Theses

Since Spring 2013, our department has offered a Thesis option to our Masters students. A thesis, unlike a project, consists on original research done by the student. Below are the theses our students have produced.

Recent Theses

Student: Summer Al-Hamdani
Advisor: Khang Tran
Graduation Date: May 2021
Thesis Title: Zero Distribution of Binomial Combinations of Chebyshev Polynomials of the Second Kind.
Abstract:  We consider the sequence {P_m(z)}_{m=0}^\infty, which is a binomial combination of the well-known Chebyshev polynomials of the second kind; they have all real zeros on the interval (-1,1). We prove that there exists a constant C (independent of m) such that the number of zeros of P_m(z) outside of the interval (-1,1) is at most C for all m in N.

Student: Jagdeeep Basi
Advisor: Carmen Caprau
Graduation Date: May 2021
Thesis Title: Quandle Coloring Quivers of Torus Knots.
Abstract: Quandles provide an algebraic perspective to the general goal of knot theory of classifying and distinguishing mathematical knots.  We study patterns of dihedral quandles under the lens of quandle colorings and quandle coloring quivers.   This work classifies the full quandle coloring quivers of (p,2)-torus knots as a base case for the full quandle coloring quivers of general (p,q)-torus knots.

Student: Samuel Cleofas
Advisor: Oscar Vega
Graduation Date: May 2021
Thesis Title: Hyperplanes Arrangements over Finite Fields.
Abstract: This work studies blocking sets of the complement of a hyperplane arrangement in a vector space over a finite field in a way that it generalizes the work of Settepanella on hyperplane arrangements in a finite space.

Student: Maria Diaz
Advisor: Oscar Vega
Graduation Date: May 2021
Thesis Title: The Characteristic Polynomial of a Spread in F_q^4.
Abstract: We introduce the concept of the characteristic polynomial of a spread in F_q^4 and then see what the degree of these polynomials are for a variety of spreads of F_q^4, mostly for small values of q. We focus specially on regular and Andre spreads.

Student: Erick Gonzalez
Advisor: Oscar Vega
Graduation Date: May 2021
Thesis Title: The Topological Symmetry Group of Graphs in F_Delta(K_7).
Abstract: We study the embeddings of C_{13} and H_8, and determine which automorphism groups of these graphs can be induced by groups of homeomorphisms of R^3.

Student: Bradley Scott
Advisor: Carmen Caprau
Graduation Date: May 2021
Thesis Title: Minimal Generating Sets of Oriented Reidemeister-Type Moves for Knot and Spatial Trivalent Graph Diagrams.
Abstract: It has recently been shown by Polyak that all oriented versions of Reidemeister moves for knot diagrams can be generated by a set of just 4 oriented Reidemeister moves, and no fewer than 4 moves generate them all. We expand upon Polyak's work by proving the existence of an additional 11 minimal generating sets of oriented Reidemeister moves for oriented knot diagrams, and we prove that these 12 sets represent all possible minimal generating sets. Then we consider the Reidemeister-type moves that relate oriented spatial trivalent graph diagrams and prove the minimality of a generating set of 10 such moves.

Past Theses

Student: Erica Sawyer
Advisor: Mario Banuelos
Graduation Date: December 2020
Thesis Title: Deep Learning Methods for Detecting Structural Variants in Related Individuals
Abstract: We implement neural networks to predict Structural Variants (SVs).  We discuss a model, which incorporates the observed genomic information of two parents and an offspring to predict locations of SVs in the genome of the child.  We discuss a generalization of this model and we investigate the performance of these models under different neural network architectures.

 

Student: Elizabeth Compton
Advisor: Oscar Vega
Graduation Date: December 2019
Thesis Title: The Power Graph of Split Metacyclic Groups
Abstract: Split metacyclic groups generalize the family of dihedral groups in a natural way. The power graph of a group is a refinement of the group lattice that focuses only on cyclic subgroups.  This thesis characterizes the power graph of split metacyclic groups using exclusively group-theoretical tools.

Student: John Jimenez
Advisor: Marat Markin
Graduation Date: December 2019
Thesis Title: On the Chaoticity of Rolewicz-Type Operators on Function Spaces
Abstract: The chaoticity of Rolewicz-type linear operators on certain function spaces is proved and their spectral structure is revealed.  

 

Student: Edward Sichel
Advisor: Marat Markin
Graduation Date: December 2019
Thesis Title: On Expansive Mappings and Non-Hypercyclicity
Abstract: We take a close look at the nature of expansive mappings on certain metric spaces (compact, totally bounded, and bounded), provide a finer classification for such mappings, and use them to characterize boundedness. We also furnish a simple straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator in a complex Hilbert space as well as of a certain collection of its exponentials.

 

Student: Jennifer Elder
Advisor: Oscar Vega
Graduation Date: May 2016
Thesis Title: Generalizing the Futurama theorem
Abstract: Every permutation x can be written as a product of cycles that have not been used in the construction of x, as long as two new elements are incorporated in the transpositions. This results generalizes Keeler's Theorem A.K.A. The Futurama Theorem.

Student: Hillary Bese
Advisor: Oscar Vega
Graduation Date: May 2015
Thesis Title: The Well-covered Dimension of the Adjacency Graph of Generalized Quadrangles
Abstract: The well-covered space of the adjacency graph of the classical generalized quadrangle W_q is trivial, for every prime power q.

Student: David Heywood
Advisor: Tamas Forgacs
Graduation Date: May 2015
Thesis Title: Multiplier Sequences of the Second Kind
Abstract: An investigation of sequences of real numbers, which - when viewed as standard diagonal linear operators on polynomials - preserve the real rootedness of polynomials with only real zeroes of the same sign.

Student: Megan Kuneli
Advisor: Oscar Vega
Graduation Date: May 2014
Thesis Title: Spreads and Parallelisms
Abstract: Study of partitions of the lines in a projective plane into lines that partition the points in such plane.

Student: Katherine Urabe
Advisor: Carmen Caprau
Graduation Date: May 2014
Thesis Title: The Dubrovnik Polynomial of Rational Knots
Abstract: Finding a closed form expression for the Dubrovnik polynomial of a rational knot or link diagram in terms of the entries of its associated vector. The resulting closed form allows a Mathematica program which efficiently computes the Dubrovnik polynomial of rational knots and links.

Student: Karen Willis
Advisor: Oscar Vega
Graduation Date: May 2014
Thesis Title: Blocking Polygons in Finite Projective Planes
Abstract: Study of configurations of points in a finite projective plane that do not allow the existence of polygons that are disjoint from such set.