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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 R or 𝕡Qp, 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 Q this is often much more difficult. The Hasse-Minkowski theorem is a powerful tool that draws a connection between equivalence classes in local fields with equivalence classes in global fields. In this presentation, we will focus on the particularly interesting case of equivalence classes of ternary forms.

Kira Parker
Mentor: Braxton Osting
Ranking Methods in Competitive Climbing
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 2ℓ2-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 EΩ inclusions for which these bounds are sharp. The EΩ 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 3R3 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 3R3, we achieve equivalent graphs to certain interesting Dynkin diagrams.
The finite groups of automorphisms of 3R3 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 n4n≤4, wherein all of the graphs were trees. Surprisingly, this does not hold for the projective cover of S6S6, which is again planar.


 

Last Updated: 10/2/20