develop a computational model to study the compaction, network topology and elastic response of hydrogel as a function of crosslink density. Our simulations start with a covalently bonded polymer network, to which we introduce additional crosslinks by binding metal cations to reactive groups distributed along the polymer chains. We find that these crosslinks increase the compaction of the polymer network in two ways: (i) by crosslinking neighboring groups on the same polymer chain and thereby shortening the effective length of polymer chains, and (ii) by linking together two or more distinct polymer chains. These two effects combine to overall hydrogel contraction and stiffening. Our results show that the elastic modulus of the hydrogel increases significantly due to the additional crosslinks, in agreement with recent experimental observations. With the help of computer simulations, we find the relations between parameters of our model and chemical characteristics of the hydrogel such as the modulus, the compaction of hydrogel, or the average number of reactive groups bound to a single crosslinker. We analyze geometric and topological characteristics of the hydrogel, such as the time evolution of distance between groups in the hydrogel, or the proportion of crosslinks that are retained, broken or newly formed during the course of simulations. These characteristics help us better understand the internal structure of the hydrogel and explain experimental observations such as the compaction of the hydrogel when metal crosslinkers are introduced. Despite its simplicity, the model qualitatively captures the important chemical properties of the crosslinkers.

Local orthogonal rectification (LOR) provides a natural and useful geometric frame for analyzing dynamics relative to manifolds embedded in flows. LOR can be applied to any embedded base manifold in a system of ODEs of arbitrary dimension to establish a corresponding system of LOR equations for analyzing dynamics within the LOR frame. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. Additionally, we illustrate the utility of LOR by showing a wide range of application domains.

In the plane, we use the LOR approach to derive a novel definition for rivers, long-recognized but poorly understood trajectories that locally attract other orbits yet need not be related to invariant manifolds or other familiar phase space structures, and to identify rivers within several example systems.

In higher dimensions, we apply LOR to identify periodic orbits and study the transient dynamics nearby. In the LOR method, %in $\R^n$ for any $n$,

the standard approach of finding periodic orbits by solving for fixed points of a Poincar\'{e} return map is replaced by the solution of a boundary value problem with fixed endpoints, and the computation provides information about the stability of the identified orbit. We detail the general method and derive theory to show that once a periodic orbit has been identified with LOR, the LOR coordinate system allows us to characterize the stability of the periodic orbit, to continue the orbit with respect to system parameters, to identify invariant manifolds attendant to the periodic orbit, and to compute the asymptotic phase associated with points in a neighborhood of the periodic orbit in a novel way.

Finally, we generalize the definition of rivers beyond planar systems, and demonstrate a fundamental connection between canard solutions in two-timescale systems and generalized rivers. We will again use a blow-up transformation on the LOR equations, which provides a useful decomposition for studying trajectories' behavior relative to the embedded base curve.