# REU Symposium

These meetings are held at the end of semester and showcase the research that is being done by undergraduates in our department.

REU Symposium Archive

## Mathematics Department

Undergraduate Research Symposium Spring 2020

Jackson Turner

Mentor: Elena Cherkaev and Dong Wang

Title: Computation of Eigenfunctions on Various Domains

Kayla Stewart

Mentor: Ken Golden

Title: Closing the Gap: How Sea Ice Algae Live or Die within Gap Layers

Winston Stucki

Mentor: Karl Schwede and Marcus Robinson

Title: Hibi Rings

Sriram Gopalakrishnan

Mentor: Gil Moss

Title: Iwahori-Hecke Operators for Automorphic Forms on Definite Unitary Groups

Yuhui Yao

Mentor: Christopher Hacon

Title: MMP Overview

James Eckstein

Mentor: Yeketerina Epshteyn

Title: Numerical Algorithms for the Automatic Processing of Image Data, Specifically Grain Growth

Nancy Lyu

Mentor: Jingyi Zhu

Title: Building Optimal Portfolios without Relying on the Inverse of the Covariance Matrix for the Returns

## Mathematics Department

Undergraduate Research Symposium Summer 2020

Nancy Lyu

Mentor: Jingyi Zhu

Title: Building Optimal Portfolios without Relying on the Inverse of the Covariance Matrix for the Returns

Sriram Gopalakrishnan

Mentor: Gil Moss

Title: Iwahori-Hecke Operators for Automorphic Forms on Definite Unitary Groups

Mathematics Department

Undergraduate Research Symposium Spring 2019

Tuesday, April 30 9:45-11:30 in LCB 121

9:45-10:00 Wei Yao

Mentor: Karl Schwede

Polynomial evaluation over finite and rational fields through matrices in Macaulay2

10:00 - 10:15 Thomas White

Mentor: Christopher Janjigian

Simulating the inhomogeneous corner growth model

10:15-10:30 Camille Humphries

Mentor: Yekaterina Epshteyn, Qing Xia

Fast Numerical Algorithms for Models with Nonlinear Diffusion

10:30-10:45 Charlotte Blake

Mentor: Yekaterina Epshteyn

Efficient Numerical Algorithms for Automatically Processing Data with Application
to Materials Science

10:45-11:00 Gabrielle Legaspi

Mentor: Yekaterina Epshteyn

Coarsening Models

11:00-11:15 Cassie Buhler

Mentor:Fred Adler

Mathematical Modeling of Adaptive Therapy in Prostate Cancer

11:15-11:30 Justin Baker

Mentor: Elena Cherkaev

Optimal Transportation Networks

Abstracts

Wei Yao

Mentor: Karl Schwede

Polynomial evaluation over finite and rational fields through matrices in Macaulay2

Questions in algebraic geometry such as the birationality of a map between varieties
and the codimension of the singular locus of a variety can be answered through relatively
straightforward computations of ranks of certain matrices. Macaulay2 is a software
widely used by algebraic geometers and algebraists to perform such computations among
many others. However, over a polynomial ring with multiple variables and undetermined
degree, these computations turn out to be slow. We implement a particular step in
this computation via selecting minors in order to speed up the process.

Thomas White

Mentor: Christopher Janjigian

Simulating the inhomogeneous corner growth model

The goal of this research was to simulate the inhomogeneous corner growth model with
exponential weights. This is an important model in the Kardar Parisi Zhang universality
class from physics. In order to do this, I simulated geodesics to map interactions
in the discrete plane and normalized passage times, which have known behavior. As
a next step, I am investigating Tracy-widom fluctuations at the interface between
the concave and linear parts of the shape function. The fluctuations require further
investigation, and other variations to the corner growth model are still open to study.

Camille Humphries

Mentor: Yekaterina Epshteyn, Qing Xia

Fast Numerical Algorithms for Models with Nonlinear Diffusion

Diffusion equations, and, in particular, nonlinear diffusion models play an important
role in many areas of science and engineering. In general, solutions to such models
cannot be obtained analytically. Hence, there is a need for accurate and efficient
algorithms that can deliver approximate solutions to these models. We consider Difference
Potentials Method for numerical solution of various diffusion models. We will investigate
the accuracy, stability, and efficiency of the developed algorithms, and discuss related
future work.

Charlotte Blake

Mentor: Yekaterina Epshteyn

Efficient Numerical Algorithms for Automatically Processing Data with Application
to Materials Science

Our research focused on developing robust numerical algorithms that take images of
crystal grains as the input and automatically output relevant data, including information
about grain area, perimeter, and number of neighbors. In this presentation, we will
review the work accomplished in the Fall, as well as discuss the corrections and extensions
made to the algorithms this semester. We will also present the obstacles that appeared
as a part of the design of such algorithms and how they were resolved. Special focus
will be given to the aspects of the algorithms related to the computational geometry
questions of corner identification and polygon approximation of boundaries. Finally,
we will examine the efficiency of algorithms and possible improvements.

Gabrielle Legaspi

Mentor: Yekaterina Epshteyn

Coarsening Models

Cellular networks exhibit behavior on many different length and time scales and are
generally metastable. Among numerous examples of cellular networks are polycrystalline
materials/microstructures. Most technologically useful materials arise as polycrystalline
mi- crostructures, composed of a myriad of small crystallites or grains, the cells,
separated by interfaces or grain boundaries. Coarsening results from the growth and
rearrangement of the crystallites, which may be viewed as the anisotropic evolution
of a large metastable system. Our project will investigate and compare different coarsening
models. We will employ numerical simulations, mathematical analysis and data analytics
to study and improve these models.

Cassie Buhler

Mentor:Fred Adler

Mathematical Modeling of Adaptive Therapy in Prostate Cancer

Prostate cancer is a hormonally driven cancer. These cancer cells need androgen, a
class of male sex hormone, to survive and grow. Standard treatments for prostate cancer
target androgens. This type of therapy is denoted as hormone therapy. Yet, for patients
with recurring cancer, hormone therapy is not effective because cancer cells become
testosterone independent over time, and consequently, the cells gain resistance and
do not respond to therapy. There are studies that suggest therapy administered in
intervals, as opposed to continuous treatments, could prevent this occurrence. We
have analyzed mathematical models of dynamic biological systems for prostate cancer
progression in order to explore the effect of treatment timing to find the most effective
therapy in delaying the inevitable emergence of testosterone resistant cells.

Justin Baker

Mentor: Elena Cherkaev

Optimal Transportation Networks

Models of swarming behavior aid in disaster planning, direct the actions of warehouse
robots, and can map the foraging characteristics of insects. The Monge-Kantorovich
formulation of the optimal transportation problem models the best direct route for
individual agents in a swarm. The presented work investigates optimal transportation
and duality in a Monge-Kantorovich formulation. We reduce the Monge-Kantorovich formulation
to the linear programming problem which can be efficiently solved numerically. However
this formulation is not applicable in the case where the domain of travel is restricted,
so that agents must travel along a particular network of paths. We extend the Monge-Kantorovich
formulation to a network of paths which differs from transfer over a direct route.
We show that this extended formulation can also be reduced to the linear programming
problem. Finally, we investigate various applications of both the direct and network
transfer using numerical simulations.

## Mathematics Department

Undergraduate Research Symposium Summer 2019

Boyana Martinova

Mentor: Karl Schwede

Title: Applications of Linear Algebra for Varieties over Algebraic Fields in Macaulay2

Camille Humphries

Mentors: Yekaterina Epshteyn and Qing Xia

Title: Marble Degradation in 2D

Eric Brown

Mentor: Fernando Guevara Vasquez

Titles: Rational interpolation for the Helmholtz equation

Yousef Alamri

Mentor: Alla Borisyuk

Title: Quantifying the Glutamate Effect on Calcium Responses in Astrocytes

Yuhui Yao

Mentor: Karl Schwede

Title: Polynomials in Macauly2 Report

## Mathematics Department

Undergraduate Research Symposium Fall 2019

Emma Coates

Mentor: Christel Hohenegger

Title: A Numerical Study on Radially Symmetric Containers Optimizing Sloshing Frequencies

Jackson Turner

Mentors: Elena Cherkaev and Dong Wang

Title: Computation of Eigenfunctions on Various Domains

Camille Humphries

Mentors: Yekaterina Epshteyn and Qing Xia

Title: Marble Degradation in 2D

Boyana Martinova

Mentor: Karl Schwede

Title: Applications of Linear Algebra for Varieties over Algebraic Fields in Macauly2

Taylor Yates

Mentor: Sean Lawley

Title: Extreme Statistics of Diffusion

Kayla Stewart

Mentor: Ken Golden

Title Ding! Exploring the "Dinner Bell" for Algal Blooms

Mathematics Department

Undergraduate Research Symposium Spring 2018

Thursday, April 26 2:45 to 4:30 pm in LCB 215

2:45-3:00 Jack Garzella

Mentor: Fernando Guevara-Vasquez

Using Spring and Mass Networks to Model an Elastic Body

2:00- 3:15 Han Le

Mentor: Tom Alberts

Spiked sample covariance matrices

3:15-3:30 Hannah Choi

Mentor: Sean Lawley

Modeling predator-prey dynamics using mean first passage time analysis

3:30-3:45 Katelyn Queen

Mentor: Fred Adler, Jason Griffiths

Dynamic Modelling of Subclones in Patients with Late-Stage Breast and Ovarian Cancers

3:45-4:00 Hannah Waddel

Mentor: Fred Adler

The Community Ecology of the Music Canon

4:00-4:15 Charlotte Blake

Mentor: Andrej Cherkaev

Approximation of Bistable Spring Chain Dynamics

4:15-4:30 Delaney Mosier

Mentor: Ken Golden

Poisson Equation Model for Sea Ice Concentration Fields in a Changing Climate

Abstracts

Charlotte Blake

Mentor: Andrej Cherkaev

Approximation of Bistable Spring Chain Dynamics

In this presentation, we address the problem of approximating the dynamics of a chain
of bistable springs with one monotonic nonlinear spring. We assume piecewise linearity
of the bistable springs, and identical springs in the chain separated by small masses.
From there, we investigate the possible modes of behavior of the system and determine
the energy bounds for each mode. We then determine the asymptotic approximations and
examine the approximation with a two-spring system. Throughout the presentation, we
reference numerical simulations of chains.

Hannah Choi

Mentor: Sean Lawley

Modeling predator-prey dynamics using mean first passage time analysis

Animal movement structures interactions between individuals, the environment and other
species, and therefore determines resource consumption, reproductive output, place
of shelter, and survival of an individual. Most mathematical models seeking to understand
animal movement focus on the distribution of animals and resources within a landscape.
We seek to better understand movement through modeling the behavior of a single animal.
We assume that animal pathways exhibit diffusive behavior and calculate the mean first
passage time (MFPT) for an animal, a predator, to find a prey under various assumptions
and the so-called intermittent search strategy.

Jack Garzella

Mentor: Fernando Guevara-Vasquez

Using Spring and Mass Networks to Model an Elastic Body

Given a network of masses and springs, we can easily find out how those springs will
interact when nodes are displaced. However, given just the interactions of nodes on
the boundary of the network, can we find out the values of all the spring constants?
It turns out, in certain situations we can, using an iterative algorithm. Moreover,
we can small recoverable networks to approximate bigger networks, and even a continuous
elastic body.

Han Le

Mentor: Tom Alberts

Spiked sample covariance matrices

I will review the statistical theory of sample covariance matrices for random vectors,
focusing on the case where the size of the sample is comparable to the dimension of
the vectors. This is the situation increasingly encountered in big data. Spiked models
are a way of analyzing if one component of the vector has a variance that is much
larger than the others in this regime.

Delaney Mosier

Mentor: Ken Golden

Poisson Equation Model for Sea Ice Concentration Fields in a Changing Climate

The Arctic and Antarctic sea ice packs are critical regions for examining the effect
and implications of a rising global temperature. Melting polar sea ice produces ecological
dilemmas, an increased absorption of solar radiation, and rising sea levels that threaten
coastlines. Due to their significance to the evolution of our climate system, Earth’s
sea ice packs must be accurately represented in global climate models. Our study aims
to enhance the representation of sea ice by examining the evolution and behavior of
the sea ice concentration field as a function of both time and space. We will develop
a mathematical model for the concentration field based on the Poisson equation. We
will begin by exploring efficient numerical methods for solving the forward problem
in one and two spatial dimensions. Our investigation of novel low-order models of
the concentration field and its evolution will give us tools to mathematically analyze
the changes in sea ice concentration under global warming. Our study will also begin
to encompass the inverse problem for the Poisson equation, utilizing satellite data
on the sea ice concentration field from 1979 to present. Given the discrete form of
the concentration field on a two-dimensional lattice, we will invert for the field
(the right hand side) representing the distribution of sources and sinks of ice concentration.
In the analogous electrostatic problem, data on the electric potential is inverted
to identify regions of positive and negative electrical charge. Our source-sink field
will similarly represent sources and sinks in ice concentration - regions in which
ice is primarily either freezing and converging or melting and diverging, respectively.
We will explore the evolving structure of this field as the global climate has warmed
and analyze the patterns we discover in shifting “hot” and “cold” spots, as well as
in overall or homogenized behavior.

