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Congruences Between Coefficients Of A Class Of Eta-Quotients And Their Applications To Combinatorics, Shashika Petta Mestrige

LSU Doctoral Dissertations

Ramanujan in $1920$s discovered remarkable congruence properties of the partition function $p(n)$. Later, Watson and Atkin proved these congruences using the theory of modular forms. Atkin, Gordon, and Hughes extended these works to $k$-colored partition functions. In $2010$, Folsom-Kent-Ono and Boylan-Webb proved the congruences of $p(n)$ by studying a $\ell$-adic module associated with a certain sequence of modular functions which are related to $p(n)$.

Primary goal of this thesis is to generalize the work of Atkin, Gordon, Hughes, Folsom-Kent-Ono, and Boylan-Webb about the partition function to a larger class of partition functions. For this purpose we study a closely related …

Rigid Connections On The Projective Line With Elliptic Toral Singularities, Alisina Azhang

LSU Doctoral Dissertations

We generalize two studies of rigid $G$-connections on $\pp$ which have an irregular singularity at origin and a regular singularity at infinity with unipotent monodromy: one is the work of Kamgarpour-Sage which classifies rigid homogeneous Coxeter $G$-connections with slope $\frac{r}{h}$, where $h$ is the Coxeter number of $G$, and the other is the work of Chen, which proves the existence of rigid homogeneous elliptic regular $G$-connections with slope $\frac{1}{m}$, where $m$ is an elliptic number for $G$. In our work, similar to Chen, we look for rigid homogeneous elliptic regular $G$-connections, but we allow the slope to have a numerator …

Algorithms Related To Triangle Groups, Bao The Pham

LSU Doctoral Dissertations

Given a finite index subgroup of $\PSL_2(\Z)$, one can talk about the different properties of this subgroup. These properties have been studied extensively in an attempt to classify these subgroups. Tim Hsu created an algorithm to determine whether a subgroup is a congruence subgroup by using permutations \cite{hsu}. Lang, Lim, and Tan also created an algorithm to determine if a subgroup is a congruence subgroup by using Farey Symbols \cite{llt}. Sebbar classified torsion-free congruence subgroups of genus 0 \cite{sebbar}. Pauli and Cummins computed and tabulated all congruence subgroups of genus less than 24 \cite{ps}. However, there are still some problems …

An Equivalence Between Contact Gluing Maps In Sutured Floer Homology: A Conjecture Of Zarev, Charles Ryan Leigon

LSU Doctoral Dissertations

We show that the contact gluing map of Honda, Kazez, and Matic has a natural algebraic description in bordered sutured Floer homology. In particular, we establish Zarev's conjecture that his gluing map on sutured Floer homology is equivalent, in the appropriate sense, to the contact gluing map. This further solidifies the relationship between bordered Floer theory and contact geometry.

Noncommutative Tensor Triangular Geometry And Its Applications To Representation Theory, Kent Barton Vashaw

LSU Doctoral Dissertations

One of the cornerstones of the representation theory of Hopf algebras and finite tensor categories is the theory of support varieties. Balmer introduced tensor triangular geometry for symmetric monoidal triangulated categories, which united various support variety theories coming from disparate areas such as homotopy theory, algebraic geometry, and representation theory. In this thesis a noncommutative version will be introduced and developed. We show that this noncommutative analogue of Balmer's theory can be determined in many concrete situations via the theory of abstract support data, and can be used to classify thick tensor ideals. We prove an analogue of prime ideal …

On Properties Of Weil Sums Of Binomials, Liem P. Nguyen

LSU Doctoral Dissertations

This dissertation explores questions regarding the Weil sum of binomials, a finite field character sum originated from information theory. The Weil spectrum counts distinct values of the Weil sum through invertible elements in the finite field. The value of these sums and the size of the Weil spectrum are of particular interest, as they link problems in information theory, coding theory, and cryptography to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we prove the Vanishing Conjecture of Helleseth ($1971$) on the presence of zero values in the Weil spectrum and …

Applications Of Nonstandard Analysis In Probability And Measure Theory, Irfan Alam

LSU Doctoral Dissertations

This dissertation broadly deals with two areas of probability theory and investigates how methods from nonstandard analysis may provide new perspectives in these topics. In particular, we use nonstandard analysis to prove new results in the topics of limiting spherical integrals and of exchangeability.

