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The balanced minimum evolution (BME) polytope is a structure representative of a problem in biology, in particular in the study of phylogenetic trees. In this scope, the polytope is used to answer the question of how a set of species are related to one another. In this paper we explore generalized instances of the BME polytope for networks. For one of these generalized BME polytopes we focus on the discovery of new facets and their corresponding equations, while for the other we give the facets of the polytope and discuss the relationship that they have to another well known polytope outside of the field of biology. Furthermore, we also provide the dimension reducing equalities that were discovered which hold for every BME polytope and then prove their existence.

In 2011, Gao and Heindl proposed a family of Multivariate Public Key Cryptosystems by combining the triangular scheme and the oil-vinegar scheme. The new design was claimed to be secured under known attacks. Besides that, they also used the Medium-Field Multivariate Public Key Cryptosystem as an example of their general frame and explained how it works. Later, by introducing several Diophantine equations into their design, they presented the Diophantine Equations Multivariate Public Key Cryptosystem (DEMPKC) with three sets of suggested parameters and the claimed security level were high.In this paper, we present our cryptanalysis on DEMPKC. Our cryptanalysis uses embedded surfaces associated with the DEMPKC and shows the attack can break the system efficiently. Our work provides an example of more general embedded surfaces other than linearization type of equations can be very useful to attack cryptosystems.

An abelian network is a collection of communicating automata whose state transitions and message passing each satisfy a local commutativity condition. The foundational theory for abelian networks was laid down in a series of papers by Bond and Levine (2016), which mainly focused on networks that halt on all inputs. In this dissertation, we extend the theory of abelian networks to nonhalting networks (i.e., networks that can run forever). A nonhalting abelian network can be realized as a discrete dynamical system in many different ways, depending on the update order. We show that certain features of the dynamics, such as minimal period length, have intrinsic definitions that do not require specifying an update order. We give an intrinsic definition of the \emph{torsion group} of a finite irreducible (halting or nonhalting) abelian network, and show that it coincides with the critical group of Bond and Levine (2016) if the network is halting. We show that the torsion group acts freely on the set of invertible recurrent components of the trajectory digraph, and identify when this action is transitive. This perspective leads to new results even in the classical case of sinkless rotor networks (deterministic analogues of random walks). In Holroyd et. al (2008) it was shown that the recurrent configurations of a sinkless rotor network with just one chip are precisely the unicycles (spanning subgraphs with a unique oriented cycle, with the chip on the cycle). We generalize this result to abelian mobile agent networks with any number of chips. We give formulas for generating series such as \[ \sum_{n \geq 1} r_n z^n = \det (\frac{1}{1-z}D - A ) \] where $r_n$ is the number of recurrent chip-and-rotor configurations with $n$ chips; $D$ is the diagonal matrix of outdegrees, and $A$ is the adjacency matrix. A consequence is that the sequence $(r_n)_{n \geq 1}$ completely determines the spectrum of the simple random walk on the network.

In this research paper, I apply the Susceptible-Infected-Virus (SIV) model to the Human Immunodeficiency Virus (HIV). The SIV model is a compartmental model describing within-host dynamics of viral infections; I analyze it in both its deterministic (in which constants are assumed to be known exactly) and stochastic (in which the death rate of the healthy cells is represented by a random variable) forms. First, I give analytical solutions to two simplified versions of the deterministic model. Next, I apply numerical methods to the full deterministic and stochastic systems. The results give an illustrative picture of HIV in-host population dynamics in the absence of treatment. They also demonstrate how randomness can impact the progression of the disease.

The delivery of oxygen by blood through blood vessels is a critical process thatenables cells and tissue to maintain functionality. This thesis focuses on this process at asmall scale in the body. In particular, it examines the flow of blood through capillaries inthe brain and how that blood diffuses oxygen into the surrounding tissue. These twomodels, blood flow and oxygen diffusion, are modeled and the two models are coupled.Two different implementations of this coupled model are used to solve the partialdifferential equation (PDE). First, an implementation in MATLAB based on the FiniteElement Method (FEM) was written for this paper. Additionally, an implementation inC++ is used which is based on Green’s Function Methods. This implementation waswritten by Secomb and can be found at github [3]. These implementations are used toexamine oxygen transport and the effects of heterogeneity on it (differences in oxygencontent of blood vessels that are close to each other). Heterogeneity tends to occuralongside diseases. We aim to show that this heterogeneity decreases the total oxygen insurrounding tissue. Additionally, we aim to show what the consequences of heterogeneityand the decrease in oxygen in the tissue would be on nearby neurons.

In this paper, we will explore the origins, applications, and properties of these sums and one of their generalizations. We seek to explain what these sums represent and how they behave by exploring some of their arithmetic properties. In addition, we hope to show the reader why one should care about these sums. We do this by presenting two important areas in which these sums appear: number theory and the study of enumerating lattice points inside of polytopes.

It is well documented that undergraduate students struggle with the more formal and abstract concepts of vector space theory in a first course on linear algebra. Some of these students continue on to classes in abstract algebra, where they learn about algebraic structures such as groups. It is clear to the seasoned mathematician that vector spaces are in fact groups, and so linear transformations are group homomorphisms with extra restrictions. This study explores the question of whether or not students see this connection as well. In addition, I probe the ways in which students’ stated understandings are the same or different across contexts, and how these differences may help or hinder connection making across domains. Students’ understandings are also briefly compared to those of mathematics professors in order to highlight similarities and discrepancies between reality and idealistic expectations. The data for this study primarily comes from clinical interviews with ten undergraduates and three professors. The clinical interviews contained multiple card sorts in which students expressed the connections they saw within and across the domains of linear algebra and abstract algebra, with an emphasis specifically on linear transformations and group homomorphisms. Qualitative data was analyzed using abductive reasoning through multiple rounds of coding and generating themes. Overall, I found that students ranged from having very few connections, to beginning to form connections once placed in the interview setting, to already having a well-integrated morphism schema across domains. A considerable portion of this paper explores the many and varied ways in which students succeeded and failed in making mathematically correct connections, using the language of research on analogical reasoning to frame the discussion. Of particular interest were the ways in which isomorphisms did or did not play a role in understanding both morphisms, how students did not regularly connect the concepts of matrices and linear transformations, and how vector spaces were not fully aligned with groups as algebraic structures.

