What follows is a list of some of the projects that faculty members in the department of mathematics have suggested as suitable for undergraduate research projects. Some of the projects that are listed were suggested some time ago, but faculty typically are still willing to work with students on these or a related project.
Students who wish to participate can register and receive credit for an independent study, or, may be able to obtain URA funding to get paid to work on these projects.
Details of the project requirements will be worked out between the faculty supervisor and the student. Some of these projects require little background and are suitable for freshmen or sophomores, while others require knowledge of linear algebra, ordinary differential equations, or group theory. This list is by no means exclusive: any student with a particular interest in some area of research is encouraged to seek out a faculty supervisor, or to contact the URA Program Coordinator for help on finding a suitable faculty member.
Quick links to detailed descriptions below
- Tom Kennedy, Bridges and Self Avoiding Random Walks
- Stephen Gagola How are Loops like Groups?
- Jim Cushing, Modeling ecology competition
- Chris Bergevin, Modeling Octoacoustic Emissions
- Moysey Brio. Computation for optical pulse propagation in dispersive, active and nonlinear media.
- Joceline Lega, Projects in Mathematical Modeling (off page)
- Doug Pickrell. Scissors congruence.
- Robert Indik. Phase reconstruction from near and far field images.
- D. E. Myers, R. Harris, M. K. O'Rourke and S. Rogan. Use of GIS in analyzing environmental cancer risks as a function of geographic scale.
- Dan Madden. Mahler measures of polynomials of small degree.
- Tom Kennedy. Simulating self-avoiding random walks.
- T.W. Secomb. Modeling microvascular networks.
- David Lomen. Mathematical modeling.
- Marek Rychlik. Integrable and non-integrable billiards.
- Marek Rychlik. Speech and attractors.
- Marek Rychlik. Neural networks and genetic algorithms.
- Joseph Watkins. Population biology — modeling and education.
Bridges and Self Avoiding Random Walks
Advisor: Tom Kennedy
Prerequisites: Basic probability concepts, such as the density function of a random variable. The project will involve computer simulations, but the program to generate self avoiding walks already exists. The programming needed for the project can be done in Matlab, so some familiarity with Matlab (or an eagerness to learn) would be helpful.
Summary: To construct a random walk, at each time increment you randomly choose one of the four directions (north, east, south or west) and take a step in that direction. In a self-avoiding random walk (SAW) you are not allowed to visit the same place more than once. A SAW confined to the upper half plane can be written as a concatenation of "bridges." The bridges are separated by horizontal lines with the property that the SAW crosses the line only once. A problem of current interest to researchers working on SAW's is the statistical properties of the bridges. For example, what is the average height of a bridge? How are these heights distributed? How does this distribution relate to recent results on bridges in the Schramm-Lowener evolution (SLE)? The goal of this project is to understand the theorem on the decomposition of the SAW into bridges, understand the distribution of the bridges through simulations and then write a research paper about their distribution and its relation to bridges in SLE.
How are Loops like Groups?
Advisor: Stephen Gagola
Prerequisites: Linear algebra, some familiarity with groups would help but is not required.
Summary: If we start with real numbers and add a square root of -1, we get the complex numbers but we loose ordering. You can throw in another square root of -1 in order to form something called the quaternions. In this case you loose commutativity, namely, ab is not always ba. The set of invertible elements of the quaternions form what we call a group. The study of such groups led to important fundamental results in the theory of finite groups (an incredibly useful and rich area of mathematics). By extending the quaternions a similar way you can enlarge to get the octonions, but now you loose even the associative law for multiplication. However, there remains a weaker law of associativity called the Moufang identity ( a((bc)a)=(ab)(ca) ). In this project we will investigate some of the marvelous properties that hold for the set of inverible elements of split octonion algebras which we call a loop (Moufang loop). My current research involves finding which universal truths (theorems) for groups still hold for Moufang loops. See www.theoremoftheday.org and myweb.lsbu.ac.uk/~whittyr/MathSci/TheoremOfTheDay/GroupTheory/Loops/TotDLoops.pdf
Modeling ecological competition
Advisor: Jim Cushing
Prerequisites: A first course on differential equations & some familiarity with matrices. Some proficiency with a programming language would be useful (MatLab, Maple, Mathematica, etc.).
Summary: A fundamental goal in ecology is to gain an understanding of how ecosystems are built and maintained. In this regard, it is important to understand how several species can coexist while competing for limited resources or otherwise interfering with each other’s well being. To attain diversity within an ecosystem numerous species must find ways to coexist while in ecological competition. The life cycle strategy adopted by individuals (the timing of developmental stages, when and how often to reproduce, and so on) is an important component that bears on the ultimate survival or extinction of a species. To give but one example: why do some species of cicadas adopt a life cycle in which adults emerge only once in every 17 years? How is this a successful strategy? Why do other species not adopt such a strategy, but instead overlap their generations? Mathematical models for the dynamics of competing species that include a description of the life cycles of individuals can help determine which strategies result in increased chances for coexistence and which lead to probable extinction.
