Catherine Sawyer, Tyler Jarvis, Mathematics Introduction The purpose of this project was to develop a method by which sets of multivariate polynomials with known roots could be rapidly generated. Generating such polynomials would be of great use in testing root finding algorithms, yet remains an open problem Methodology Polynomial interpolation is a well-studied problem in the […]

## Time evolution study of viscous conservation laws

David Todd and Dr. Blake Barker, Department of Mathematics Introduction The goal of this project was to create code for a generic numerical solver utilizing a finite difference scheme. Prior to the creation of this code, those wishing to study the evolution of traveling waves in STABLAB, a MATLAB based program specifically designed for studying […]

## Weak Synchronization in Excitatory – Inhibitory Neuronal Networks

Eliason, Joel Weak Synchronization in Excitatory-Inhibitory Neuronal Networks Faculty Mentor: Benjamin Webb, Mathematics Introduction One ubiquitously observed dynamic phenomenon in the nervous system is that of weak synchronization or clustering, a behavior in which a large group of neurons in a population will all fire synchronously and then fall out of synchronization. When isolated to […]

## Spectral Graph Theory for Weighted Digraphs

Alexander Zaitzeff and Faculty Mentor: Jeffrey Humpherys, Mathematics For digraphs weighted and unweighted, one important application is ranking: Given a directed graph, whether it be the Internet or a social network, which node (representing a web page or a person) is the most important? There are many different methods to find answer this question. A […]

## The Fourier Coefficients of Modular Forms

Kyle Pratt and Dr. Paul Jenkins, Department of Mathematics Modular forms are complex analytic functions with remarkable properties. Modular forms possess interesting and surprising connections to many different branches of mathematics. For example, it is well-known that Andrew Wiles’ proof of Fermat’s Last Theorem, a conjecture that had been unresolved for more than three centuries, […]

## Zeros of Poincare Series of Level 2

Andrew Haddock, Paul Jenkins, Mathematics Introduction Poincaré series are a certain type of modular form. Modular forms are complex-valued functions that satisfy certain symmetry properties. There are many different types of modular forms, and one way to classify modular forms is by their level, such as 1, 2, 3, etc. They are of much interest […]

## Transpose Symmetry Groups of Noninvertible Polynomials

Nathan Cordner and Dr. Tyler Jarvis, Department of Mathematics Introduction Mirror symmetry is an area of mathematical research that stems from theoretical physics, particularly from string theory. Solutions of problems in mirror symmetry yield not only interesting mathematical results, but also have important theoretical implications for high energy particle physics. In Landau-Ginzburg mirror symmetry, there […]

## Defining the Transpose Group in Landau-Ginzburg Mirror Symmetry

Lisa Bendall, Dr. Tyler Jarvis, Math Department Introduction. Mirror symmetry is a phenomenon first observed in theoretical physics, which has garnered interest among mathematicians. The Landau-Ginzburg mirror symmetry conjecture proposes two algebraic structures which are isomorphic, or in some sense “mirror” each other. These structures are built from polynomials and corresponding symmetry groups. Much research […]

## QUASIGROUP STRUCTURES ARISING FROM PARTIAL LATIN SQUARES

William Cocke and Dr. Rodney Forcade, Mathematics Department Latin squares represent one of the oldest elements of modern algebra, despite being over- shadowed by the structurally superior subclass of nite groups. Every nite group generates a Latin square, an n by n grid wherein n distinct element appear exactly once in each row and column. […]

## Up Congruences of Modular Functions Modulo Powers of Primes

Michael Griffin and Dr. Paul Jenkins, Department of Mathematics Modular forms are constructs of complex analysis that possess many intricate connections to widely-separated branches of mathematics. In the most well-known application of these functions, Andrew Wiles established a connection between the Fourier coefficients of modular forms and elliptic curves–objects of analytic geometry–in order to prove […]

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