## Mathematics Department

Undergraduate Research Symposium

Fall 2016

## December 14, 3 to 5:15 pm in LCB 323

**3-3:15 Noble Williamson**

Mentor: Gordan Savin

Hasse Minkowski Theorem for Ternary Quadratic Forms (Intro to Research)

**3:15-3:30 Kira Parker**

Mentor: Braxton Osting

Ranking Methods in Competitive Climbing (Intro to Research)

**3:30-3:45 Rebecca Hardenbrook**

Mentor: Jon Chaika

Interval Exchange Transformations (NSF funded)

**3:45-4:00 Max Carlson**

Mentors: Christel Hohenegger and Braxton Osting

Introducing Surface Tension to the Free Surface Sloshing Model

**4:00-4:15 Dietrich Geisler**

Mentor: Aaron Bertram

Construction of the Simple Lie Algebras from Associated Dynkin Diagrams

**4:15-4:30 Michael Zhao**

Mentor: Gordan Savin

Hermitian Forms and Orders of Quaternion Algebras

**4:30-4:45 Peter Harpending**

Mentor: Elena Cherkaev

The general solution of fractional-order linear homogeneous ordinary differential equations

**4:45-5:00 Caleb Webb**

Mentor: Maxence Cassier

A classical model for infinite, periodic, nonreciprocal media with dissipative elements

**5:00-5:15 Jake Madrid**

Mentor: Sean Lawley

The Effects and Implications of Rapid Rebinding in Biochemical Reactions

## Abstracts

**Noble Williamson**

Mentor: Gordan Savin

Hasse Minkowski Theorem for Ternary Quadratic Forms

Mentor: Gordan Savin

Hasse Minkowski Theorem for Ternary Quadratic Forms

A major facet of the study of quadratic forms is establishing equivalence classes among such forms. In local fields such as $\mathbb{R}$ or $\mathbb{Q_p}$, this problem tends to be fairly simple because there exist complete sets of invariants in those fields that are fairly easy to check. However, over global fields like $\mathbb{Q}$ this is often much more difficult. The Hasse-Minkowski theorem is a powerful tool that draws a connection between equivalence classes in local fields with equivalence classes in global fields. In this presentation, we will focus on the particularly interesting case of equivalence classes of ternary forms.

**Kira Parker**

Mentor: Braxton Osting

Ranking Methods in Competitive Climbing

Mentor: Braxton Osting

Ranking Methods in Competitive Climbing

Ranking and rating methods are used in all aspects of life, from Google searches to sports tournaments. Because all ranking methods necessarily have advantages and disadvantages, USA Climbing, the organizer of national climbing competitions in US, has changed their ranking method three times in the past seven years. The current method marked a drastic step away from the other two in that it failed to meet the Independence of Irrelevant Alternatives criterion and was neigh impossible for spectators to calculate on their own. We will examine this method, comparing it to older USA Climbing methods as well as other methods from literature, and determine if its use is reasonable or not.

**Rebecca Hardenbrook**

Mentor: Jon Chaika

Interval Exchange Transformations (NSF funded)

Mentor: Jon Chaika

Interval Exchange Transformations (NSF funded)

In the study of interval exchange transformations (IETs), particular interest is found in rank two IETs, mainly minimal rank two IETs. Boshernitzan proves in his paper on rank two IETs that all minimal rank 2 IETs are uniquely ergodic, a property that is important in the study of topological dynamical systems. This talk will give a quick introduction to how one can develop an algorithm to find aperiodicity of an IET and discuss interest in expanding this algorithm to find minimality.

**Max Carlson**

Mentors: Christel Hohenegger and Braxton Osting

Introducing Surface Tension to the Free Surface Sloshing Model

Mentors: Christel Hohenegger and Braxton Osting

Introducing Surface Tension to the Free Surface Sloshing Model

Building on the numerical methods for the simple free surface sloshing model we developed last semester, we now introduce surface tension to the model. Using the finite element method with two function spaces of different dimensions, this model becomes a coupled system of equations corresponding to a steklov eigenvalue problem. Extending matlab code provided by Professor Hari Sundar, we have developed a numerical method to compute the eigenvalues and eigenfunctions corresponding to a sloshing free surface under the effects of surface tension for a variety of container geometries.