Katelyn Queen

Mentor: Fred Adler, Jason Griffiths

Dynamic Modelling of Subclones in Patients with Late-Stage Breast and Ovarian Cancers

During the course of chemotherapy treatment in cancer patients, genetically related
groups of cells in the tumors called subclones can become resistant to treatments.
This study uses mathematical and statistical models to understand this process, and
eventually, we hope, to find ways to delay or even reverse the development of the
chemo-resistant tumors associated with breast and ovarian cancers. Our data come from
metastatic tumor cells from the pleura of patients with breast and ovarian cancer
collected before, during, and after different types of treatments. These cells are
then genetically sequenced with a method called single-cell RNA seq that gives the
genetic composition in detail. Our current choice of mathematical tool is ordinary
differential equations (ODE) models, which we are using to watch phenotypic changes
during cancer growth which lead to resistance of tumors. These models will allow us
to derive, simulate and analyze a dynamic model of subclone changes during the transformation
of a chemo-sensitive tumor to a chemo-resistant tumor, which in turn would create
the opportunity to then block or reverse this transformation in patients with late-stage
breast and ovarian cancers. As an added complexity, the progression of cancers can
be facilitated by non-cancerous cells like white blood cells. We will extend our data
analysis techniques to determine how cancer subclones interact with other cell types,
and how these interactions might be used to delay the transition to a chemo-resistant
state.

Hannah Waddel

Mentor: Fred Adler

The Community Ecology of the Music Canon

Symphony orchestras today have access to a musical canon stretching back at least
four hundred years and play a critical role in its establishment. Full symphony orchestras
are expensive to operate and have limited performance time, which only allows a finite
number of songs and composers to be considered canonical, and more music exists than
time to perform it. Community ecology describes the structure and interactions of
species competing for limited resources. The structure of the musical canon and its
dissemination lends itself to analysis using ecological methods and models. Treating
composers or songs as species, we are using quantitative ecological methods to characterize
the ways that music interacts in the canon, with a particular focus on the ways that
composers and pieces enter and remain in the canon. The canon follows some common
ecological patterns, where a small number of species constitute the bulk of the biomass
in an ecosystem and most species are rare. In the canon, a few "warhorse" composers
dominate the repertoire while most new composers barely receive performance time.
Our project provides a cross-disciplinary analysis of the western canon of music using
mathematical and ecological methods. The efficacy of ecological models in describing
the behavior of the canon lent insights into what analogous processes occur in the
field of music composition that cause a composer to either fade into obscurity or
become enshrined through centuries of performance.

Mathematics Department

Undergraduate Research Symposium Summer 2018

Thursday, August 23, 2:15 to 3:45 pm, room JTB 120

2:15-2:30 Jack Garzella

Mentor: Fernando Guevara-Vasquez

Understanding Spring Netowrk Approximations for Continuous Elastic Bodies

2:30- 2:45 Adam Lee

Mentor: Daniel Zavitz, Alla Borisyuk

Relationship between the connectivity of directed networks with discrete dynamics and their attractors.

2:45-3:00 Dylan Johnson

Mentor: Karl Schwede, Daniel Smolkin, Marcus Robinson

Searching for Rings with Uniform Symbolic Topology Property

3:00-3:30 Faith Pearson and Dylan Soller

Mentor: Anna Romanova, Peter Trapa

Cracking Points of Finite Gelfand Pairs

3:30-3:45 Audrey Brown

Mentor: Alla Borisyuk

Intro to research: Analysis of Mice Olfactory Response Data

Abstracts

Audrey Brown

Mentor: Alla Borisyuk

Intro to research: Analysis of Mice Olfactory Response Data

In the olfactory system, the original neural response is produced when odors bond
to an olfactory receptor neuron (ORN) in the nose. The response then travels through
glomeruli, and mitral cells (in mammals). The original input to ORN’s is dependent
on the type of odor--some ORN types will respond more to one odor than others. The
similarity of ORN response was investigated in recent data from eight mice. A response
to each odor across ORN's was considered as a high-dimensional vector. Similarity
between each pair of odors was calculated using cosine similarity. The odors were
then grouped by their chemical class. By comparing the distributions and averages
of the odor similarity using histograms, box plots, and matrices, it was determined
that the general trend across mice is that similarity between odor responses is higher
within a chemically similar group than between chemical groups. It was also determined
by comparing odor response to overall activity that there is not a clear correlation
between activity level and odor similarity.

Jack Garzella

Mentor: Fernando Guevara-Vasquez

Understanding Spring Netowrk Approximations for Continuous Elastic Bodies

What is the best way to approximate a continuous elastic body? The differential equations
governing such systems are not well-known, or necessarily solved, even in the 2D case.
We study ways of trying to approximate an elastic body, and determine the Lamé parameters
of a substance based on only data gathered from the boundary of the object. We first
use discrete Spiring Networks, and conclude that this method doesn't work well. We
then use a Finite Element discretization, and conclude that this is a better approach.

Adam Lee

Mentor: Daniel Zavitz, Alla Borisyuk

Relationship between the connectivity of directed networks with discrete dynamics
and their attractors.

Inspired by neural activity, we examine simplified directed networks with discrete
dynamics. These simplified dynamics allow nodes to have an active, neutral, and refractory
state. After each discrete time step, the signal moves from active nodes to nodes
in the neutral state. Eventually, the activity forms a stable, cyclic pattern called
an attractor or dies. We examine the relationship between the connectivity of these
networks and the number and size of their attractors.

Dylan Johnson

Mentor: Karl Schwede, Daniel Smolkin, Marcus Robinson

Searching for Rings with Uniform Symbolic Topology Property

Using recent work from D. Smolkin and J. Carvajal-Rojas, we seek to identify new commutative,
Noetherian rings with Uniform Symbolic Topology Property, abbreviated USTP. First,
we show that, for k a field, the toric ring k[x,y] has USTP. Then, we consider k[w,x,y,z]/(wx-yz),
and make progress toward showing that this ring also has USTP. Both are toric rings
already known to have the property, but we provide an alternative proof using a different
method, one which we hope to extend to rings unknown to have USTP.

Faith Pearson and Dylan Soller

Mentor: Anna Romanova, Peter Trapa

Cracking Points of Finite Gelfand Pairs

This presentation examines the representation theory of finite Gelfand pairs, which
are algebro-combinatorial objects of interest in finite group theory and harmonic
analysis. In this presentation, we will construct our Gelfand pairs of interest, and
define the cracking point of a finite group. We will present a new smaller upper bound
for cracking points of certain groups, and also present our significant progress in
finding the cracking points of all symmetric groups.

Mathematics Department

Undergraduate Research Symposium Fall 2018

Monday, December 10, 12:15 to 1:45 pm in LCB 222

12:15-12:30 Justin Baker

Mentor: Elena Cherkaev

Designed Swarming behavior Using Optimal Transportation Networks

12:30-12:45 Charlotte Blake

Mentor: Yekaterina Epshteyn

Efficient Numerical Algorithms for Automatically Processing Data with Application
to Materials Science

12:45-1 Audrey Brown

Mentor: Alla Borisyuk

Analysis of Mice Olfactory Response Data

1-1:15 Dylan Johnson

Mentor: Karl Schwede, Daniel Smolkin, Marcus Robinson

Searching for Rings with USTP

1:15-1:30 Dylan Soller

Mentor: Anna Romanov

Finite Gelfand Pairs

1:30-1:45 Eric Allen

Mentor: Lajos Horvath

To be Stationary or to be Non-Stationary, GARCH simulations in RStudio

ABSTRACTS

Eric Allen

Mentor: Lajos Horvath

To be Stationary or to be Non-Stationary, GARCH simulations in RStudio

The GARCH (Generalized Autoregressive Conditional Heteroskedastic) models are used
to model stock volatility. Stock volatility is defined as how much and how frequently
a stock price changes in the stock market. We run GARCH simulations in order to analyze
a times series of data which represent these incremental changes. A time series is
a series of values taken at successive, equally- spaced times. The time series represents
a sequence of discrete-time data. I used packages, “rugarch” and “fgarch” in RStudio
to run stationary and non-stationary GARCH simulations. Multiple simulations were
conducted with fixed alpha, beta, and omega to produce plots of the time series and
density function. Simulations to estimate values for alpha, beta, and omega in order
to predict the accuracy of the “ugarchspec” function were also conducted.

Justin Baker

Mentor: Elena Cherkaev

Designed Swarming Behavior Using Optimal Transportation Networks

Different models of swarming behavior can be used to study, analyze, and optimize
the behavior of large populations, building evacuation by emergency response teams,
and model groups of robots, people and animals. The current project considers the
problem of designing the swarming behavior, and formulates this problem as an optimal
transportation problem. We formulate the optimal transportation problem as a discretized
linear programming problem. We use the dual problem to maximize efficiency of the
designed transportation network. Finally, we numerically compute the solution using
Python and develop a visualization of the network and solution for several hypothetical
models

Charlotte Blake

Mentor: Ekaterina Epshteyn

Efficient Numerical Algorithms for Automatically Processing Data with Application
to Materials Science

Our research focused on developing robust numerical algorithms that take images of
crystal grains as the input and automatically output relevant data, including information
about grain area, perimeter, and number of neighbors. In this presentation, we will
discuss the process of obtaining each type of data, including the difficulties along
the way. We will also present the obstacles that appeared as a part of the design
of such algorithms and how they were resolved. Special focus will be given to the
aspects of the algorithms related to the computational geometry questions of corner
identification and polygon approximation of boundaries.

Audrey Brown

Mentor: Alla Borisyuk

Analysis of Mice Olfactory Response Data

1-1:15 Dylan Johnson

Mentor: Karl Schwede, Daniel Smolkin, Marcus Robinson

Searching for Rings with USTP

1:15-1:30 Dylan Soller

Mentor: Anna Romanov

Finite Gelfand Pairs

1:30-1:45 Eric Allen

Mentor: Lajos Horvath

To be Stationary or to be Non-Stationary, GARCH simulations in RStudio

ABSTRACTS

Eric Allen

Mentor: Lajos Horvath

To be Stationary or to be Non-Stationary, GARCH simulations in RStudio

The GARCH (Generalized Autoregressive Conditional Heteroskedastic) models are used
to model stock volatility. Stock volatility is defined as how much and how frequently
a stock price changes in the stock market. We run GARCH simulations in order to analyze
a times series of data which represent these incremental changes. A time series is
a series of values taken at successive, equally- spaced times. The time series represents
a sequence of discrete-time data. I used packages, “rugarch” and “fgarch” in RStudio
to run stationary and non-stationary GARCH simulations. Multiple simulations were
conducted with fixed alpha, beta, and omega to produce plots of the time series and
density function. Simulations to estimate values for alpha, beta, and omega in order
to predict the accuracy of the “ugarchspec” function were also conducted.

Justin Baker

Mentor: Elena Cherkaev

Designed Swarming Behavior Using Optimal Transportation Networks

Different models of swarming behavior can be used to study, analyze, and optimize
the behavior of large populations, building evacuation by emergency response teams,
and model groups of robots, people and animals. The current project considers the
problem of designing the swarming behavior, and formulates this problem as an optimal
transportation problem. We formulate the optimal transportation problem as a discretized
linear programming problem. We use the dual problem to maximize efficiency of the
designed transportation network. Finally, we numerically compute the solution using
Python and develop a visualization of the network and solution for several hypothetical
models

Charlotte Blake

Mentor: Ekaterina Epshteyn

Efficient Numerical Algorithms for Automatically Processing Data with Application
to Materials Science

Our research focused on developing robust numerical algorithms that take images of
crystal grains as the input and automatically output relevant data, including information
about grain area, perimeter, and number of neighbors. In this presentation, we will
discuss the process of obtaining each type of data, including the difficulties along
the way. We will also present the obstacles that appeared as a part of the design
of such algorithms and how they were resolved. Special focus will be given to the
aspects of the algorithms related to the computational geometry questions of corner
identification and polygon approximation of boundaries.