This thesis is a representation of my contribution to the paper of the same name I co-author with Dr. Wasilkowski. It deals with linear problems defined on γ-weighted normed spaces of functions with infinitely many variables. In particular, I describe methods and discuss results for ε-truncation and ε-superposition methods. I show through these results that the ε-truncation and ε-superposition dimensions are small under modest error demand ε. These positive results are derived for product weights and the so-called anchored decomposition.

Diffusion-weighted image(DWI) and tractography have been developed for decades and are key elements in recent, large-scale efforts for mapping the human brain. The two techniques together provide us a unique possibility to access the macroscopic structure and connectivity of the human brain non-invasively and in vivo. The information obtained not only can help visualize brain connectivity and help segment the brain into different functional areas but also provides tools for understanding some major cognitive diseases such as multiple sclerosis, schizophrenia, epilepsy, etc. There are lots of efforts have been put into this area. On the one hand, a vast spectrum of tractography algorithms have been developed in recent years, ranging from deterministic approaches through probabilistic methods to global tractography; On the other hand, various mathematical models, such as diffusion tensor, multi-tensor model, spherical deconvolution, Q-ball modeling, have been developed to better exploit the acquisition dependent signal of Diffusion-weighted image(DWI). Despite considerable progress in this area, current methods still face many challenges, such as sensitive to noise, lots of false positive/negative fibers, incapable of handling complex fiber geometry and expensive computation cost. More importantly, recent researches have shown that, even with high-quality data, the results using current tractography methods may not be improved, suggesting that it is unlikely to obtain an anatomically accurate map of the human brain solely based on the diffusion profile. Motivated by these issues, this dissertation develops a global approach that incorporates anatomical validated geometric shape prior when reconstructing neuron fibers. The fiber tracts between regions of interest are initialized and updated via deformations based on gradients of the posterior energy defined in this paper. This energy has contributions from diffusion data, shape prior information, and roughness penalty. The dissertation first describes and demonstrates the proposed method on the 2D dataset and then extends it to 3D Phantom data and the real brain data. The results show that the proposed method is relatively immune to issues such as noise, complicated fiber structure like fiber crossings and kissing, false positive fibers, and achieve more explainable tractography results.

In this paper, for the current algebra associated with the Lie algebra $\lie{so}_{2n}(\mathbb{C})$, we connect particular CV-modules, indexed by pairs of integral weights $(\lambda,\mu)$ that satisfy particular conditions, with generalized Demazure modules which were introduced by [Naoi]. In particular, we show an isomorphism of a CV-module $V(\xi(\lambda,\mu))$ and the module generated by the tensor product of generating vectors $v_\lambda$, $v_\mu$ for the local Weyl modules $W_{\loc}(\lambda)$ and $W_{\loc}(\mu)$ respectively, i.e. $V(\xi(\lambda,\mu))\cong \langle v_\lambda \otimes v_\mu\rangle \subset W_{\loc}(\lambda) \otimes W_{\loc}(\mu)$. To complete this proof, we construct short exact sequences of CV-modules. Moreover, through this construction, we obtain a Demazure character formula for the CV-module $V(\xi(\lambda,\mu))$.

In this paper, we give an overview of the classical Temperley-Lieb algebra, reviewing some of the basic results about the algebra. Then we will discuss the construction of the generalized Temperley-Lieb algebras from Coxeter groups, and the connection between fully commutative elements of the Coxeter group and the reduced elements of the corresponding Temperley-Lieb algebra. Finally, we will look at the diagrammatic representations of some finite-dimensional Temperley-Lieb algebras, specifically those of type B, D and E, and describe their bases.

This dissertation consists of two parts. First, we define and study positional marked patterns. A positional marked pattern tau is a permutation where one of elements in tau is underlined. Given a permutation sigma, we say that sigma_i is tau-match at position i if tau occurs in sigma in such a way that sigma_i plays the role of the underlined element in the occurrence. We let pmp_tau(sigma) denote the number of position i which sigma is tau-match. This defines a new class of statistics on permutations, where we study such statistics and prove a number of results. In particular, we prove that two positional marked patterns (1)23 and (1)32 give rise to two statistics that have the same distribution. The equidistibution phenomenon also occurs in other several pairs of patterns like 1(2)3 and 1(3)2, which we prove in this dissertation. The second part of the dissertation focuses on the Whitney numbers of Dowling lattices. In the papers [2, 3], Benoumhani defined two polynomials Fm,n,1(x)and Fm,n,2(x). Then, he defined Am(n,k) and Bm(n,k) to be the polynomials satisfyingFm,n,1(x) = Sum_k=0^n Am(n,k)x^(n-k)(x+1)^k andFm,n,2(x) = Sum_k=0^n Bm(n,k)x^(n-k)(x+1)^k.In this dissertation, we give a combinatorial interpretation of coefficients of Am+1(n,k) and prove a symmetry of the coefficients, namely Am+1(n,k)|m^s = Am+1(n,n-k)|m^(n-s). We also give a combinatorial interpretation of Bm+1(n,k), prove that Bm+1(n,n-1) is a polynomial in m with non-negative integer coefficients, and prove that Bm+1(n,n-2) is a polynomial in m with non-negative integer coefficients except for the coefficient of m^(n-1) which is -(n-1) for n >=6.

The Black-Litterman model combines the market equilibrium with the investor's personal views and gives optimal portfolio weights. In this paper we will review the original Black-Litterman model, we will modify the model such that it fits in a Bayesian framework by considering the investors' personal views to be a direct prior on the means of the returns and by including a typical Inverse Wishart prior on the covariance matrix of the returns. We will then consider Leonard and Hsu's (1992) idea for a prior on the logarithm of the covariance matrix. We encountered both running time and memory allocation problems when we applied the latter version to the whole S&P 500. To overcome such computational problems, Bayesian factor models are considered for the analysis. This choice was also motivated by the strong connection between Black-Litterman and the Capital Asset Pricing Model, which itself can be seen as a factor model. Sensitivity analyses for the level of confidence that investors have in their own personal views were performed and performance of the models was assessed on a test data set consisting of returns over the month of January 2018.

In this dissertation we will first give an exposition for some topics on Perelman's entropy and some results related to the analysis of the entropy, and then present the content of two papers among the author's publication list. Chapter 1 is an introduction to Perelman's entropy and the author's main results. The statement of these main results can be found in section 1.3. Chapter 2 and chapter 3 are expository materials on Perelman's entropy and its related analytic tools; these results are included because of their importance to our main theorems. In chapter 4, we prove some estimates for the Nash entropy on ancient solutions and thereby prove a gap theorem for the asymptotic entropy. In chapter 5, we prove an assertion made by Perelman, saying that for an ancient solution to the Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to $\kappa$-noncollapsing on all scales. This proof is based on accurate gaussian upper and lower estimates for the conjugate heat kernel.