**Dietrich Geisler**

Mentor: Aaron Bertram

Construction of the Simple Lie Algebras from Associated Dynkin Diagrams

Mentor: Aaron Bertram

Construction of the Simple Lie Algebras from Associated Dynkin Diagrams

Lie Groups provide a structure to study differential equations, and so have a variety of important applications in modern mathematics and theoretical physics. Classifying the set of Lie Groups is therefore an inherintly interesting and useful question. This paper seeks to classify these groups by classifying the Lie Algebras; algebras which can be associated with a given Lie Group. Each simple Lie Algebra comes equipped with a root space and associated Dynkin Diagram; these diagrams have properties that can be used to construct a countable set of such diagrams. We will show that this set consists of the root spaces associated with the classical Lie Algebras and the 5 exceptional Lie Algebras.

**Michael Zhao**

Mentor: Gordan Savin

Hermitian Forms and Orders of Quaternion Algebras

Mentor: Gordan Savin

Hermitian Forms and Orders of Quaternion Algebras

We discuss progress on development of a quaternionic analogue of the classical correspondence between quadratic forms and lattices in the plane. I'll start with an account of this classical correspondence, and then discuss progress on the hermitian form to order direction, which involves a discriminant relation.

**Peter Harpending**

Mentor: Elena Cherkaev

The general solution of fractional-order linear homogeneous ordinary differential equations

Mentor: Elena Cherkaev

The general solution of fractional-order linear homogeneous ordinary differential equations

The general solution of an integer-order linear homogeneous or- dinary differential equation is span {e^{lambda_i t}} where {lambda_i} is the set of eigenvalues of the associated linear transformation. We present a sketch of a proof, using integral transforms, that the general solu tion to a fractional-order linear homogeneous differential equation is also span {e^{lambda_i t}}, and present a procedure for finding the eigenvalues of the transformation.

**Caleb Webb**

Mentor: Maxence Cassier

A classical model for infinite, periodic, nonreciprocal media with dissapative elements

Mentor: Maxence Cassier

A classical model for infinite, periodic, nonreciprocal media with dissapative elements

In previous work, the spectral properties of two component composite systems of finitely many degrees of freedom have been analyzed. Specifically, gyroscopic media consisting of high-loss and lossless components have been discussed. As a next step in modeling real composite materials with gyroscopic elements, we construct a general framework for analyzing periodic systems. Using a Lagrangian formalism, we derive a convenient model for studying infinite, periodic, nonreciprocal systems with arbitrarily man degrees of freedom in one (physical) dimension. We apply this model to analyze the band structure of both a lossless, single component gyroscopic system and a two component, gyroscopic, system with dissapative elements.

**Jake Madrid**

Mentor: Sean Lawley

The Effects and Implications of Rapid Rebinding in Biochemical Reactions

Mentor: Sean Lawley

The Effects and Implications of Rapid Rebinding in Biochemical Reactions

Enzyme kinetics is the study of biochemical reactions in which an enzyme modifies a substrate in some way. In this study, we will consider reactions in which a single enzyme modifies a substrate through a series of repeated reactions. The behavior of such reactions depends on whether the enzyme acts processively or distributively. Processive enzymes only need to bind once to a substrate in order to carry out a series of modifications. By contrast, distributive enzymes release the substrate after each modification. It has been shown that distributive enzymes can act processively through a process called rapid rebinding. These types of reactions can be studied through biological experiment, spatial stochastic simulation or, under certain assumptions, deterministic ODE models. We will focus on the stochastic simulations and ODE models, and under what conditions these two models agree.