Audrey Brown

Mentor: Alla Borisyuk

Analysis of Mice Olfactory Response Data

Relationships within mice olfactory response data was investigated in existing data
from eight mice. Previously identified patterns in odor response similarity was further
analyzed, and patterns in glomerular response similarity was analyzed using clustering
methods. The first method used was hierarchal clustering, for which several different
linkage and distance measurements were experimented with. The second method used was
K-means clustering Finally, Gaussian distribution clustering was used. For odor response
clustering, it was determined that, though with some variability, clustering patterns
tend to follow previously observed patterns in odor response similarity. For clustering
by glomerular response, different clustering methods were experimented with. Future
plans include using glomerular clustering to identify glomeruli from mouse to mouse,
and testing correlation between glomerular response and glomerular anatomical position.

Dylan Johnson

Mentors: Karl Schwede, Daniel Smolkin, Marcus Robinson

Searching for Rings with USTP

Using recent work from D. Smolkin and J. Carvajal-Rojas, we seek to identify new commutative,
Noetherian rings with Uniform Symbolic Topology Property, abbreviated USTP. For k
a field, we show that the toric rings k[x,y], k[x,y,z], and k[w,x,y,z]/(wx-yz) have
USTP. All are toric rings already known to have the property, but we provide an alternative
proof using a different method, one which we hope to extend to rings unknown to have
USTP. Indeed, we also classify the 2 and 3 dimensional toric rings for which Smolkin's
and Carvajal-Rojas's method may show that he rings have USTP. Finally, we consider
a specific type of toric ring, called Hibi rings, and work on applying this method
to them in larger dimensions.

Dylan Soller

Mentor: Anna Romanov

Finite Gelfand Pairs

Finite Gelfand paris are algebro-combinatorial objects that arise naturally in various
areas of mathematics including statistics, coding theory, and combinatorics. In this
presentation, we will discuss ways in which finite Gelfand pairs are related to Markov
chains and association schemes. We will also define the “cracking point” of a finite
group, and discuss new results including the cracking points of the symmetric groups.

Mathematics Department

Undergraduate Research Symposium Spring 2017

Monday, May 1. 9:15-11:30 am in LCB 323

9:15-9:30 Sarah Melancon

Mentor: Adam Boocher

Intro to Research: Commutative Algebra in the Polynomial Ring

9:30 -9:45 Dylan Soller

Mentor: Adam Boocher

Intro to Research: commutative algebra

9:45- 10:00 Max Carlson

Mentor: Christel Hohenegger, Braxton Osting

Improving the Numerical Method for Approximating Solutions to the Free Surface Sloshing
Model

10:00-10:15 Weston Barton

Mentor: Tom Alberts

A Direct Bijective Proof of the Hook Length Formula

10:15-10:30 Michael (Barrett) Williams

Mentor: Elena Cherkaev

Ice Diffusivity

10:30 -10:45 Willem Collier

Mentor: Arjun Krishnan

The RSK Algorithm and the Marcenko-Pastur Law

10:45-11:00 Peter Harpending

Mentor: Elena Cherkaev

The Geometry of Fractional Calculus

11:00- 11:15 Rebecca Hardenbrook

Mentor: Ken Golden

Bounds on the thermal conductivity of sea ice in the presence of fluid convection

ABSTRACTS

Weston Barton

Mentor: Tom Alberts

A Direct Bijective Proof of the Hook Length Formula

The hook length formula, discovered in 1954 by J. S. Frame, G. de B. Robinson, and
R. M. Thrall, gives the number of standard Young tableaux of a given shape. The original
proof was rather complicated, as have many of the proofs developed since. In 1997
Novelli, Pak, and Stoyanovskii in developed a direct bijection between the set of
Standard Young Tableaux and the set of ordered pairs consisting of one non-standard
Young tableaux and a set of new objects—Hook functions. This bijection may provide
insight into counting subclasses of standard Young tableau, such as those with a particular
Schützenberger Path.

Max Carlson

Mentor: Christel Hohenegger and Braxton Osting

Improving the Numerical Method for Approximating Solutions to the Free Surface Sloshing
Model

By further improving the numerical free surface sloshing model to include non-axisymmetric
containers and fully 3D computations, the limitations of the current method have become
clear. This semester, I explore parallel programming algorithms to improve the performance
and accuracy of the numerical model I've developed previously. In addition, with the
type of containers that can be analyzed being expanded, I will discuss parametric
container shapes and how they can be used to further analyze the model.

Willem Collier

Mentor: Arjun Krishnan

The RSK Algorithm and the Marcenko-Pastur Law

The Robinson-Schensted-Knuth (RSK) algorithm is a bijection between matrices with
non- negative entries and pairs of Young Tableaux. The lengths of the rows of the
Young diagram obtained through the RSK algorithm serve as eigenvalues of a certain
growth process on the plane called last-passage percolation. When the matrix consists
of geometrically distributed random variables, it has been shown that the empirical
law of the lengths of the rows of the Young diagram converges to the Marchenko-Pastur
distribution. We explore this convergence numerically, and see if it generalizes to
random variables that are not geometrically distributed.

Rebecca Hardenbrook

Mentor: Ken Golden

Bounds on the thermal conductivity of sea ice in the presence of fluid convection

Sea ice forms the thin boundary layer between the ocean and atmosphere in the polar
regions of Earth. As such, ocean-atmosphere heat exchange is largely controlled by
the thermal conductivity of sea ice. However, frozen seawater is a complex composite
material consisting of an ice host containing brine and air inclusions. Moreover,
the volume fraction, geometry and connectivity of the brine inclusions changes significantly
with temperature, and if the temperature of the sea ice exceeds a critical temperature,
fluid can flow through the ice which can enhance thermal transport. Calculating the
thermal conductivity of sea ice is thus a very challenging problem requiring sophisticated
mathematics and computation, and little is known from either an experimental or theoretical
perspective about this key parameter. The underlying equation describing the physics
of this system is known as the advection-diffusion equation. Here we exploit Stieltjes
integral representations for the effective or homogenized thermal conductivity of
sea ice in the presence of a fluid flow field to obtain rigorous bounds on this key
geophysical parameter. In particular, we assume a periodic convective flow field and
analytically calculate the moments of the spectral measure for the system, which is
at the heart of the integral representation. The moments are then used to obtain the
first bounds on the thermal conductivity of sea ice which incorporate fluid velocity
effects.

Peter Harpending

Mentor: Elena Cherkaev

The Geometry of Fractional Calculus

Fractional calculus is the study of differentiation and integration to non-integer
orders. We explore geometrical interpretations of frac- tional calculus operations,
and compare them to their integer-order counderparts. This analysis leads to a rather
beautiful result regard- ing paths in the spectrum of the fractional derivative operator.

Sarah Melancon

Mentor: Adam Boocher

Commutative Algebra in the Polynomial Ring

Given a surface described by parametric equations, we may want to determine a set
of polynomials which vanish on the surface. We will learn how to find these polynomials
using ideas from commutative algebra such as rings, ideals, and varieties.

Barrett Williams

Mentor: Elena Cherkaev

Ice Diffusivity

Marginal ice zones (MIZ) are areas of ice that are in the transition from an ice shelf
to the open ocean. MIZ can be comprised of large and small ice floes, which can be
found in both the Arctic and Antarctic Oceans. We will be starting with the diffusion
equation and will build a model that will represent the ice concentration data that
we have, using our model we will be able to find diffusivity. This allows us to test
the accuracy of our method by comparing the results from the simulated ice concentration
data and the actual data collected. The results of this general problem will show
us how the ice diffuses on a spatial basis, allowing us to see how the MIZ move spatially
based on the current concentrations.

Mathematics Department

Undergraduate Research Symposium Fall 2017

December 11, 11:15am to 2pm in LCB 215

11:15-11:30 Scott Neville

Mentor: Arjun Krishnan

An explicit bijection from particular Kostka numbers and Permutations with specific
Longest Increasing Subsequences.

11:30-11:45 Sarah Melancon

Mentor: Adam Boocher

Modules and Free Resolutions

11:45-12:00 Peter Harpending

Mentor: Elena Cherkaev

Numerical Fractional Calculus

12:00-12:15 Bo Zhu and Chong Wang

Mentor: Sean Lawley

Branching Process Model of Breast Cancer

12:15-12:30 Hannah Choi

Mentor: Sean Lawley

First Passage Time and its Ecological Applications

12:30-12:45 Break

12:45-1:00 Hannah Waddel

Mentor: Fred Adler

The Community Ecology of the Music Canon

1:00-1:15 Katelyn Queen

Mentor: Fred Adler

Dynamic Modelling of Subclones in Patients with Late-Stage Breast and Ovarian Cancers
to Block and Reverse Chemo-resistant Tumors

1:15 -1:30 Barrett Williams

Mentor: Jingyi Zhu

Diffusion Equation with Cross-Derivative Term with Variable Coefficients

1:30-1:45 Alex Henabray

Mentor: Christel Hohenegger

Flow Around Axisymmetric Biconcave Shapes

1:45-2:00 Max Carlson

Mentor: Christel Hohenegger, Braxton Osting

A Numerical Solution to the Free Surface Sloshing Problem with Surface Tension

Abstracts

Scott Neville

Mentor: Arjun Krishnan

Kostka Numbers and Longest Increasing Subsequences

The Kostka Numbers appear in several natural combinatorial problems, such as symmetric
polynomials or partitions. They also count the number of Young tableaux with a given
shape and content. If we consider a pair of tableaux with the same shape, the RSK
algorithm lets us convert them into a permutation. This bijection sends the width
of our Young Tableaux to the length of the longest increasing subsequence of our permutation.
If we look at permutations with a specific longest increasing subsequence, like 1,2,
then we empirically see that the number of such permutations is equal to the number
of non crossing permutations. This seems to generalize. We generalize an existing
proof to convert pairs of Young tableaux with certain width into a single Young tableau,
with a specific shape and content.

Sarah Melancon

Mentor: Adam Boocher

Modules and Free Resolutions

Free resolutions are central to the field of commutative algebra. Computing a free
resolution gives us lots of information about a module, and there are a variety of
fascinating open questions about the properties of free modules. We will discuss what
a free resolution is, what it can tell us, and what kinds of problems commutative
algebraists would like to solve.

Peter Harpending

Mentor: Elena Cherkaev

Numerical Fractional Calculus

The fractional derivative is a generalization of the ordinary derivative, which allows
differentiation to arbitrary real order. We present a numerical algorithm for computing
fractional derivatives efficiently, and for solving some fractional partial differential
equations. We also present a proof-of-concept program.

Bo Zhu and Chong Wang

Mentor: Sean Lawley

Branching Process Model of Breast Cancer

We develop a branching process model for breast cancer growth and change accounting
for three types of cell populations: Primary (cells in the breast), Lymph (live cells
in nearby lymph nodes), and Metastatic (cells transplanted on other remote organs).
Then we dig up the data online, use the data to find what is the best window of opportunity
to recover from the breast cancer.

Hannah Choi

Mentor: Sean Lawley

First Passage Time and its Ecological Applications

TBA

Hannah Waddel

Mentor: Fred Adler

The Community Ecology of the Music Canon

Symphony orchestras today have access to a musical canon stretching back at least
four hundred years and play a critical role in its establishment. Full symphony orchestras
are expensive to operate and have limited performance time, which only allows a finite
number of songs and composers to be considered canonical. Most works by new composers,
if performed at all, never get performance time and enter the canon. Community ecology
describes the structure and interactions of species that are competing for limited
resources. Treating composers or songs as species, we are using quantitative ecological
methods to characterize the ways that music interacts in the canon, with a particular
focus on the ways that composers and pieces enter or remain in the canon. The canon
already appears to follow some common species composition patterns, where a small
number of species constitute the bulk of the biomass in an ecosystem and most species
are rare. In our data from 9 symphony orchestras, over performances of 207,000 pieces
by 296 composers, 91 composers had less than 120 performances of their work while
Mozart alone accounted for nearly 12,000 performances. This result lends us confidence
that some ecological methods may be useful to describe the canon. The efficacy of
ecological models in describing the behavior of the canon lends insights into what
processes occur in the field of music composition that cause a composer to either
fade into obscurity or become enshrined through centuries of performance.