This dissertation discusses the Frankel conjecture and the Kähler-Ricci flow approach to it. Frankel conjecture (first proved by Mori and Siu-Yau independently) states that every compact Kähler manifold of positive bisectional curvature is biholomorphic to the complex projective space. On the other hand, the Ricci flow introduced by Hamilton was used by Bando and Mok to generalize Siu-Yau's theorem to nonnegative bisectional curvature case. It's natural to ask if there is a primarily flow proof of the original Frankel conjecture.The convergence of Kähler-Ricci flow on compact manifolds with positive bisectional curvature would imply such a proof, however not yet been completed. The advances closest to this target might be a series of papers by Phong-Song-Sturm-Weinkove along with the improvements by Cao-Zhu and Zhang. We are going to survey their works in this thesis, and also cover some new result proved by the author.This thesis is organized as follows. In chapter one, we first collect some fundamental facts of Kähler geometry, and then go over the convergence theory of Kähler-Ricci flow on Fano manifolds built on stability conditions. In chapter two, we review known results on bisectional curvature, and then relate the curvature to the former stability conditions. Finally we will state and prove our new result.

This paper describes some possible equilibrium configurations of a string rotating in a certain class of force fields, which have properties motivated by the inverse square gravitational force. Specifically it is shown that for a given number n, there exists a sufficient rotational rate such that any equilibrium configuration with no more than n zeros is guaranteed to exist for any force field in the class considered.

The bounded cohomology of a group G with coefficients in a normed G-module V was first systematically studied by Gromov in 1982 in his seminal paper [Gro82] in connection to the minimal volume of manifolds. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for the most basic groups: There is no finitely generated group G for which the full bounded cohomology with real coefficients is known except where it is known to vanish in all degrees (see [Mon06]). In this thesis we discuss several new constructions for classses in bounded cohomology. There is a well-known interpretation of ordinary group cohomology in degrees 2 and 3 in terms of group extensions. We establish an analogous interpretation in the context of bounded cohomology. This involves certain maps between arbitrary groups called quasihomomorphisms, which were defined and studied by Fujiwara and Kapovich in [FK16]. A key open problem is to compute the full bounded cohomology of a non-abelian free group F with trivial real coefficients. It is known that the bounded cohomology in dimension n is trivial for n = 1 and infinite dimensional for n = 2, 3, but essentially nothing is known about for n ≥ 4. For n = 4, one may construct classes by taking the cup product between two 2-classes, but it is possible that all such cup-products are trivial. We show that all such cup-products do indeed vanish if both classes are induced by the quasimorphisms of Brooks or Rolli.

This paper is an educational approach to group characters through examples which introduces the beginner algebraist to representations and characters of finite groups. My hope is that this exploration might help the advanced undergraduate student discover some of the foundational tools of Character Theory. The prerequisite material for this paper includes some elementary Abstract and Linear Algebra. The basic groups used in the examples are intended to excited a student into exploration of groups they understand from their undergraduate studies. Throughout the section of examples there are exercises used to check understanding and give the reader opportunity to explore further. After taking a course in Abstract Algebra one might find that groups are not concrete objects. Groups model actions, rotations, reflections, movements, and permutations. Group representations turn these abstract sets of objects into sets of n X n matrices with real or complex entries, which can be easily handled by a computer for any number of calculations.

In algebraic geometry, seemingly geometric problems can be solved using algebraic techniques. Some of the most basic geometric objects we can study are polynomial curves in the plane. In this paper we focus on the intersections of two curves. We address both the number of times two curves intersect at a given point, counting multiplicity (whatever that means), and the total number of intersections of the curves, again counting multiplicity. The former is known as the intersection number of the curves at the point. This concept, although geometrically motivated, can be described in algebraic terms; it is this relationship which makes it such a powerful concept. The paper concludes with an important application of the intersection number, Bezout's Theorem. This ubiquitous theorem gives a beautifully concise solution to the total number of intersections, given sufficiently nice assumptions on the curves and the ambient space.

This paper will examine observables in the bc system, a two-dimensional free conformal field theory. We begin by encoding the bc system into the BV formalism following procedures of Costello and Gwilliam. This will allow us to construct the factorization algebra of observables for the bc system. The cohomology of the factorization algebra recovers the observables themselves. In cohomology, we will compute the commutation relations and factorization algebra structure maps for observables supported on disks and annuli. These structure maps will be used to prove the equivalence of the factorization algebra and vertex algebra structures for the bc system. This proof provides a rigorous derivation of the free fermionic vertex algebra starting from the action functional of the bc system. Using this equivalence, we will provide a dictionary to translate the action of the Virasoro algebra to the language of factorization algebras. Also in this paper, we examine the bc system in four-dimensions. We construct its factorization algebra and show that its observables are anti-commutative. Lastly, we prove that the global observables of the bc system are one-dimensional on a compact manifold of complex dimension one or two.

In 1934, H. Whitney presented a series of papers which discussed how to determine whether a function or a jet of order m is the restriction of a Cm-function on Rn. In the first paper of the series, Whitney's Extension Theorem was proved. In the latter, Whitney answered special cases of the following questionQuestion(Whitney's Extension Theorem, WEPn,m) Let f be a continuous function from a closed subset E of Rn to R.How can we determine whether f is the restriction of a Cm-function on Rn?In this dissertation, we work in o-minimal expansions of real closed ordered fields. Definable versions of Whitney's Extension Theorem and Whitney's Extension Problems will be discussed in this context. Definable set-valued maps are also studied; a definable version of Michael's Selection Theorem will be proved and used, in combination with a definable version of Whitney's Extension Theorem, to give a positive answer to a definable version of WEPn,1.In addition to the above problems, we also discuss smoothing problems. This is inspired by a series of papers by A. Fischer.In this series, a construction of a definable Cm-approximation of a definable locally Lipschitz function is provided. Here, we also work in an o-minimal expansion of a real closed field and relax the condition further to just continuous.