In the former area, our methods allow us to represent finite dimensional Gaussian measures in terms of marginals of measures on hyperfinite-dimensional spheres in a certain strong sense, thus generalizing some previously known results on Gaussian Radon transforms as limits of spherical integrals. This first area has roots in the kinetic theory of gases, which is …

Dna Curve Classification With Unsupervised Learning, Ben Thomas

Honors Theses

No abstract provided.

Connectivity Of Matroids And Polymatroids, Zachary R. Gershkoff

LSU Doctoral Dissertations

This dissertation is a collection of work on matroid and polymatroid connectivity. Connectivity is a useful property of matroids that allows a matroid to be decomposed naturally into its connected components, which are like blocks in a graph. The Cunningham-Edmonds tree decomposition further gives a way to decompose matroids into 3-connected minors. Much of the research below concerns alternate senses in which matroids and polymatroids can be connected. After a brief introduction to matroid theory in Chapter 1, the main results of this dissertation are given in Chapters 2 and 3. Tutte proved that, for an element e of a …

Knots And Links In Overtwisted Contact Manifolds, Rima Chatterjee

LSU Doctoral Dissertations

Suppose $(\M,\xi)$ be an overtwisted contact 3-manifold. We prove that any Legendrian and transverse link in $(\M,\xi)$ having overtwisted complement can be coarsely classified by their classical invariants. Next, we defined an invariant called the support genus for transverse links and extended the definition of support genus of Legendrian knots to Legendrian links and prove that any coarse equivalence class of Legendrian and transverse loose links has support genus zero. Further, we show that the converse is not true by explicitly constructing an example. We also find a relationship between the support genus of the transverse link and its Legendrian …

Stochastic Navier-Stokes Equations With Markov Switching, Po-Han Hsu

LSU Doctoral Dissertations

This dissertation is devoted to the study of three-dimensional (regularized) stochastic Navier-Stokes equations with Markov switching. A Markov chain is introduced into the noise term to capture the transitions from laminar to turbulent flow, and vice versa. The existence of the weak solution (in the sense of stochastic analysis) is shown by studying the martingale problem posed by it. This together with the pathwise uniqueness yields existence of the unique strong solution (in the sense of stochastic analysis). The existence and uniqueness of a stationary measure is established when the noise terms are additive and autonomous. Certain exit time estimates …

Nonlinear Potential Analysis On Sobolev Multiplier Spaces, Keng Hao Ooi

LSU Doctoral Dissertations

We characterize preduals and Kothe duals to a class of Sobolev multiplier type spaces. Our results fit in well with the modern theory of function spaces of harmonic analysis and are also applicable to nonlinear partial differential equations. As a maneuver, we make use of several tools from nonlinear potential theory, weighted norm inequalities, and the theory of Banach function spaces to obtain our results. After characterizing the preduals, we establish a capacitary strong type inequality which resolves a special case of a conjecture by David R. Adams. As a consequence, we obtain several equivalent norms for Choquet integrals associated …

Towards A General Framework For Practical Quantum Network Protocols, Sumeet Khatri

LSU Doctoral Dissertations

The quantum internet is one of the frontiers of quantum information science. It will revolutionize the way we communicate and do other tasks, and it will allow for tasks that are not possible using the current, classical internet. The backbone of a quantum internet is entanglement distributed globally in order to allow for such novel applications to be performed over long distances. Experimental progress is currently being made to realize quantum networks on a small scale, but much theoretical work is still needed in order to understand how best to distribute entanglement and to guide the realization of large-scale quantum …

A Conjecture On The Irregularity Function For Local Geometric Langlands Parameters And The Formal Frenkel-Gross Connection, Andrew Alaniz

LSU Doctoral Dissertations

For a simple complex algebraic group $G$, M. Kamgarpour and D. Sage have shown that the adjoint irregularity of an irregular singular flat $G$-bundle on the formal punctured disc is bounded from below by the rank of $G$, moreover the rank is realized by the formal Frenkel-Gross connection. This is a geometric analog of a conjecture of Gross and Reeder on the swan conductor of arithmetic local Langlands parameters. In this work, we explore an interesting combinatorial problem which arises when trying to consider the minimal value of the irregularity function with respect to an arbitrary representation of $G$.