Katelyn Queen

Mentor: Fred Adler

Dynamic Modelling of Subclones in Patients with Late-Stage Breast and Ovarian Cancers
to Block and Reverse Chemo-resistant Tumors

(By Katelyn J. Queen, Dr. Fred Adler, Dr. Jason Griffiths) During the course of chemotherapy
treatment in cancer patients, genetically related groups of cells in the tumors called
subclones can become resistant to treatments. This study uses mathematical and statistical
models to understand this process, and eventually, we hope, to find ways to delay
or even reverse the development of the chemo-resistant tumors associated with breast
and ovarian cancers. Our data come from metastatic tumor cells from the pleura of
patients with breast and ovarian cancer collected before, during, and after different
types of treatments. These cells are then genetically sequenced with a method called
single-cell RNA seq that gives the genetic composition in detail. Our main mathematical
tool will be integral projection models, which can track continuous change in the
stage of disease progression over time, and will be used for the first time to map
phenotypic changes in metastatic cells. These methods will allow us to derive, simulate
and analyze a dynamic model of subclone changes during the transformation of a chemo-sensitive
tumor to a chemo-resistant tumor, which in turn would create the opportunity to then
block or reverse this transformation in patients with late- stage breast and ovarian
cancers. As an added complexity, the progression of cancers can be facilitated by
non-cancerous cells like white blood cells. We will extend our data analysis techniques
to determine how cancer subclones interact with other cell types, and how these interactions
might be used to delay the transition to a chemo-resistant state. Currently, the project
is focused on a simple, dynamic, ordinary differential equations model of cancer,
that entertains two different cancer cell states as well as therapy and the immune
system.

Barrett Williams

Mentor: Jingyi Zhu

Diffusion Equation with Cross-Derivative Term with Variable Coefficients

TBA

Alex Henabray

Mentor: Christel Hohenegger

Fluid Flow Around Biconcave Surfaces

Fluid mechanics is defined as the mathematical study of fluids and their behavior
at rest and in motion. Scientists use fluid mechanics to explore many natural phenomena,
including the red spot on Jupiter and the behavior of tornados. Mathematicians who
specialize in this field are often concerned with developing mathematical models and
deriving equations that accurately describe fluid behavior, especially around obstacles.
English mathematician Sir George Stokes derived the equation that describes viscous
fluid flow around a sphere, and other mathematicians have expanded on Stokes’ work
with ellipsoidal objects . Modeling fluid flow around obstacles with irregular shapes
is very challenging, and these problems typically do not have analytical solutions.
The purpose of this research project is to study fluid flow around biconcave shapes,
using infinite series and Gegenbauer polynomials. These models will then be examined
using MATLAB to determine how accurate they model the fluid’s behavior.

Max Carlson

Mentor: Christel Hohenegger, Braxton Osting

A Numerical Solution to the Free Surface Sloshing Problem with Surface Tension

I will be presenting a scalable, parallel numerical method for computing approximate
solutions to the free surface sloshing problem with surface tension and discussing
the performance of this method.

## Mathematics Department

Undergraduate Research Symposium Spring 2016

## May 3, 12:30-2:45pm - Room LCB 323

12:30-12:45 Tyler McDaniel

Mentor: Braxton Osting

Utah’s Pathways to Higher Education: A Quantitative, Critical Analysis

12:45-1:00 Chong Wang

Mentor: Don Tucker

Eradicating Ebola

1:00-1:15 Dan Armstrong

Mentor: Sean Lawley

Methods for Modeling Neurite Growth Driven by Vesicular Dynamics

1:15-1:30 Curtis Houston

Mentor: Jyothsna Sainath

Mapping Counts of Death in League of Legends

1:30-1:45 Yushan Gu

Mentor: Firas Rassoul-Agha

Rare Events for Non-Homogenous Markov Chains

1:45-2:00 Naveen Rathi and Gerardo Rodriguez-Orellana

Mentors: Owen Lewis and Leif Zinn-Björkman

A Mechanical Model for Simulating the Cell Motility of a Visoelastic Cell

2:00-2:15 Nathan Willis and Olivia Dennis

Mentors: Owen Lewis and Leif Zinn-Björkman

Mathematical Model of Cell Motility

## ABSTRACTS:

**Tyler McDaniel**

Mentor: Braxton Osting

Utah’s Pathways to Higher Education: A Quantitative, Critical Analysis**Abstract:** Using data from the 2008 cohort of Utah high school graduates, this project analyzes
the effect of demographic factors (race, class, mobility, language, geography) and
past achievement (ACT scores, AP scores, GPA) on the individual’s likelihood of attaining
success in higher education (college entry, GPA, graduation). Data is obtained from
the Utah System of Higher Education, and consists of 41,303 observations (students).
The goal is to describe college pipelines in terms of access for racial, class, mobile,
language-learner and geographic groups. What groups are advantaged in the college-application
process? Equity will be assessed using a critical pedagogy framework, in which education
has the potential to liberate students from systemic oppression (Friere, 1970, Harris,
2014).

Principal Components Analysis (PCA) was used in order to choose variables of importance
and model underlying factors. Principal Components Analysis (PCA) is appropriate for
reducing a dataset in order to analyze its most important variables. ACT Scores were
the most important variable in all PCA models. Further, ACT Math, Science and Composite
scores were consistently the variables with the highest loadings in our models. This
result affirms education literature stating that math and science scores are particularly
important in the college-going process.

Additionally, college pathways will be described using a Random Forest (RF) decision
tree algorithm. The RF model has been used widely across academic fields and returns
the predictive power of variables included. Further, the RF model is predictive, using
random samples of predictor variables to generate decision trees that form “forests.”
The RF model is ideal for college access data because it has robust algorithms for
dealing with missing data (Hardman et. al, 2013).

**Chong Wang**

Mentor: Don Tucker

Eradicating Ebola**Abstract:** Build a realistic, sensible, and useful model that considers not only the spread of
the disease, the quantity of the medicine needed, possible feasible delivery systems
(sending the medicine to where it is needed), (geographical) locations of delivery,
speed of manufacturing of the vaccine or drug, but also any other critical factors
I consider necessary as part of the model to optimize the eradication of Ebola.

**Dan Armstrong**

Mentor: Sean Lawley

Methods for Modeling Neurite Growth Driven by Vesicular Dynamics**Abstract:** Neurite growth requires both membrane expansion via vesicle exocytosis and cytoskeletal
dynamics. Mathematicians have modeled neurite growth by simulating the dynamics of
vesicle motion and microtubule interaction at the boundary of the growing neurite.
We examine different methods, both stochastic and deterministic, for modeling neurite
growth and compare the results. Stochastic models use a coarse-graining method to
eliminate intermittent dynamics and derive a single SDE that describes vesicle motion.
In past studies, this coarse-grained SDE is derived using vesicle motion in two-dimensional
cells. We use a method that produces a more accurate SDE because it is derived using
vesicle motion in a three-dimensional cell. Using this SDE we generate models of growing
neurites, which predict different neurite growth regimes depending on cytoskeletal
dynamics.

**Curtis Houston**

Mentor: Jyothsna Sainath

Mapping Counts of Death in League of Legends**Abstract:** In the video game League of Legends, player deaths greatly impact the tide of each
match. In this project we take match data from League games and use a bootstrap method
to find counts of kills across the map in order to better understand where players
are likely to die. We then separate kills by time in order to consider how these counts
change throughout the course of the game and across the map. In particular, we find
that kills tend to move toward the middle lane of the map and spread further apart
up and down the lanes as the game progress.

**Yushan Gu**

Mentor: Firas Rassoul-Agha

Rare Events for Non-Homogenous Markov Chains**Abstract:** This project is about simulating rare events for non-homogenous Markov Chains. When
a fair coin is flipped, the probability that we get heads is 1/2. When we flip this
fair coin n times, the expected number of heads is n/2. Getting 0.7n heads is a rare
event for this model. There is a very small probability that this rare event occurs.
Therefore, a huge number of simulations is required in order to generate such a rare
event. Using the theory of large deviations and the notion of entropy one can calculate
the distribution of the sequence of iid coin tosses, conditioned on the rare event.
Then one can use this to generate rare events. Now, every sample is rare event. The
same can be done for homogenous Markov Chains. However, in applications such as climate
models, one often has non-homogenous Markov Chains. The final goal of this project
is finding the distribution of the process conditioned on rare events, for non-homogenous
Markov Chains. More precisely, we consider two Markov Chains with state space {0,
1} and the Markov Chain that alternates between the two. We will find the distribution
of this alternating process, conditional on it having an unusual number of ones.

**Naveen Rathi and Gerardo Rodriguez-Orellana**

Mentors: Owen Lewis and Leif Zinn-Björkman

A Mechanical Model for Simulating the Cell Motility of a Visoelastic Cell**Abstract:** In modern biology, cell locomotion is a key issue that is under constant investigation
by both theoreticians and experimentalists. By understanding how a cell’s internal
rheology and active contraction, as well as interactions with its surrounding environment
affect the speed of the movement, it is possible to gain deeper insight into the mechanics
of cell locomotion. In this research, we developed a mechanical model of a viscoelastic
cell and numerically simulate to determine how variations in the time-dependent interactions
affect migration velocity. In particular, we investigated how the relative phase of
time-dependent adhesion and contraction affects the speed of the modeled cell.

**Nathan Willis and Olivia Dennis**

Mentors: Owen Lewis and Leif Zinn-Björkman

Mathematical Model of Cell Motility**Abstract:** Cell motility is a vital process in a wide array of biological contexts including
immune response, embryonic development, and wound healing, as well as the spread of
cancer cells. Following previous studies, we develop a one-dimensional partial differential
equation which models a motile amoeboid cell by balancing internal body forces with
drag against the underlying substrate. We numerically simulate this model using Finite
Differences and the Forward Euler method. We investigate the profile and coordination
of adhesion between the cell and substrate. Specifically, we are interested in how
the coordination of adhesion relative to active contraction within the cell affects
the behavior of a motile cell.

## Mathematics Department

Undergraduate Research Symposium Summer 2016

## August 26, 3pm to 5pm in LCB 225

**3:15-3:30 Jimmy Seiner**

Mentor: Adam Boocher

Algebra, Geometry, and Syzygies**3:30-3:45 Anran Chen**

Mentors: Travis Mandel and Yuan Wang

Geometry of surfaces**3:45- 4:00 Noble Sage Williamson**

Mentor: Gordan Savin

Quadratic Forms and Their Significance to Number Theory**4:00-4:15 Stephen McKean**

Mentor: Stefan Patrikis

Inverse Galois Problem**4:15-4:30 Jacob Madrid**

Mentor: Sean Lawley

An Improved Numerical Method for the Stochastic Simulation of the Diffusion Process
with a Partially Absorbing Boundary**4:30-4:45 Max Carlson**

Mentors: Christel Hohenegger and Braxton Osting

Constructing a Numerical Solution to Sloshing Free Surface using FEniCS**4:45- 5:00 Olivia Dennis**

Mentor: Braxton Osting

Information Theory

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

A major facet of the study of quadratic forms is establishing equivalence classes
among such forms. In local fields such as ℝR or ℚ𝕡Qp, 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 ℚQ 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

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)

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

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

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

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

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

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

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.

## Mathematics Department

Undergraduate Research Symposium Spring 2015

## Session 1/2 - Monday, May 4 - 1pm-3:10pm - Room LCB 222

01:00-01:30pm, Anand Singh and Braden Schaer, A Model for Restricted Diffusion of
Evoked Dopamine

Mentors: Sean Lawley and Heather Brooks

Abstract: A simple gap diffusion model is commonly used for short, unaided intercellular
transport. One prominent example is dopamine diffusion in neural synapses, where intercellular
space is often a crowded environment consisting of extracellular networks, biological
waste, macromolecules, and other obstructions. The traditional model fails to represent
many unique characteristics of restricted diffusion in complex cellular environments.
We compare existing mathematical models of gap and restricted diffusion against data
acquired by fast-scan voltammetry of evoked dopamine in the striatum of rat brains.
Finally, we propose a more physically realistic model that accounts for the spatial
structure of this system.

(Supported by Mathematical Biology RTG grant)

01:30-01:50pm, Oliver Richardson, Precise Timing in a Neural Field Model

Mentors: Sean Lawley and Heather Brooks

Abstract: Building upon the the work of Veliz-Cuba et al, this project addresses building
a system of differential equations that lead to the precise encoding of neural firing
event timings into the weighting between neurons. However, instead of dealing with
a discrete case and a global inhibitory neuron, we have generalized this to the continuous
neural field setting, and built inhibition directly into the weight itself.