We examine the Carlson-Simpson Lemma (VW(k,l)), which is the combinatorial core of the Dual Ramsey Theorem, from the perspective of Reverse Mathematics. Our results include the following:Working in the system BSigma_2^0, we carry out the construction of a failure of the ordered version of the Carlson-Simpson Lemma OVW(k,l), which was introduced in a paper by Miller and Solomon. This observation implies that we can construct such a recursive counterexample in the model of SRT_2^2 that was discussed in a recent paper by Chong, Slaman and Yang. It follows that SRT_2^2 does not prove OVW(k,l) over RCA_0.We also show that the strength of the principle VW(k,l) is independent of the number of colors l being used.By proving that VW(k,l) is not conservative over RCA_0 for arithmetical sentences, we conclude that VW(k,l) is not provable from any theory that is conservative over RCA_0 for arithmetical sentences.

This dissertation consists of three papers I have written or helped write during my time at UC Berkeley. The three papers all fall under the common theme of exploring connections between free probability and planar algebras.In Chapter 2, an amalgamated free product algebra, $\cN(\Gamma)$, is constructed for any connected, weighted, loopless graph $\Gamma$, and its isomorphism class can be nicely read off based on the weighting data on the graph. This construction was heavily influenced by an algebra appearing in in work of Guionnet, Jones and Shlyakhtenko, and it is used, along with some standard embedding arguments, to show that the factors of Guionnet, Jones, and Shlyakhtenko appearing in are isomorphic to $L(F_{\infty})$ when the planar algebra is infinite depth.Chapter 3 is the paper ``Rigid $C^{*}-$tensor categories of bimodules over interpolated free group factors" which was co-authored with Arnaud Brothier and David Penneys. In this paper, we establish an unshaded planar algebra structure (with multiple colors of strings) which can be used to model a countably generated rigid $C^{*}-$tensor category, $C$. We use this to construct a category of bifinite bimodules over a $II_{1}$ factor $M_{0}$, and we show that this category is equivalent to $C$. Finally, we use the work in Chapter \ref{chap:graph} to show that the factor $M_{0}$ is isomorphic to $L(F_{\infty})$.Chapter 4 is a note regarding multishaded planar algebras. The problem studied consists of placing the ``Fuss-Catalan" potential on a specific kind of subfactor planar algebra $Q$, which is the standard invariant for and inclusion $N \subset M$ with an intermediate subfactor $P$. This potential is best understood by augmenting $Q$, forming a bigger multishaded planar algebra $\mathcal{P}$. The isomorphism classes of the algebras $M_{\alpha}$ associated to $\mathcal{P}$ will be computed explicitly. While the isomorphism class of the von Neumann algebras $N_{k}^{\pm}$ associated to $Q$ are still not yet known, they will be shown to be contained in a free group factors and contain a free group factors. This potential is shown to yield a nice free product expression for the law of $\cup$, an element which plays a critical role in understanding algebras that arise from this construction. Many of the ideas in this chapter influenced the work in the Chapter 3.

This dissertation is a collection of four papers involving embedded contact homology (ECH). ECH is a three-manifold invariant recently defined by Michael Hutchings. The dissertation covers both technical results about ECH (for example a proof that the absolute grading in ECH is a topological invariant) and applications of ECH to Reeb dynamics and symplectic embedding problems. All four papers were written while the author was a graduate student at UC Berkeley.

Given a linear equation, the least integer n, provided it exists, such that for every t-coloring of the natural numbers {1, 2, . . . , n} there exists a monochromatic solution to the equation is called the t-color Rado number for the equation. In this paper, the 2-color Rado number for each of the following two equations is established:2x1 + 2x2 + c = x3and 2x1 + 2x2 + 2x3 + c = x4 where c is an integer. Given a linear equation, the least real number b, provided it exists, such that for every t-coloring of the interval [1, b] there exists a monochromatic solution to the equation is called the t-color continuous Rado number for the equation. In this paper, the 2-color continuous Rado number for the following family of equations is also established: a1x1 + a2x2 + · · · + am-1 xm-1 + c = xm where m is an integer such that m ≥ 3, a1, a2, . . . , am-1 are natural numbers, and c is a real number.

Cluster randomized trials are increasingly popular in epidemiological and medical research. When analyzing the data from such studies it is imperative that the hierarchical structure of the data be taken into account. Multilevel logistic regression is used to analyze clustered data with binary outcomes. Previous literature shows that a greater number of clusters is more important than a large number of subjects per cluster. This paper investigates if it is possible to compensate for the increased bias found for parameter estimates when the number of clusters is decreased. A simulation study was conducted where the absolute percent relative bias for each parameter estimate with 5 to 49 clusters and 10, 20, 30, 60, 90, 120, 150, 180, and 210 subjects per cluster were compared to the bias found for corresponding parameter estimates when the number of clusters was 50 with 10 subjects per cluster. Maximum Likelihood, Restricted Maximum Likelihood, and Generalized Estimating Equation methods, with multiple Intraclass Correlation Coefficients were examined. For Maximum Likelihood estimates, results show that it is possible to account for the effects of few clusters by increased sample size when examining fixed effect parameter estimates. For variance components, it was not possible to fully compensate under all conditions, but in general, the trend found was that increasing the number of subjects per cluster either results in decreased bias or the bias plateaued after a certain sample size. Further investigation is needed on Restricted Maximum Likelihood and Generalized Estimating Equation estimates, but results show that they do not perform well when the number of subjects per cluster is few. The results of this study are very informative for researchers who are limited to few clusters.

This study examined the effectiveness of the iPad app Mathspace on high school students’ algebra computation and skill maintenance. The participants in this study were four ninth grade students enrolled in beginning algebra at a large, urban high school. All four students were struggling to obtain grade-level algebra skills and were reported by their teachers as the lowest performing students in their math classes. This study used a multiple probe design across participants. In the intervention condition, participants used Mathspace to learn and practice algebra skills related to exponents then answered 10-item paper and pencil algebra probes to measure progress. Once students reached mastery criteria of at least 90% accuracy on three consecutive trials, they entered the maintenance phase, where students completed intermittent probes to measure the retention of the skills learned in the intervention. The results of this study indicated that all three participants who completed the intervention improved their algebra calculation skills as a result of using Mathspace. Maintenance data obtained after the completion of the intervention indicated a decrease in scores for all three participants, but for the majority of students, their scores remained above baseline levels. Implications for practitioners and directions for future research are included.