(Supported by Mathematical Biology RTG grant)

01:50-02:10pm, Marie Tuft, Quantitative Analysis of Virus Trafficking in a Biological
Cell

Mentors: Sean Lawley and Heather Brooks

Abstract: For a virus to successfully infect a host cell it must travel from the cell
wall to the nucleus by hijacking that cell's existing transport system. This motion
occurs as two iterated steps: passive diffusion through cell cytosol and active transport
along microtubule networks. An existing model shows that this process can be approximated
as a stochastic differential equation in the limit as the number of microtubules goes
to infinity. We propose a different model which reduces the complex viral trajectory
to a simpler finite state Markov process. Preliminary results show this approximation
to be superior to the existing model across several modes of comparison.

(Supported by Mathematical Biology RTG grant)

02:10-02:30pm, Hitesh Tolani, Dependence of Disease Transmission on Contact Network
Topology

Mentors: Braxton Osting and Damon Toth

Abstract: Despite much attention in recent years, there are a number of open questions
related to epidemic dynamics for many infectious diseases, such as measles and ebola.
Consequently, intervention strategies for these diseases are inadequate. As transmission
occurs through direct contacts between infected and susceptible individuals, epidemic
models must include properties of the host population's social structure. For example,
individuals with more social contacts are more likely to transmit (or receive) an
infection. It follows that the network topology of a social network has an impact
on epidemic dynamics, such as growth rate and final outbreak size. In this work, we
studied the influence of network topology on epidemic dynamics using agent-based simulations
on synthetic social networks that were constructed using the contact history of students
at a local elementary school. (No students were hurt in these simulations.) I'll discuss
the results of these simulations and conclude with some preliminary results on improved
intervention strategies.

02:30-02:50pm, Yuji Chen, The Google rank page

Mentor: Peter Alfeld

Abstract: I will describe and discuss the original algorithm that Google used to rank
web pages.

02:50-03:10pm, Alex Beams, Implications of Antibiotic Use for Co-Infections when a
Fitness Trade-Off for Resistance is Present

Mentor: Fred Adler

Abstract: How much does indiscriminate antibiotic use promote the spread of antibiotic-resistant
infections in a population? Assuming a fitness trade-off for resistance exists, it
is possible for an antibiotic-vulnerable strain to outlast a resistant type within
an untreated host carrying both. That means prudent treatment schemes can potentially
manage resistance levels in the population-at-large. We use a compartmental ODE model
incorporating a class of co-infected individuals to analyze the epidemiology. The
Next Generation Operator method gives R0R0 for the two strains, and we perform invasion analysis for both types under different
treatment schemes. Haphazard antibiotic use favors resistant strains by treating both
singly-infected and co-infected people, while shrewd treatment targets those carrying
the vulnerable strain only. According to the model, inattentively treating co-infected
individuals significantly promotes resistance in a population.

Session 2/2 - Monday May 4 - 3:30pm-5:30pm - Room LCB 222

03:30-03:50pm, Jacob Madrid, Study of a Link Invariant- Stabilized Hat Heegaard Floer
Homology

Mentor: Sayonita Ghosh Hajra

Abstract: A knot can be thought of as a tangled rope with ends attached. One of the
fundamental ideas in studying knots and links is determining whether two projections
are equivalent. A knot invariant is a property of a knot, which is the same for equivalent
knot projections. When two invariants give two different objects, then it can be said
that the knots are different. However, invariants cannot be used to distinguish between
knots if they result in the same object. In todays presentation, we will talk about
another knot and link invariant called "Stabilized Hat Heegaard Floer Homology". This
invariant is a combinatorial algorithm introduced by A. Stipsicz. Here we will present
this algorithm and describe a code to change a "grid diagram" into an "extended grid
diagram".

03:50-04:10pm, Tianyu Wang, Maxima of Correlated Gaussian Random Variables

Mentor: Tom Alberts

Abstract: We consider the maximum of correlated Gaussian random variables and examine
the methods for computing the mean and distribution of the maximum. We are particularly
interested in how these quantities change as a function of the correlation structure.
We will focus on the 2 dimensional case.

04:10-04:30pm, Mackenzie Simper, The Stochastic Heat Equation on Markov Chains

Mentor: Tom Alberts

Abstract: Consider a casino with several gambling tables and a gambler who chooses
to move among them randomly following the dynamics of a Markov chain. The heat equation
describes the evolution of the probabilities for the gambler being at a given table
at a given time. While he is at each table the gambler gains or loses a random amount
of money, and the stochastic heat equation describes the evolution of his expected
fortune over all possible trajectories between the tables. The fundamental solution
to the stochastic heat equation is described by a matrix-valued stochastic differential
equation (SDE). We explore various properties of this process of random matrices,
including the evolution of the norm, the determinant, and the trace of the matrix.

04:30-04:50pm, Michael Zhao, Spectra of Random Graph Models

Mentor: Braxton Osting

Abstract: Given a graph GG from ``real-world'' data, let G0G0 be a graph generated from a random graph model, e.g. BTER or Chung-Lu. Let σ(L(H),l)σ(L(H),l) be the vector of the smallest ll eigenvalues of the graph Laplacian of a graph HH, and let A(H)A(H) be the adjacency matrix. Then we consider the constrained optimization problem of
adding or removing edges to G0G0 to create a graph GtGt where σ(L(Gt),l)σ(L(Gt),l) is closer to σ(L(G),l)σ(L(G),l) than σ(L(G0),l)σ(L(G0),l), in the sense of ℓ2ℓ2-norm, and ||A(G0)−A(Gt)||1=M||A(G0)−A(Gt)||1=M for some preset MM. The performance of the algorithm is evaluated for a variety of random graph models,
for differing values of ll and MM, and also when L(G)L(G) is replaced by A(G)A(G).

04:50-05:10pm, Jeremy Allam, Low-Energy Satellite Transfer from Earth to Mars

Mentor: Elena Cherkaev

Abstract: A new type of satellite transfer that uses half the amount of fuel as conventional
transfers has been discovered. This transfer, called a low-energy transfer, proved
to work in 1991 when a Japanese satellite successfully went in orbit around the moon
using this technique. Since then, more research has been conducted to prove that a
low-energy transfer can be accomplished between the moons of Jupiter. In the case
of this presentation, it is shown that a low-energy transfer is possible from Earth
to Mars using similar techniques as between the Jovian moons.

05:10-05:30pm, Anthony Cheng, Percolation Theory for Melt Ponds on Arctic Sea Ice

Mentor: Kenneth Golden

Abstract: Extreme losses in summer Arctic sea ice pack, a leading indicator of climate
change, have led to the need for significant revisions of global climate models. Part
of the efforts to improve these models is an increased emphasis on sea ice albedo
(reflectance), which is closely related to the formation of melt ponds on the sea
ice. This project investigates the novel use of percolation theory to model melt pond
formation. Innovative methods were implemented to calculate the percolation probability
as a function of pp, the probability of an edge being open. The results were analyzed to determine the
critical threshold value pcpc, which was compared with the known value of 0.50 for a two dimensional square lattice.
This model was then adapted to determine the critical area threshold for melt ponds
on sea ice as 0.4845, which had not been numerically determined previously. This threshold
allows for simple detection of percolation, and thus also changes in the albedo and
the melting rate of the sea ice. The calculation of this parameter is a first step
to creating more dynamic models for the growth and disappearance of melt ponds over
time.

## Mathematics Department

## Undergraduate Research Symposium Fall 2015

## December 16 - 2:30pm-5:00pm - Room LCB 323

2:30-2:45, **Stephen McKean**, *Fluid Dynamics and Traffic Flow**Mentor:* Don Tucker*Abstract:* The flow of liquids and gasses is well explained through physical laws and theories.
However, in terms of mathematical theories and predictions, traffic flow is not understood
to the same extent as the physics of matter flow. Is there a correlation between matter
and traffic, and can the physical theories of fluid dynamics be generalized to the
abstracted case of traffic flow?

2:45-3:00 , **Siben Li**, *Statistical Analysis of Lightcurves**Mentor:* Lajos Horvath*Abstract:* A Type Ia supernova (SN Ia) is the explosion of a carbon–oxygen white dwarf. The observed
brightness can be used to measure the distance to the exploded star. These measurements
can be used to deduce the expansion of the Universe. The analysis of the measurements
led to the discovery that the Universe is dominated by an unknown form of energy that
acts in the opposite sense of energy (“dark matter”). Our analysis is based on lightcurves,
and we will assume a model to perform our analysis. In this research we would like
to study the following question: Is the model supported by the data? We try 3 different
methods to answer this question: likelihood method, least squares, and weighted least
squares. If not, what is the better model?

3:00-3:15, **Michael Zhao**, *Maximal Orders of Quaternion Algebras**Mentor:* Gordan Savin*Abstract:* Suppose we have a base field kk, a quadratic extension LL and a quaternion algebra HH. Let LOL and HOH be two maximal orders of LL and HH. I'll discuss an order is, and the differences between orders in the split and non-split
case (i.e. when HH is a matrix algebra or a field). Additionally, I'll discuss \emph{optimal} embeddings,
which are embeddings f:L−>Hf:L−>H with f(L)∩H=f(L)f(L)∩OH=f(OL). Finally, I'll present some preliminary results concerning the relation between integral
binary hermitian forms and maximal orders.

3:15-3:30, **Jake Madrid**, *A Study of Knot and Link Invariants**Mentor:* Sayonita Ghosh Hajra *Abstract:* A knot is the embedding of a closed curve in space. One of the fundamental problems
in knot theory is to distinguish knots. Knot invariants are objects assigned to different
knot projections and are invaluable in solving this problem. In this talk, we will
explore several of these knot invariants and will also discuss Vassiliev invariants.
Finally, we will discuss and analyze an algorithm used to compute an invariant called
Stabilized Hat Heegaard Floer homology.

3:30-3:45, **Hunter Simper**, *Is the maximum number of singular points for an irreducible tropical curve the same
as a complex curve?**Mentor:* Aaron Bertram*Abstract:* A brief overview of definitions and concepts in the tropical setting. Followed by
an introduction to the above problem by examining how key differences between the
tropical version of Bezout and the usual one preclude the direct translation of the
common proof for this fact.

4:00-4:15 , **Mackenzie Simper**, *Bak-Sneppen Backwards**Mentor:* Tom Alberts*Abstract:* The Bak-Sneppen model is a Markov chain which serves as a simplified model of evolution
in a population of spatially interacting species. We study the backwards Markov chain
for the Bak-Sneppen model and derive its corresponding reversibility equations. We
show that, in contrast to the forwards Markov chain, the dynamics of the backwards
chain explicitly involve the stationary distribution of the model, and from this we
derive a functional equation that the stationary distribution must satisfy. We use
this functional equation to derive differential equations for the stationary distribution
of Bak-Sneppen models in which all but one or all but two of the fitnesses are replaced
at each step.

4:15-4:30, **Jammin Gieber**, *Eigenvalues in Chaos**Mentor:* Elena Cherkaev*Abstract:* I will be talking about how the eigenvalues change in the Lorenz strange attractor
and about some of the bifurcations and where they occur along the flow.

4:30-4:45, **Chong Wang**, *Surprising Mathematics of Longest Increasing Subsequences**Mentor:* Tom Alberts*Abstract:* I will talk about patience sorting and the Robinson-Schensted algorithm.

4:45-5:00, **Tauni Du **, *The Role of mRNA Decay in a Genetic Switch**Mentor:* Katrina Johnson and Fred Adler*Abstract:* Genes can be switched on or off by regulatory proteins. For example, two genes may
each synthesize a protein that downregulates the other gene, creating a repressor-repressor
switch that has two stable steady states: one being when the first gene is "on" and
the second gene is repressed, and the other in which the situation is reversed. Previous
study of a model by Cherry and Adler showed that the existence of such a switch depends
on factors including the shapes of the two repression functions. Adding to the model
an additional point of control - mRNA dynamics - resulted in further restrictions
upon the parameters that can lead to a functional genetic switch. Specifically, compared
to the original model, it is twice as difficult for a system that takes into account
second-order mRNA decay to have a working switch.

Mathematics Department

Undergraduate Research Symposium Spring 2014

Monday, April 28 Session - 2pm-3:40pm - LCB 121

02:00-02:40pm, Ryan Durr and Michael Senter, Mean-square displacement and Mean first
passage time in fluids with memory

Mentor: Christel Hohenegger

Abstract: Random motion in water is well studied and understood. However, questions
arise when attempting to extend Brownian models to media other than water which exhibit
some viscosity. We will present research on the effects of varying parameters on particle
behavior in such media. We investigated the influence of Kernel number and type on
the mean-squared displacement of particles, as well as present on its influence on
mean first passage time.

02:40-03:00pm, Wantong Du, Crime Rate Analysis

Mentor: Davar Khoshnevisan

Abstract: Crime in the United States has been declining steadily since the early to
mid-1990s. Violent crime rate and property crime rate have decreased by 49% and 42%,
respectively, during the last two decades. Homicide rate, in particular, is at its
lowest in nearly fifty years. I attempt to model the homicide rates and forecast the
rates on a national level.

03:00-03:20pm, Logan Calder, A trading strategy for auto-correlated price processes

Mentor: Jingyi Zhu

Abstract: Classical Brownian motion is the foundation of many stock price models.
Due to the independence nature of Brownian motion, without major corrections, these
models cannot account for autocorrelation observed in some price processes. Replacing
classical Brownian motion with fractional Brownian motion has been one suggested remedy,
but this poses serious theoretical and practical difficulties. If a price process
did follow from some autocorrelated process, such as fractional Brownian motion, knowing
the history of a stock could help one to make better bets. How might this be used
in a trading strategy? We simulated autoregressive and fractional Brownian motion
processes. Based on positively autocorrelated simulations, we will show one strategy
that can lead to favorable gains. When we tested this strategy on real data, we found
that quickly executing the strategy is crucial to making a profit.

03:20-03:40pm, Michael Primrose, Imaging with waves

Mentors: Fernando Guevara Vasquez and Patrick Bardsley

Abstract: We studied the problem of imaging a few point scatterers in 2D with waves
emanating from point sources located on a linear array. The data we use for imaging
are the waves recorded at a few locations that coincide with the source locations.
This setup is very similar to the setup used for ultrasound imaging in medical applications.
We considered both homogeneous and random media. For homogeneous media, we use the
Born approximation (i.e. linearization) to model wave propagation. For random media,
we first considered the problem of generating random media with known mean and correlation,
and instead of using a full wave solver we used the so called travel time approximation.
We then used the Kirchhoff migration method to image a few scatterers using data corresponding
to media that were either homogeneous or random.

Tuesday April 29 Session - 2pm-2:40pm - LCB 323

02:00-02:20pm, Kouver Bingham, Reflection Groups and Coxeter Groups

Mentor: Mladen Bestvina

Abstract: A pervasive and very beautiful type of group in group theory is one known
as a Reflection Group. One of the most popular groups learned about in elementary
group theory is the symmetric group on nn letters, and this group can be viewed as a reflection group. Reflection groups have
rich historical roots and are crucial in classifying polygonal tessellations of surfaces.
This talk will serve as an introduction to these groups. We'll look at several intriguing
pictures of triangle tessellations of the plane, the sphere, and the hyperbolic plane,
and even give the complete classification of these.

(Supported by Algebraic Geometry and Topology RTG grant)

02:20-02:40pm, Drew Ellingson, Towards Intersection Computations in Deligne-Mumford
Compactification of Mg,nMg,n

Mentor: Steffen Marcus

Abstract: Programs to calculate intersection numbers on Mg,nMg,n are well established, but none of these programs satisfactorily manage boundary classes.
To extend the functionality of these programs, we can reduce these computations to
problems in combinatorics on decorated graphs. This talk will begin with an introduction
to moduli spaces and the moduli space of curves, and will conclude with a general
discussion of the calculations involved in intersection computations on the Deligne-Mumford
compactification.

(Supported by Algebraic Geometry and Topology RTG grant)

Wednesday April 30 Session - 2pm-4:20pm - LCB 121

02:00-02:20pm, Sage Paterson, Dendritic Growth: Diffusion-Limited Aggregation and
other Fractal Growth Models

Mentor: Elena Cherkaev

Abstract: Dendritic growth patterns arise in a wide variety of natural phenomena,
including dielectric breakdown and particulate aggregations. Various models of this
type of fractal growth have been well studied, including diffusion-limited aggregation
(DLA), Laplacian growth models, and others. Diffusion-limited aggregation is a process
where random walking particles cluster together to form dendritic trees. We present
research on this algorithm, and the experiments and methods we developed. Specifically
we investigated methods of controlling the density and fractal dimensions of the DLA
clusters. We also vary the starting geometry for the growth, and model anisotropic
growth. We experiment with using chaotic deterministic maps in place of pseudo-random
number generators and observe nearly identical results. We also compare DLA to other
models of fractal growth.

02:20-02:40pm, Camille Humphries, Numerical Methods for Conservation Laws and Shallow
Water Models

Mentors: Yekaterina Epshteyn and Jason Albright

Abstract: Conservation laws are widely used in many areas of science and engineering.
In this talk, I will describe a modern class of numerical methods called central schemes
[1] that are designed to accurately approximate solutions to conservation laws. First,
to illustrate the capabilities of these schemes, we will look at two models: the linear
advection equation and Burger's equation. In the case of non-linear conservation laws,
central schemes are implemented in conjunction with slope-limiters and high-order
time discretizations (such as Strong-Stability Preserving Runge-Kutta Methods [5]).
We will conclude this talk with an introduction to a current area of research, in
which numerical methods based on central-upwind schemes [2,3,4] are being used to
solve shallow water systems that model ocean waves, hurricane flood surges, and tsunamis.

[1] A. Kurganov, E. Tadmor, "New High Resolution Central Schemes for Nonlinear Conservation
Laws and Convection-Diffusion Equations", Journal of Computational Physics 160 (2000),
241-282.

[2] A. Kurganov, S. Noelle, G. Petrova, "Semi-Discrete Central-Upwind Schemes for
Hyperbolic Conservation Laws and Hamilton-Jacobi Equations", SIAM Journal on Scientific
Computing 23 (2001), 707-740.

[3] A. Kurganov, D. Levy, "Central-Upwind Schemes for the Saint-Venant System", Mathematical
Modelling and Numerical Analysis, 36 (2002), 397-425.

[4] S. Bryson, Y. Epshteyn, A. Kurganov, G. Petrova, "Well-Balanced Positivity Preserving
Central-Upwind Scheme on Triangular Grids for the Saint-Venant System", Mathematical
Modelling and Numerical Analysis, 45 (2011), 423-446.

[5] S. Gottlieb, C.-W. Shu, E. Tadmor, "High order time discretization methods with
the strong stability property", SIAM Review 43 (2001) 89-112.

02:40-03:00pm, Nathan Briggs, Multimaterial optimal composites for 3D elastic structures

Mentor: Andrej Cherkaev

Abstract: The problem of multimaterial optimal elastic structures has already been
investigated. Namely given a strong and expense, a weak and inexpensive, and void
an optimal structure can be computed. This structure consists of various composite
microstructures. We expand this problem to 3D by finding the optimal microstructures.
By laminating pure materials together, computing effective fields and properties,
and enforcing the Hashin Shtrikman bounds an optimal composite microstructure can
be found for any admissible eigenvalues of the stress tensor. The problem is considered
for the specific case when the cost functional of the weak material touches the quasiconvex
envelope formed by the strong material and void.

[1] A. Cherkaev and G. Dzierzanowski. Three-phase plane composites of minimal elastic
stress energy: High-porosity structures. International Journal of Solids and Structures,
50, 25-26, pp. 4145-4160, 2013.

[2] A. Cherkaev. Variational Methods for Structural Optimization. Springer

[3] Z. Hashin, S. Shtrikman. A variational approach to the theory of the elastic behavior
of multiphase materials. J. Mech. Phys. Solids, 1963, Vol. 11

03:00-03:20pm, Nathan Briggs, Immersed Interface Method as a numerical solution to
the Stefan Problem

Mentor: Yekaterina Epshteyn

Abstract: The IIM algorithm in 1D [1] is based on well-known Crank-Nicolson algorithm
on a fixed grid with adjustments made to account for discontinuities of the derivative
at the interface, as well as adjustments to account for the moving boundary. In particular
it is shown what happens when the interface crosses grid points. The method is extended
to 2D as well, and results and conclusions are presented.

Finally the Stefan Problem is relaxed so the solution at the interface is no longer
known, but the jump discontinuity (if any) at the boundary is known. The problem of
shape optimization is formulated as the moving boundary problem similar to Stefan
Problem, and the possibility of using the IIM to solve it is explored.

[1] Zhilin Li and Kazufumi Ito. The immersed interface method, volume 33 of Frontiers
in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia,
PA, 2006. Numerical solutions of PDEs involving interfaces and irregular domains.

(Math 4800: Selected Numerical Algorithms and Their Analysis)

03:20-03:40pm, Stephen Durtschi, Evolutionary Algorithms Applied to a Pathing Game

Mentor: Yekaterina Epshteyn

Abstract: An approximate solution to the Traveling Salesman problem has been successfully
developed using both Genetic Algorithms and Swarm Algorithms. This work applies similar
techniques to attempt to find optimal paths for the board game Ticket to Ride, a game
which requires players to choose a best path between cities. Strengths and weaknesses
of both methods are discussed.

(Math 4800: Selected Numerical Algorithms and Their Analysis)

03:40-04:00pm, Wenyi Wang, Imaging defects in a plate with waves

Mentors: Fernando Guevara Vasquez and Dongbin Xiu

Abstract: This project is about detecting and imaging damage (such as cracks) in a
plate by using ultrasonic waves. The wave source is an ultrasonic transducer which
is carried by a robot that can move on the plate. The data used for imaging are the
waves recorded at a receiver, which is another ultrasonic transducer carried by the
robot. We simulated wave propagation in the plate by keeping only the first two Lamb
modes and using the Born or linearization approximation. The imaging was done using
Kirchhoff migration and assumed an a priori known path for the robot on the plate.
The application of this research is to aircraft structural health monitoring and is
done in collaboration with Thomas Henderson (School of Computing, University of Utah).

04:00-04:20pm, Wyatt Mackey, Determinant of genuine representations of the spin cover
of SnSn

Mentor: Dan Ciubotaru

Abstract: Deriving the character table for the symmetric group can be done in many
ways. One interesting way is by writing it as a product of matrices. One result of
this, if we are careful about the construction of the matrices, is a simple formula
for the determinant of the character table. This paper presents such a construction.
We then investigate the determinant of the character table of a spin cover of the
symmetric group, and present preliminary computations, and an interesting conjecture
for further computations. We finally investigate possible methods of proof for our
conjecture by relating it back to the character table of the symmetric group.

Mathematics Department

Undergraduate Research Symposium Fall 2014

Monday, December 15 - 10:30am-12:30pm - Room LCB 323

10:30-10:50am, Alex Henabray, Numerical Solutions for the Heat Equation

Mentors: Yekaterina Epshteyn and Jason Albright

Abstract: Differential equations play an important role in science and engineering
because they are employed to model many natural phenomena, including diffusion. Diffusion
is defined as the movement of a substance (flow of heat energy, chemicals, etc.) from
regions of high concentration to regions of low concentration. In this talk, I will
consider the heat equation, which is one example of a model for diffusion processes.
For instance, I will use the heat equation to model heat transfer along a metal rod
and I will present two methods to approximate solutions of the heat equation numerically.

10:50-11:10am, Nathan Simonsen, Simulation of Texas hold'em

Mentor: Stewart Ethier

Abstract: TBA

11:10-11:30am, Michael Primrose, Imaging with waves in random media

Mentor: Fernando Guevara Vasquez

Abstract: We explored different imaging functionals to determine the placement of
wave sources in a random medium. We were interested in generating images that had
good resolution and statistical stability. The first method we explored was Kirchhoff
migration. This method varied greatly from one realization of the medium to the next.
Time reversal was implemented next with better results, however, the method assumes
a perfect knowledge of the medium which is unrealistic. Then we used interferometric
imaging which exploits the statistical stability of the cross-correlation to generate
images. The images lacked range resolution. To get a better range resolution, we worked
with coherent interferometry (CINT). CINT introduces additional constraints on the
cross-correlations resulting in better range resolution. The statistical stability
of CINT was then studied as we varied the constraints.

(Supported by NSF DMS-1411577)

11:30-11:50am, Wenyi Wang, Imaging in a homogeneous aluminum plate by using ultrasonic
waves

Mentor: Fernando Guevara Vasquez

Abstract: This project is about detecting and imaging damage (such as cracks) in a
plate by using ultrasonic waves. The waves are generated by a source (an ultrasonic
transducer) that is part of a robot that can move on the plate. The waves traveling
in the plate are recorded at a receiver (another ultrasonic transducer) that is also
carried by the robot. The imaging method we use is Kirchhoff migration and we do a
rigorous resolution study of this imaging method for two different configurations
of the source/receiver pair, assuming the robot follows a straight path and that certain
length scalings hold. Our results reveal that one of the setups gives images with
much better resolution than the other one. Imaging on other paths is illustrated with
numerical experiments. The application of this research is to aircraft structural
health monitoring and is done in collaboration with Thomas Henderson (School of Computing,
University of Utah).