This dissertation entails four different topics related to coding theory.While much effort has gone recently into studying codes over ring alphabets (rather than the traditional field alphabets), we change this perspective slightly and focus not necessarily on the algebraic structure of the alphabet but in the ancillary structure of the ambient (the algebraically enhanced environment from which codes are drawn.) Doing so, we determine, for example, that when the ambient is a Frobenius algebra a version of the MacWilliams identities holds between the weight distribution of the appropriate codes and their properly defined duals.We further the consideration of rational power series and sequential codes, which had mostly been studied over field alphabets, to the case when the alphabet is a commutative local ring with nilpotent radical. Perhaps the most striking feature of our study is the introduction of a modified division algorithm that is based not on the degree of polynomials but on their codegree (degree of the lowest degree non-zero term.) As was the case in the initial papers on the subject, we explore criteria to recognize the Kronecker Criterion as well as multivariable rational functions among arbitrary multivariable power series with coefficients in a commutative local ring with nilpotent radical.The duals of sequential codes are the so-called polycyclic codes. We have studied intrinsic notions of duality based on the codegree of polynomials with sights to have a fully intrinsic perspective while studying codes in such ambients.We continue the ongoing explorations to recognize commutative local finite rings. Our study considers the feasibility of extending results in the literature for rings with 16 elements to rings with p4 elements (p an arbitrary prime.)

We study the existence and stability of singularity formation in nonlinear wave equations.First, we formulate the problem of mode stability of self-similar blowup solutions to a class of nonlinear wave equations. We then develop a method of proving mode stability in this context and illustrate it on two important examples, the spherically symmetric Yang-Mills equation and equivariant wave maps into a sphere, for which the problem was open for almost a decade. Our method is broadly applicable and provides a general approach to stability problems related to self-similar solutions of nonlinear wave equations. This part of the thesis is based on two research papers that already appeared in print, see [13,16].Next, we study the singularity formation in wave maps into a negatively curved target manifold. More precisely, we consider wave maps on (1+d)-dimensional Minkowski space. For each dimension d≤8 we construct a negatively curved, d-dimensional target manifold that allows for the existence of a self-similar wave map which provides a stable blowup mechanism for the corresponding Cauchy problem. In particular, we illustrate how the problem of mode stability plays a decisive role in the proof of the (nonlinear) stability of self-similar blowup. This part is based on a joint work with Roland Donninger [19].

The symmetric group of degree n, denoted as S_n, is defined to be the group of permutations on a set with cardinality n, thereby consisting of n! total permutations. In this paper, we bound the maximal irreducible character degree of S_n in terms of smaller degrees of the irreducible characters. Specifically, we demonstrate that for every integer n ≥ 106,∑{χ ∈ Irr(S_n)} {χ(1) < b(S_n)} χ(1)² > 4b(S_n)²where Irr(S_n) represents the set of irreducible complex characters of S_n, b(S_n) represents the largest degree of a character of Irr(S_n), and χ(1) represents the degree of a character of Irr(S_n).

This dissertation deals with topics in convex analysis and its application in quantum information theory.In the introduction, we collect definitions and background material from analysis, convex geometry and quantum information theory that will be used throughout the paper. Part I addresses problems on mathematical aspects of quantum information. In Chapter 2, we prove the reverse triangle inequality for quantum states. It is a fundamental property of quantum states which is also interesting on its own. It turns out that it is crucial in deriving a quantitative upper bound of quantum violation for the (generalized) Bell inequality with 2 × n inputs. In Chapter 3 we calculate the exact values of quantum violations for all extremal Bell correlation inequalities that appear in the setups involving up to four measurements; they are all not greater than v2, the Tsirelson bound for the 2×2 setup. While various authors investigated these inequalities via numerical methods, our approach is analytic. We also include tables summarizing facial structure of Bell polytopesin low dimensions. The content of these two chapters appears in [59] and [65].Part II concentrates on convex geometric analysis. Recently, many notions and re- sults in convex geometry have been extended from the class of convex bodies to classes of functions. In Chapter 4, we introduce floating sets for convex, not necessarily bounded subsets in Rn. This allows us to define floating functions for appropriate classes of func- tions and measures. We establish the asymptotic behavior of the integral difference of alog-concave function and its floating function which in turn leads to a new affine invariant for functions. The content of these results appears in [66].

The first steps in defining a notion of spherical tropicalization were recently takenby Tassos Vogiannou in his thesis and by Kiumars Kaveh and Christopher Manonin a related paper. Broadly speaking, the classical notion of tropicalization concernsitself with valuations on the function field of a toric variety that are invariant underthe action of the torus. Spherical tropicalization is similar, but considers insteadspherical G-varieties and G-invariant valuations.The core idea of my dissertation is the construction of the extended tropicalizationof a spherical embedding. Vogiannou, Kaveh, and Manon only concern themselveswith subvarieties of a spherical homogeneous space G/H. My thesis describes how totropicalize a spherical embedding by tropicalizing the additional G-orbits of X andadding them to the tropicalization of G/H as limit points. This generalizes work doneby Kajiwara and Payne for toric varieties and affords a means for understanding howto tropicalize the compactification of a subvariety of G/H in X.The extended tropicalization construction can be described from three differentperspectives. The first uses the polyhedral geometry of the colored fan and the secondextends the Grobner theory definition given by Kaveh and Manon. The third methodworks by embedding the spherical variety in a specially-constructed toric variety,tropicalizing there with the standard theory, and then applying a particular piecewise-projection map. This final perspective introduces a novel means for tropicalizing a homogeneous space that allows us to prove several statements about the structure of a spherical tropicalization by transferring results from the toric world where more is known.We also suggest a definition for the tropicalization of subvarieties of a homogeneousspace whose defining equations have coefficients with non-trivial valuation. Allthe previous theory has been done in the constant coefficient case, i.e. when thecoefficients of the defining equations all have trivial valuation.

Our goal in this paper is to lay the foundation for a theory of smooth automorphic forms and representations on local and adelic reductive groups G that does not rely on the K-finite or classical unitary theories. Specifically, we consider forms that are smooth but not necessarily K-finite, that is, whose orbits under a maximal compact subgroup are not necessarily contained in finite dimensional spaces. To do so, we introduce spaces of automorphic forms of "fixed type" à la Borel and Jacquet, imposing locally convex topologies to gain control of these now infinite dimensional spaces. Along the way we present a novel treatment of group norms on G, emphasizing their compatibility both with the structure of G as a reductive group and a restricted product of reductive groups over local fields.Motivated by the work of Wallach in the 1980s, we relate the asymptotic behavior of our representations to the growth of the automorphic forms that generate them. We show that our smooth representations behave analogously to the Casselman-Wallach globalizations of representations of real reductive groups. We then introduce a new algebra of "Schwartz functions" on G and demonstrate it serves the same role for smooth representations as the Hecke algebra does for their K-finite counterparts.Ultimately, our aim is to illustrate how smooth representations serve the same role for K-finite automorphic representations as Casselman-Wallach globalizations do for Harish-Chandra modules. Our tensor product theorem for smooth automorphic representations completes the picture of these representations as generalizations of Casselman-Wallach globalizations. Therein we show the "factors" of such a representation (ϖ, V) at the archimedean places are precisely the Casselman-Wallach globalizations corresponding to archimedean factors in the underlying Harish-Chandra module of K-finite vectors in V.

Dynamical systems as models of complex biological systems are powerful tools that have been used to study problems in biology. This dissertation first discusses a problem in the theory of dynamical systems on the definition of topological entropy. Then dynamical systems are applied in two different fields of biology: proliferation of monolayer epithelia and the spread of infections and information.The notion of topological entropy is a measure of the complexity of a dynamical system. It can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution ε within T time units. It can then be formally defined as a limit of a limit superior that involves either covering numbers, or separation numbers, or spanning numbers. If covering numbers are used, the limit superior reduces to a limit. While it has been generally believed that the latter may not necessarily be the case when the definition is based on separation or spanning numbers, no actual counterexamples appear to have been previously known. Here we fill this gap in the literature byconstructing such counterexamples.We then use dynamical systems to study the proliferation of epithelia. Epithelia are sheets of tightly adherent cells that line both internal and external surfaces in metazoans (multicellular animals). Mathematically, a cell in an epithelial tissue can be modeled as a k-sided polygon. Empirically studied distributions of the proportions of k-sided cells in epithelia show remarkable similarities in a wide range of evolutionarily distant organisms. Multiple types of mathematical models have been proposed to explain this phenomenon. Among the most parsimonious of such models are topological ones that take into account only the number of sides of a given cell and the neighborhood relation between cells. The model studied here is a refinement of a previously published such model (Patel et al., PLoS Comput. Biol., 2009). While using the same modeling framework as that paper, we introduce additional options for the choice of the endpoints of the cleavage plane in the simulated development of epithelial tissues. Some of these options appear to be more biologically realistic approximations of known mechanisms that influence the cleavage plane orientation. A comparative analysis of comprehensive simulations for relevant combinations of both previously studied and our newly designed options is reported here. We compared the outcomes with the empirical distributions of the same five organisms as in (Patel et al., 2009), Drosophila (fruit fly), Hydra (closely related to jellyfish), Xenopus (clawed frog), Cucumber, and Anagallis (a flowering plant). We found that combinations of some of our new options consistently gave better fits with each of these data sets than combinations of previously studied options for choosing the cleavage plane.We also examine ODE models of epidemic spreading with a preventive behavioral response that is triggered by awareness of the infection. Previous research on such models has mostly focused on the impact of the response on the initial growth of an outbreak and the existence and location of endemic equilibria. Here we study the question whether this type of response is sufficient to prevent future flare-ups from low endemic levels if awareness is assumed to decay over time. In the ODE context, such flare-ups would translate into sustained oscillations with significant amplitudes. Our results show that such oscillations are ruled out in Susceptible-Aware-Infectious-Susceptible models with a single compartment of aware hosts, but can occur if we consider two distinct compartments of aware hosts who differ in their willingness to alert other susceptible hosts.

With the recent surge of infectious disease outbreaks across college campuses in the United States, there is a mounting need to ascertain strategies that best mitigate an outbreak’s size and duration. Thus, this paper (i) creates a model to quantify the impact of university-level responses and (ii) determines which control interventions are most effective. The model is among the first in the field of epidemiology to overcome three crucial issues: stochastic variation in small populations, missing or unobserved case data, and changes in disease transmission rates post-intervention. To test the model and assess various interventions, we use data from the 2016 mumps outbreak at Harvard University. The results suggest that universities should design more aggressive diagnostic procedures and stricter isolation policies to decrease infectious disease incidence on campus.

The following dissertation is a combination of three papers, emanating from two separate and mostly disjoint projects. The first project is an investigation of when the product of arbitrary infinite matrices is defined and associative. In the case where infinite matrices represent endomorphisms on an infinite-dimensional vector space, this question is trivial since multiplication of these matrices amounts to composition of the endomorphisms, a well-defined and associative operation. In the case of infinite matrices in general, this question becomes very nontrivial. In Chapter 2 we will explore this question of associativity and give necessary and sufficient conditions for a set of three infinite matrices A, B, and C to have an associative product. Afterwards we will give some applications for this characterization.The second project which makes up this dissertation is an exploration of R-algebras which have the property that there exists a basis B for the algebra which consists solely of strongly regular elements. This work was originally inspired by [26] and [27] which investigated algebras which have bases consisting solely of invertible elements. In the third chapter, it is shown that these strongly regular elements are also satisfy a slightly weaker version of invertibility which is dubbed “local invertibility.” Because local invertibility does not presuppose the existence of a multiplicative identity, we may examine algebras which fail to have a unit. Among algebras which satisfy the local invertibility property are all finite dimensional unital algebrs over a division ring D, and many examples of infinite matrix algebras such as the R-algebra of infinite matrices with finitely many nonzero entries, and the column-finite, and row-and-column finite matrix algebras.The fourth chapter is work towards determining when Leavitt path algebras have the invertibility and local invertibility properties. In [28–30], the authors completely classify when the Leavitt path algebra arising from a finite graph is invertible. I follow up on these author’s research in two ways. First, using the author’s analysis of “source loops,” I show that so long as the finite graph has no non-isolated source loops, the Leavitt path algebra is locally invertible. Secondly I give various conditions under which certain elements of Leavitt path algebras are locally invertible. Then I give some graph theoretic characterizations which are sufficient for local invertibility of the Leavitt path algebra. Among other characterizations it is shown that every von Neumann regular Leavitt path algebra and every directly finite Leavitt path algebra is locally invertible.Finally in the fifth chapter I will outline some avenues for future research on the problems described above, in many cases giving partial results towards the resolutions of the questions.