11:50-12:30pm, Jin Zhao and Yu Tao, Construction of Optimal Portfolio Strategies

Mentor: Jingyi Zhu

Abstract: One of common practices in financial industry to calculate correlation between
different stock returns is to use daily closed prices. In this project, we propose
new methods to compute correlation matrix that uses real stock data based on transaction
level. We use both Brownian bridge and linear interpolation to interpolate prices
of all stocks at any given specific time.

Tuesday December 16 - 1pm-3pm - Room LCB 225

01:00-01:20pm, Trevor Dick, On the ideal dynamic climbing rope

Mentor: Davit Harutyunyan

Abstract: We consider the rope climber fall problem, the simplest formulation of which
is when the climber falls from a point being attached to one end of the rope and the
other end of the rope is attached to the rock. The problem of our consideration is
to minimize the maximal value of the force that the climber feels during the fall,
which is the tension of the rope at the point attached to the climber. Given the initial
height of the rock attached point, the altitude and the mass of the climber and the
length of the rope, we find the so called best dynamic rope in the framework of nonlinear
elasticity.

01:20-01:40pm, Michael Senter, Numerical investigation of First Passage Time through
a Fluid Layer

Mentor: Christel Hohenegger

Abstract: Modeling the motion of passive particles in a viscous fluid is well studied
and understood. Extensions to passive motion in a complex fluid which exhibits both
viscous and elastic properties have been developed in recent years. However, questions
remain on the characterization of mean-square displacement and mean first passage
for different theoretical models. We will present a statistically exact covariance
based algorithm implemented in parallel C++ to generate particle paths to answer these
questions.

(Supported by NSF DMS-1413378)

01:40-02:00pm, Michael Zhao and Nathan Briggs, An inverse problem: finding boundary
fields which produce breakdown

Mentor: Graeme Milton

Abstract: We study the inverse problem of finding the lower bound on the maximum value
the electric field/stress of a two-phase body of unknown interior geometry, fields
which exceed these bounds will cause breakdown. We apply the splitting method to obtain
bounds without using variational principles. Knowing the materials' properties (density,
threshold for breakdown) and weight of the object, we are able to determine necessary
criteria for breakdown. We also find optimal EΩEΩ inclusions for which these bounds are sharp. The EΩEΩ inclusions are computed by finding geometries such that the field is uniform in the
inclusion. The geometry of the inclusions can then be determined by enforcing boundary
conditions and physical constraints.

(Math 4800: An inverse problem: finding boundary fields which produce breakdown)

02:00-02:20pm, Curtis Houston, Algebraic varieties associated to simple statistical
models

Mentor: Sofia Tirabassi

Abstract: TBA

02:20-02:40pm, Braden Schaer and Anand Singh, A Model for Restricted Diffusion of
Evoked Dopamine

Mentors: Sean Lawley and Heather Brooks

Abstract: A simple gap diffusion model is commonly used for short, unaided intercellular
transport. One prominent example is dopamine diffusion in neural synapses, where intercellular
space is often a crowded environment consisting of extracellular networks, biological
waste, macromolecules, and other obstructions. The traditional model fails to represent
many unique characteristics of restricted diffusion in complex cellular environments.
We compare existing mathematical models of gap and restricted diffusion against data
acquired by fast-scan voltammetry of evoked dopamine in the striatum of rat brains.
Finally, we propose a more physically realistic model that accounts for the spatial
structure of this system. (Presented by Braden Schaer.)

(Supported by Mathematical Biology RTG grant)

Wednesday December 17 - 10:50am-12:30pm - Room LCB 323

10:50-11:10am, Marie Tuft and Oliver Richardson, Mathematical model insights into
neurobiology

Mentors: Sean Lawley and Heather Brooks

Abstract: TBA

(Supported by Mathematical Biology RTG grant)

11:10-11:30am, Alexander Beams, Epidemiology of dual-strain bacterial infections

Mentors: Fred Adler and Damon Toth

Abstract: Antibiotic resistance in bacteria has big consequences for medicine and
public health. A compartmental deterministic ODE model (a not-too-distant relative
of the familiar SIS) shows how two different bacterial strains interact, and what
makes this model stand out is the assumption that people can be infected with both
strains at once. In some scenarios this class of infectives can help antibiotic resistance
persist when it otherwise would not. Other situations lead to lower overall levels
of resistance than would otherwise be expected. The idea of people being infected
with two strains at once has empirical evidence supporting it, but in the model it
creates an inconsistency when analyzed in the context of evolutionary epidemiology.
When strains are highly differentiated the inconsistency may not be a critical issue,
but future work will nevertheless deal with finding a satisfactory fix for the model's
apparent shortcomings. An area of future study includes describing the dynamics when
a hospital is embedded in a community.

11:30-11:50am, Hitesh Tolani, Modeling the dynamics of influenza transmission in school-age
population of Utah

Mentors: Fred Adler and Damon Toth

Abstract: Dynamics of infectious disease transmission vary radically between smaller
vs. larger sized populations. Phenomenological models that well predict the average
outbreak sizes in cities generally do a poor job of simulating outbreaks in schools,
hospitals etc. Infectious diseases transmit primarily through direct contact between
individuals. Phenomenological models are usually based on simplifying assumptions
which overlook the population native contact-network dynamics. The Center for Disease
Control and Prevention, (CDC), quantified a contact-network in their project entitled
"Contacts among Utah's School-age Population", (CUSP). Mechanistic simulations on
CUSP network reveal that network dynamics dominate outbreaks in smaller populations.
In this project we're using the CUSP network to identify essential features of a contact-network.
These are necessary in order to construct simpler random-graph based epidemic models
which accurately simulate influenza outbreaks in school-age population of Utah.

11:50-12:10pm, Yang Lou, The PAM method for interface tracking

Mentor: Qinghai Zhang

Abstract: Interface tracking is an essential problem in numerically simulating multiphase
flows. At present, three main methods have been developed, namely front tracking methods,
level set methods and VOF methods. Despite the success of these interface tracking
methods, these methods have some shortcomings. A relatively new interface tracking
method which evolved from VOF methods is called the polygonal area mapping (PAM) method,
which represents material areas explicitly as piecewise polygons, traces characteristic
points on polygon boundaries along pathlines, and calculated new material regions
inside interface cells via polygon-clipping algorithms in a discrete manner. Translation
test and vortex test are performed to test the accuracy and efficiency of the PAM
method on a structured rectangular grid with a uniform grid size hh.

12:10-12:30pm, Kejia Zhu and Logan Calder, Damage in Lattice Model of Materials

Mentor: Andrej Cherkaev

Abstract: Lattice structures are often used to create discrete models of materials.
For engineering purposes, a lot of mathematics has been developed to model the behavior
of different types of materials as they experience different forms of stress: heat,
pressure, impacts, etc. Experiments have helped establish standards for when a material
may be critically damaged. What may be lacking though, is a model that describes the
changes in a material as it receives damage and a measure for how close a partially
damaged material is to critical damage. This paper will report our attempt to construct
a model for the distribution of forces on a stressed periodic lattice and for the
progression of damage as the lattice is over stressed. We do not attempt to describe
the progress of damage with in each edge, only the progress of damage within the lattice.
That is, we lay the ground work to investigate pattern of broken edges while the lattice
is over stressed. We will expound on our question and only discuss some of the ground
work necessary to begin simulations.

Mathematics Department

Undergraduate Research Symposium Spring 2013

Friday, April 26 Session - 2pm-4pm - LCB 323

02:00pm-02:20pm, Alessandro Gondolo, Characterization and Synthesis of Rayleigh-damped
elastodynamic networks

Mentor: Fernando Guevara Vasquez

Abstract: We consider damped elastodynamic networks where the damping matrix is assumed
to be a non-negative linear combination of the stiffness and mass matrices (Rayleigh
damping). We give here a characterization of the frequency response of such networks.
We also answer the synthesis question for such networks, i.e., how to construct a
Rayleigh damped elastodynamic network with a given frequency response. The characterization
and synthesis questions are sufficient and necessary conditions for a frequency dependent
matrix to be the frequency response of a Rayleigh damped network. We only consider
the non-planar case.

02:20pm-02:40pm, Ben DeViney, Suction Feeding on Multiple Prey

Mentor: Tyler Skorczewski

Abstract: Suction feeding is one of the most common forms of predation among aquatic
feeding vertebrates. There are several instances in nature and technology that show
numerous bodies in close proximity working together to reduce the effects of drag.
Here we simulate a suction feeding attack on multiple prey items in an attempt to
understand how the fluid dynamics changes between single and multiple prey cases.
To model a suction feeding attack, we numerically find solutions to the incompressible
Navier-Stokes equations on Chimera overset grids. In comparing the prey capture times
across various cases, our results show that additional prey have no effect. From these
results, we can conclude that suction feeding mechanisms are precise and highly localized
to the point where multiple prey items have no effect on the suction attack.

02:40pm-03:00pm, Alex Burringo, Carli Edwards, Trevor Myrick, Mathematical Modeling
of Molecular Motors

Mentors: Parker Childs and Ross Magi

Abstract: Kinesin and dynein are molecular motor proteins that transport cargo in
opposite directions along microtubules in cells. They are of interest in biophysics
because of their roles in DNA replication and intracellular transport. Despite considerable
experimental research concerning single motor dynamics, a clear theoretical explanation
for motor procession along cellular microtubule filaments is not present, especially
when many motors are present. Current research shows that it is common for multiple
dynein and kinesin motors to be attached to a single cargo. However, there is much
speculation as to how the cargo moves given this expected tug-of-war possibility.
To help reconcile current theories for procession, we created a stochastic tug-of-war
model of kinesin and dynein dynamics to simulate data for comparison with current
experimental data. Much of the semester was spent coding the molecular movement of
dynein and kinesin, with evaluation of stepping probabilities, diffusion force effects,
cargo-motor attachment forces, effect of multiple motors, motor-microtubule attachment
forces, and the addition of temperature variations. Because of the stochastic nature
of the code, it was run numerous times in parallel on a math department compute cluster.
Evaluation of model effectiveness will be done by statistical comparison of the simulated
and experimental results, specifically by Dr. Vershinin in the Physics Department,
to aid in the continuing research of molecular motor dynamics.

03:00pm-03:20pm, Brady Thompson, Binary Quadratic Forms

Mentor: Gordan Savin

Abstract: Binary quadratic forms are quadratic forms in two variables having the form f(x,y)=ax2+bxy+cy2f(x,y)=ax2+bxy+cy2. In this talk we'll discuss the reduction, automorphisms, and finiteness reduced
forms of a given discriminant. The main objective will be to present an interesting
proof about the relationship between automorphisms of forms and solutions to the Pell
equation.

03:20pm-03:40pm, Steven Sullivan, A Trace Formula for G2G2

Mentor: Gordan Savin

Abstract: An nn-dimensional representation of a group GG on a vector space VV is a homomorphism from GG to GL(V)GL(V). For our purposes, we consider an irreducible representation to be a representation
which cannot be decomposed into the direct sum of smaller-dimensional representations.
Let HH be a subgroup of GG. The way in which irreducible representations of GG decompose into irreducible representations of HH is called branching. In order to calculate such branching, one must first obtain
a trace formula for each conjugacy class of HH in irreducible representations of GG. In this talk, we summarize the obstacles which must be overcome when attempting
to calculate such trace formulas for the conjugacy classes of G2(2)G2(2) in irreducible representations of G2G2. Then we show the methods we used to overcome these challenges and find the trace
formulas for the sixteen conjugacy classes.

03:40pm-04:00pm, Wyatt Mackey, On the inverse Galois problem for symmetric groups
and quaternions

Mentor: Dan Ciubotaru

Abstract: The Galois group was introduced in relation to the question of solvability
by radicals of polynomial equations. The inverse Galois problem asks if, given a finite
group GG, there exists a field extension of rational numbers such that the Galois group equals GG. This projects investigate explicit field extensions when GG is the quaternionic group Q8Q8 or a symmetric group.