Our goal is to be able to determine the distance between two partitions in the same Rouquier block. This paper explores a lower bound on the number of steps required to get from the quotient of a partition lambda; to (empty set, empty set, ..., (w)). We will use the Young diagrams in the quotient of lambda and a process of right and left steps to obtain a new partition that is connected to lambda in the Brauer graph of these representations. Paired with this process we will incorporate a notation called the badness of each box in the Young diagram representation of the quotient of lambda;. This will allow us to show that the lower bound on the number of steps is less than or equal to the initial badness, b, of the starting partition divided by the maximum reduction of badness by each right and left step, 2w^2.

In this paper, we examine a certain graph related to the representation theory of the symmetric group. This graph is called the Brauer graph of a Rouquier block. It is known that the diameter d(R) of this graph satisfies the following condition: the diameter d(R) is between p - 1 and two times the ceiling function of "log base two of w" + p + 2 where p and w are certain associated parameters. Note that the preceding condition describes both a lower bound and an upper bound on the diameter d(R). We conjecture that the best upper bound for the diameter is actually between p - 1 and the ceiling function of "log base 2 of w" + p. We conjecture that two vertices in our graph have the maximum distance between them, and we prove that this distance is at most the ceiling function of "log base 2 of w" + p by exhibiting an algorithm that creates a path between these two vertices.

In this paper, we go through the steps to define the crossed product, a construction that allows us to consolidate appropriate types of dynamical systems into a C*-algebra. From there, we study how various properties of this constructed crossed product relate to those of the base system. In particular, we show the equivalence of the minimality of a classical dynamical system and the simplicity of the crossed product.

In this paper, we establish the John-Stromberg inequality in certain anisotropic BMO spaces and apply the inequality to anisotropic Holder continuous function spaces. To achieve this goal, we define the median value, BMO space and sharp maximal function first. Secondly we constructed a family of continuously expanding rectangles and give a Calderon–Zygmund type decomposition. By using these tools, we establish the John-Stromberg inequality. Finally, an application to anisotropic Holder continuous function spaces is presented.

This paper concerns forecasting the performance of the S/V Zodiac, a two-masted schooner operating as a public and private charter vessel out of Bellingham, Washington. Two aspects of performance are discussed, namely the business cycle of the Zodiac and the operating efficiency of the Zodiac. For the business cycle, it is found that the best model for the business cycle is the one-year-ahead plan developed for the vessel in or around November each year. As for the operating efficiency, an original model is developed from techniques used for forecasting power outputs of wind farms and individual wind turbines. This model uses probabilistic methods to generate a weekly “efficiency measure” of the S/V Zodiac that may be used comparatively.

Calculus is an introductory course for most students in Science, Technology, Mathematics andEducation (STEM). Limits are an essential part of the learning of calculus. It has been previouslydocumented that students tend to struggle when first learning the topic of limits. This paper is aninvestigation in to the personal concept definition of limits and how consistent a students personalconcept definition to the formal definition of the limit after they have completed courses in introductorycalculus. The study took place during a 15-week semester, with 28 students, who weretaking either Calculus III or Ordinary Differential equations. These 14 male and 14 female participantsattended the same Midwestern Public University. They were given a short, in-class, surveywhere they could demonstrate their concept definition and the operability of their concept definitionas it pertains to limits and limit-based problems. Three categories were created to signify thestudents level of operability in solving limit-based problems based on their responses to the survey.The categories were Low, Mid, and High Scoring, which demonstrated an inoperable, partially operable,or fully operable concept definition, respectively. The majority of students fell in to Lowand Mid Scoring categories, indicating their lack of operability in personal concept definition as itpertains to limits. This study suggests that students in mathematics should be encouraged to developtheir conceptual understanding and move past procedural knowledge as a way of masteringa mathematical topic, such as limits.

As we know, there are many equalities and inequalities for stochastic integrals. Some equalities and inequalities hold when the stochastic integrator has very nice characteristics, like Brownian motion. What will happen to the equalities or inequalities valid for the Brownian motion in the case of other stochastic process, as like proper tempered alpha-stable Levy process? A proper tempered alpha-stable Levy process combines both the alpha-stable and Gaussian trends. In a short time frame it is close to an alpha-stable process while in a long time frame it approximates a Brownian motion. So, we can find the keys that make analogs of these equalities and inequalities hold.First I found two equalities and two inequalities that hold for the Brownian motion integral. There are two aspects to prove them. First of all, we consider when the integrand is predictable step process. Base on that it is a finite sum, we can get it from normal random variable inequalities. On the other hand, we need to extend the situation to where integrand is a general predictable process. It involves the problem whether what we deal with is integrable. Secondly, I will research if these equalities and inequalities hold for proper tempered alpha-stable Levy process. In this step, I will find the space of functions which are integrable for proper tempered alpha-stable Levy process at first. So that we can find the predictable process which is integrable for proper tempered alpha-stable Levy process then. And I will research if these equalities and inequalities still hold for integrals of predictable step integrands with respect to proper tempered alpha-stable Levy processes. At last, I will extend the predictable step condition to general predictable condition. Based on the research about proper tempered alpha-stable Levy process, there are some tools that make these equalities and inequalities hold. I will prove that a process X(t) which is a Levy process and martingale with EX(t)^2=t satisfies these equalities and inequalities.The results are based on the the book by Kwapien and Woyczynski and paper by Rosinski.

This thesis will present an original work characterizing hyponormality and positivity of Toeplitz operators with bounded symbols on the weighted Bergman spaces of on the disk and whole complex plane.Cuckovic and Curto have recently obtained a necessary condition for the hyponormality of the Toeplitz operators with a certain harmonic symbol on the disk in C. We further extended the same problem to the weighted Bergman space and obtained the analogous result as in the unweighted Bergman space. This thesis will also survey the positivity of Toeplitz operators with bounded and unbounded symbols on the Bergman, Fock and certain Model spaces in terms of the Berezin transform of the symbol. Inspired by the paper of Zhao and Zheng, we have studied positivity of the Toeplitz operators with a bounded symbol on the Model spaces in terms of the Berezin transform. Analogous results have been obtained in the Fock space case as well.

In data rich environments we may sometimes deal with time series of infinite dimensional objects such as smooth curves, square-integrable functions or probability density functions. The so-called functional time series analysis considering such time series has been intensively developed by many authors. For stationary functional time series, we already have well-developed theoretical results. On the other hand, the literature on nonstationary functional time series is not rich enough yet; we only have a few recent papers that consider nonstationary functional time series. This dissertation deals with functional cointegrated linear processes and develops a generalization of the Granger-Johansen representation theory in Hilbert spaces and Banach spaces. Given the scarcity of the existing literature on nonstationary functional time series, I believe that my results pave the way for the development of statistical methods involving functional time series exhibiting the random walk-type nonstationarity. The first chapter concerns cointegrated linear processes in an arbitrary complex Hilbert space. We extend the notion of cointegration for time series taking values in such a Hilbert space, and provide generalized notions of I(0) and I(1) sequences. In the chapter we specifically show that the cointegrating space for an I(1) process may be sensibly defined as the kernel of the long run covariance operator of its first difference. Another main result of the chapter is a generalization of the Granger-Johansen representation theorem for I(1) autoregressive processes. We will observe that a geometric reformulation of the Johansen I(1) condition is useful to our Hilbert space setting, and it will be shown that a generalization of the Granger-Johansen theorem is derived based on this observation.The second chapter is more focused on the Granger-Johansen theory in an arbitrary complex Hilbert space setting. We provide a generalization of the representation theorems for I(1) and I(2) autoregressive processes taking values in such a Hilbert space. A big difference from the previous chapter is that we rely on rigorous analytic operator-valued function theory. The most important input for our representation theory in the chapter is the so-called analytic Fredholm theorem, which will turn out to be useful. We will demonstrate this in detail. In the last chapter, we will show that our representation theory based on the analytic Fredholm theorem can be extended to a Banach space setting. Specifically, we study the inversion of a holomorphic Fredholm operator-valued function in detail, and provide a closed-form expression of the inverse. Applying these results, we obtain our representation theory in an arbitrary complex separable Banach space. One meaningful aspect of this chapter is that we obtain a generalization of the Granger-Johansen theory without the help of rich geometric structure of a Hilbert space.

The theory of bisets has been very useful in progress towards settling the longstanding question of determining units for the Burnside ring. In 2006 Bouc used bisets to settle the question for $p$-groups. In this paper, we provide a standard basis for the unit group of the Burnside ring for groups that contain an abelian subgroups of index two. We then extend this result to groups $G$, where $G$ has a normal subgroup, $N$, of odd index, such that $N$ contains an abelian subgroups of index $2$. Next, we study the structure of the unit group of the Burnside ring as a biset functor, $B^\times$, on this class of groups and determine its lattice of subfunctors.We then use this to determine the composition factors of $B^\times$ over this class of groups. Additionally, we give a sufficient condition for when the functor $B^\times$, defined on a class of groups closed under subquotients,has uncountably many subfunctors.

In this paper I will first discuss the structure of non-standard maximal parabolics of twisted affine Lie algebras. Then I will define global Weyl modules in this setting and discuss the associated commutative associative algebra, $\bA_\lambda.$ These modules are an analog of Verma modules in the affine setting and were defined in 2001 by Chari and Pressley. Global Weyl modules have been studied in other settings such as twisted affine Lie algebras, \cite{CIK, FK, FMS} and non-standard maximal parabolics of untwisted affine Lie algebras, \cite{CKO}.

In this paper we consider the long time behavior of solutions to the cubic nonlinear Schr\"{o}dinger equation posed on the spatial domain $\mathbb{R}\times\mathbb{T}^{d}$, $1\leq d\leq4$. We first prove the local well-posedness in $C(I;L_x^2H_y^s)\cap C(I;L_{x,y}^4)$ for solutions with initial data $u_0\in H^{0,1}_xL_y^2\cap L_x^2H_y^s$. Then, for sufficiently small, smooth, decaying data, we prove global existence and derive modified asymptotic dynamics by using the wave packet method and normal form corrections. The modified scattering behavior on $\mathbb{R}\times\mathbb{T}^d$ combines the modified scattering of the cubic NLS on real line $\mathbb{R}$ with cubic NLS dynamics on torus. We also consider the corresponding asymptotic completeness problem.

In this paper, we propose a general method of defining equivariant theories in symplectic geometry using polyfolds. The construction is twofold, one is for closed theories like equivariant Gromov-Witten theory, the other is for open theories like equivariant Floer cohomology. When a compact Lie group $G$ acts on a tame strong polyfold bundle $p:W \to Z$, we construct a quotient polyfold bundle $\overline{p}:W/G \to Z/G$ if the $G$-action on $Z$ only has finite isotropy. For a general group action and if $Z$ has no boundary, then every $G$-equivariant sc-Fredholm section $s:Z\to W$ induces a $H^*(BG)$ module map $s_*: H^*_G(Z) \to H^{*-\ind s}(BG)$, which can be viewed as a generalization of the integration over the zero set $s^{-1}(0)$ when equivariant transversality holds. When $Z$ is the Gromov-Witten polyfold, $s_*$ yields a definition equivariant Gromov-Witten invariant for any symplectic manifold. We obtain a localization theorem for $s_*$ if there exist tubular neighborhoods around the fixed locus in the sense of polyfold. For open theories, we first obtain a construction for the Morse-Bott theories under minimal transversality requirement. We discuss the relations between different constructions of cochain complexes for Morse-Bott theory. We explain how homological perturbation theory is used in Morse-Bott cohomology, in particular, both our construction and the cascades construction can be interpreted in that way, In the presence of group actions, we construct cochain complexes for the equivariant theory. Expected properties like the independence of approximations of the classifying spaces and existence of action spectral sequences are proven. We carry out our construction for finite dimensional Morse-Bott cohomology using a generic metric and prove it recovers the regular cohomology. We outline the project of combining our construction with polyfold theory, which is expected to give a general construction for both Morse-Bott and equivariant Floer cohomology.In the last part, we show that for any asymptotically dynamically convex contact manifold $Y$, the vanishing of symplectic homology $SH(W)=0$ is a property independent of the choice of topologically simple (i.e. $c_1(W)=0$ and $\pi_{1}(Y)\to \pi_1(W)$ is injective) Liouville filling $W$. As a consequence, we answer a question of Lazarev partially: a contact manifold $Y$ admitting flexible fillings determines the integral cohomology of all the topologically simple Liouville fillings of $Y$.

In this paper, we study linked bilinear pairings and their associated Witt rings. An open problem is to classify all linked quaterionic pairings. The Elementary Type Conjecture [1] asserts that every finite linked quaternionic pairing can be built from symplectic pairings using direct sums and group extensions iteratively. We investigate the validity of this conjecture by studying an infinite quaternionic pairing and its sub- pairings motivated by certain structures arising in Henselian dyadic valued fields.