Monday April 29 Session - 2pm-4pm - JWB 333

02:00pm-02:20pm, Sean Quinonez, Street fighters: a model of conflict in Tetramorium
caespitum

Mentor: Fred Adler

Abstract: Mathematical models provide a powerful tool for describing and analyzing
interactions between populations of organisms. Of interest are interactions among
eusocial populations, where simple rules at the individual level translate into complex
patterns of interaction within and between colonies. We present a model that describes
conflict between colonies of the ant Tetramorium caespitum, an ideal study species
due to their abundance and status as an urbanized species. Conflicts between T. caespitum colonies
arise during seasonal competition for territory. Our mathematical model tracks flow
of ants in each of two colonies as they recruit and travel to a conflict, where ants
search and grapple with enemies. The model will examine two mechanisms of motivation,
and study how these mechanisms control the location and size of the battle. These
recruitment mechanisms introduce delays that can produce oscillations in the location
and make-up of the battle, model predictions that correspond to empirical measurements
of ant battles.

02:20pm-02:40pm, Skip Fowler, Mathematical modeling of epidemics using a probabilistic
graph

Mentor: Fred Adler

Abstract: In order to predict the number of survivors, we study a stochastic model
of an epidemic that includes with transitions infection and recovery. This approach
can be represented as a graph with probabilities on the edges between states. The
distribution of the number of survivors can be found from the set of probabilities
associated with states where the number of infected people equals 00. We find that this probability distribution breaks into two distinct components.
We compute the probability of rapid epidemic extinction with many survivors, QQ, using methods from stochastic process theory. If the epidemic takes hold, we use
an SIR model without vital dynamics, a deterministic set of differential equations
describing the spread of an epidemic, which approaches zero only when the epidemic
begins and ends. We compare the mean of the probabilities for states after QQ to determine if the mean matches the end point of the epidemic in the deterministic
SIR model, and check whether their distribution is approximately normal. We developed
computer simulations to calculate and plot the probabilities and compare with the
mathematical analysis of QQ and the SIR model. Our discoveries during the course of this project include finding
the Catalan numbers and Catalan's triangle within the graph structure and finding
a knights move based on the number of steps to any state within the graph structure.
The probabilities after QQ are not normal due to increasing kurtosis as population size and infection rate are
increased, although the mean is well-approximated by the differential equation. We
are searching for a higher dimensional system that captures this deviation from normality.

02:40pm-03:00pm, Kyle Zortman, Modeling the Progression to Invasive Cervical Cancer

Mentor: Fred Adler

Abstract: This talk will discuss the biological development of cervical cancer in
the body and the difficulties of stochastically modeling this process. Because progression
of cervical cancer is time and stage dependent, simple stochastic modeling approaches
fail. A solution is then proposed using a modification of the Gillespie Algorithm.
Simulation will be compared to real data and more deterministic solutions. Finally,
the system will be analyzed for sensitivity and applications will be discussed.

03:00pm-03:20pm, Boya Song, Horizontal fluid transport through Arctic melt ponds

Mentor: Ken Golden

Abstract: The albedo or reflectance of the sea ice pack is an important parameter
in climate modeling. Being able to more accurately predict sea ice albedo can significantly
increase the reliability of climate model projections. Sea ice albedo is closely related
to the evolution of melt ponds on the surface of Arctic sea ice. In the process of
melt pond evolution, a critical component is the drainage of surface melt water through
holes and cracks that exist naturally in ice floes. The speed and range of the drainage
is related to how easy it is for fluid to flow horizontally over the ice. This horizontal
fluid conductivity can be modeled using random resistor networks. The horizontal conductivity
of the melt pond configurations plays a key role in modeling melt pond evolution,
and therefore sea ice albedo. In this lecture we will discuss efforts at mapping melt
pond configurations onto random graphs, and then developing efficient algorithms to
calculate the effective conductivity of these random graphs.

03:20pm-03:40pm, Brady Bowen, Random Fourier Surfaces: Applications to Arctic Melt
Pond Modeling

Mentor: Ken Golden

Abstract: Arctic melt ponds are important in climate models because they modify the
level of reflectance, or albedo, of the sea ice pack. We decided to model this system
using a two dimensional random Fourier series expansion. Level sets of these random
surfaces give shapes that closely resemble the melt ponds found in the Arctic. Random
Fourier surfaces also have multiple controlled variables that can be changed to determine
which length scale dominates the generated surface. We modified these constants in
order to obtain a better fitting match of the observed melt ponds. We then calculated
area-perimeter data in order to find at what point the generated shapes went through
a fractal dimension transition, which is exhibited by real data obtained from the
Arctic melt ponds. It was also found that changing the variables which characterize
the Fourier surfaces altered where this fractal dimension shift occurred and how fast
the fractal dimension of the shapes changed.

For any questions, please contact the REU Director:

Fernando Guevara Vasquez

Office: LCB 212

Email: fguevara@math.utah.edu

Phone: 801 581 7467

Mathematics Department

Undergraduate Research Symposium Fall 2013

Monday December 16 Session - 2pm-4pm - LCB 222

02:00-02:20pm, Brady Thompson, The Discriminant Relation Formula

Mentor: Gordan Savin

Abstract: One of the most significant invariants of an algebraic number field is the
discriminant. The discriminant is an idea we're all familiar with since basic algebra,
but idea can be generalized for number fields and becomes an ubiquitous tool in number
theory. I will present general definitions and properties of the discriminant of an
algebraic number field. I will also discuss the Discriminant Relation Formula, which
is a tool that we can use to determine certain properties about a tower of fields.
These properties include finding the ring of integers in a number field and verifying
whether a field is a Hilbert class field. I will provide an example to illustrate
it's usefulness.

02:20-02:40pm, Drew Ellingson, Tropical Analogues of Classical Theorems

Mentor: Steffen Marcus

Abstract: Tropical Geometry is a field derived from the study of the worst possible
degenerations of classical Algebraic Geometry. This talk will develop the bare-bones
concepts necessary to start thinking about Tropical Geometry. We then build intuition
into the subject by stating and investigating a few tropical analogues of famous theorems
in Algebraic Geometry. We will talk about Bézout's Theorem, and then move on to the
more challenging group law on cubic curves

02:40-03:00pm, Drew Ellingson, Computation of Top Intersections on the Moduli Space
of Curves

Mentors: Steffen Marcus and Drew Johnson

Abstract: Current algorithms for computing Top Intersections on the Moduli Space of
Curves do not satisfactorily handle boundary classes. The goal of the research I have
conducted with Steffen Marcus and Drew Johnson is to implement algorithms in the mathematics
software SAGE to compute intersection numbers for arbitrary boundary strata. In this
presentation, I will introduce the Moduli Space of Curves, its compactification, and
the dual graph of a nodal curve. I will then talk about some combinatorial and graph-theoretic
problems that arise in computation.

03:00-03:20pm, Jonathan Race, Explorations in GARCH(1,1) Processes

Mentor: Lajos Horváth

Abstract: It is often the case in financial and economic data that we need models
which account for dynamic volatility, or variance. GARCH processes are a relatively
recent development in such non-linear modeling. In this presentation I will review
the application of GARCH processes and some necessary conditions for their existence.

03:20-03:40pm, Nathan Briggs, Optimal three material design on the microstructure
and macrostructure scale

Mentor: Andrej Cherkaev

Abstract: The problem of optimal three material composite as formulated and solved
by Cherkaev and Dzierzanowski is investigated [1]. Namely stress energy plus cost
is minimized for an elastic body loaded on the boundary consisting of a strong and
expensive material, a cheap but weak material, and a void. This minimization finds
the optimal microstructures by solving a multivariable nonconvex minimization problem
which is reduced to determination of the quasiconvex envelope of a multiwell Lagrangian,
where the wells represent materials' energies plus their costs; the quasiconvex envelope
represents the energy and the cost of an optimal composite [2]. After finding the
microstructures the problem of optimal design of a body with these microstructures
is investigated. The main focus is on a special case corresponding to a specific cost
of the weak material. Finally the roll of each material in the design is investigated
and applications are discussed. This work is in collaboration with Grzegorz Dzierzanowski.

[1] A. Cherkaev and G. Dzierzanowski. Three-phase plane composites of minimal elastic
stress energy: High-porosity structures. International Journal of Solids and Structures,
50, 25-26, pp. 4145-4160, 2013.

[2] A. Cherkaev. Variational Methods for Structural Optimization. Springer

03:40-04:00pm, Sophia Hudson, Exploring 2D Truss Structures Through Finite Element
Simulation

Mentor: Andrej Cherkaev

Abstract: Lattices in the Euclidean plane can be modeled as a collection of nodes
and edges, forming a graph, with nodes corresponding to intersections between the
trusses modeled by the edges. The problem of our particular interest is that of understanding
what happens to these structures when physical properties are applied to the lattice.
In this project, we apply forces to the boundary nodes of n x n truss structures,
modeled as connected collections of equilateral triangles. We use a finite element
model to understand the stresses and strains on the trusses and to visualize the displacement
of nodes and edges from their initial conditions.

Tuesday December 17 Session - 2pm-4pm - LCB 222

02:00-02:20pm, Logan Calder, Fractal Models of Finance

Mentor: Jingyi Zhu

Abstract: Fractals have already been used to solve technological problems in communication,
and the fractal power of modeling natural features is widely known. Fractals have
been used to create realistic animated landscapes and special effects in movies. The
complex features found in living organisms can be recreated by repeating a simple
pattern. Even in aspects of more modern systems, such as the risk in financial markets,
fractals provide more understanding of reality than standard models.

Typically, models of finance have been based on the normal probability distribution.
The data though, doesn't fit the model. There are two many big changes in market prices
to allow for an easy model to come from the normal distribution. The normal distribution
also doesn't allow for dependent events. Instead, the fractal dimension of market
graphs may better help us understand the dependence of price changes on each other
and therefore allow us to predict more accurately the variance of price changes over
time.

The short term and long term dependence of price changes can be determined in one
of two ways: Finding the fractal dimension of the graph of market prices over time,
or plotting the log of variance of price changes versus the log of different time
intervals.

We found that price changes in gold have short term dependence over a period of about
100 days. This allows for easier determination of variance of changes over different
time intervals.

02:20-02:40pm, Michael Senter, Random Motion in Media with Memory

Mentor: Christel Hohenegger

Abstract: Robert Brown discovered random particle motion in the 19th century. We will
discuss the model developed by Langevin to describe this motion, as well as the results
of Ornstein and Uhlenbeck. We will then proceed to look at random motion in media
with memory.

02:40-03:00pm, Camille Humphries, Numerical Methods for the Advection Equation: Comparison
of Lax-Friedrichs and Central Schemes

Mentor: Yekaterina Epshteyn

Abstract: This presentation will include an introduction to the advection equation
and will focus on two numerical approximation schemes. The structure of the Lax-Friedrichs
and Central Schemes will be presented and explained. A comparison of the accuracy
and error ratios in both schemes will be shown for a test function.

03:20-03:40pm, Ryan Durr, The Quantification of Exit Times for Varying Fluid Models

Mentor: Christel Hohenegger

Abstract: This project begins with a classic theoretical approach to modeling using
the Langevin equation. This model assumes that there is no lasting effect from the
fluid on the kinematics of a particle. The advantages of this approach is that there
is a known analytic solution that can verify the mathematical simulation. The concluding
portion of this project is to model the kinematics of a particle that is traversing
a fluid with lasting effects and to quantify its exit times. This model does not have
analytic solution. The goal of this research is to quantify the exit time for the
different fluid models.

03:40-04:00pm, Wyatt Mackey, On the McKay Graphs of the Projective Representations
of SnSn

Mentor: Dan Ciubotaru

Abstract: The McKay correspondence details a specific connection between the finite
automorphisms of ℝ3R3 and the Dynkin diagrams of certain Lie algebras. In particular, one creates the ``Mckay
Graphs" by observing multiplicities of representations in the tensor products of the
representations of a group with its spin representation, then using this to create
a weighted, directed graph. Applying this process on the projective covers of the
finite automorphisms of ℝ3R3, we achieve equivalent graphs to certain interesting Dynkin diagrams.

The finite groups of automorphisms of ℝ3R3 correspond to the cyclic groups, the dihedral groups, and A3, A4, S3,A3, A4, S3, and S4S4. This paper is interested in examining possible patterns in the McKay graphs of the
projective covers of the symmetric group SnSn for larger nn; in particular, we examine the graphs of n=5n=5 and n=6n=6. To this end, we wrote a program capable of doing all necessary computations to draw
the McKay graphs, given the character table of the group. We did not find similar
results to McKay's correspondence, however. Interestingly, we find that the McKay
graph of the projective cover of S5S5 is non-planar, quite different from the cases of n≤4n≤4, wherein all of the graphs were trees. Surprisingly, this does not hold for the projective
cover of S6S6, which is again planar.