Students in this project will have the opportunity to study and derive models of competition (and/or predation, parasitism, etc.), to analyze them mathematically, and/or to carry out simulation studies on computers.
Modeling Octoacoustic Emissions
Advisor: Chris Bergevin
Prerequisites: Math 254 or 355, Some experience with Computations (e.g. Matlab, Java or C..) is helpful
Summary: The ear acts as a
biological microphone. Its most basic function is to sense incoming
acoustic stimuli from the world around us and convert that mechanical
information into electrical signals which travel via the auditory nerve
to the central nervous system. Thus, it might seem a bit surprising
that a healthy ear actually produces and emits sound (which can be
measured via a microphone in the ear canal). These sounds are called
otoacoustic emissions (OAEs) and can arise either spontaneously or by
an evoking external stimulus. These emissions are generated in the
inner ear (cochlea) and are presumably a result of the mechano-electro
and electro-mechano transduction mechanisms responsible for converting
sound into neural impulses. Since the cochlea is encased in very hard
bone and the sensory structures are highly sensitive (i.e. easily
subject to damage), direct physiological observation of the inner ear
is difficult. However, OAEs provide a non-invasive window into the
inner ear and allow us to better understand the underlying physiology.
One of the striking features of evoked OAEs is there non-monotonic
dependence upon the input stimulus intensity. The basis for this
nonlinear behavior is currently poorly understood. This project will
focus on elucidating the physiological mechanisms underlying the
non-monotonic behavior via a computational approach. An initial step
will be to look at predictions of various proposed models and compare
those to real OAE data. This project is well suited for students who
are interested in applied math and its applications in biology/sensory
physiology. While a background in biology is not necessary, some
experience with computation (e.g. basic coding in Matlab) will be
helpful.
Scissors congruence
Advisor: Doug Pickrell
Prerequisites: A good grasp of vector calculus and linear algebra.
Summary: Given a polygon in the plane, one can compute the area by cutting the polygon into triangles and rearranging them into a rectangle. Thus every polygon in the plane is scissors congruent to a rectangle of equal area. In the 1840's Gauss asked if an analogous statement was true for a 3 dimensional simplex (i.e. a solid with planar faces) in 3-space. In 1900 Dehn proved that the answer is “no”, by showing that another number (Dehn invariant, the sum of the product of the lengths of edges and corresponding dihedral angles), in addition to volume, is preserved under scissor congruence. Sixty years later Jensen proved that two 3-simplices are scissors congruent if and only if they have the same volume and Dehn invariant. In turn Jensen asked if the analogous statement was true for simplices in spheres and other non-Euclidean geometries. Many of these problems are unresolved.
In this project we will investigate these and related issues involving volume of simplices in Euclidean and spherical 3-space.
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Phase reconstruction from near and far field images
Advisor: Robert Indik
Prerequisites: Linear algebra, and strong familiarity with complex number. Computer experience helpful.
Summary: Light such as that that comes from lasers can be described in terms of intesity and phase. When images are captured, they record the intesity of the light. If the phase can be found as well, it is possible to recover full information about the light, and to undo the effects of poor focusing or of blurring due to turbulence in the air. I am investigating the practicality of reconstructing phase information from images taken in two different focal planes (the near filed and the far field).
This problem can be recast into the question: Given the magnitude of a complex function f(x,y), and the magnitude of its Fourier transform, can one reconstruct the original function?
If one specializes to the case of discrete functions and images such as can be captured electronically, the question becomes: Given the magitudes of the complex entries in an N by N array, and the magnitudes of the discrete Fourier transform of that array, can one reconstruct the complex array?
Students participating in this project will explore these questions initially in very small test cases, and test out potential algorithms for efficiently solving for phase.
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Use of GIS in analyzing environmental cancer risks as a function of geographic scale
Advisors: Donald Myers (Mathematics), Robin Harris (College of Public Health, Program in Epidemiology), Mary Kay O'Rourke (Division of Community and Environment), and Séumas Rogan (Epidemiology Program). (Please contact Donald Myers to discuss this project.)
Prerequisites: Linear algebra, computer experience (word processing, spreadsheet, statistical software or willingness to learn, GIS or willingness to learn), probability theory and statistics desirable.
Summary: The Atlas of Cancer Mortality in the United States: 1950-1994 tabulates the distribution of cancer in the United States by county. The creation and utility of the atlas requires adoption of various assumptions. In particular, it implies that the cancer risk is equal across a county, and that the reported cancer risk represents the cancer risk for all residents of that county. If the county is fairly homogeneous in both the characteristics of the underlying population and exposure risks, then this assumption may be reasonable. However, for many Western U.S. states, this assumption may be inappropriate. These states often are divided into only a few counties, each covering large geographic spaces and having uneven distributions of populations. These characteristics make it unlikely that county-level statistics fairly represent the range of actual county experiences.
The overall goal of this research project, is to examine the geographic variation in the association between cancer risk and arsenic. Arsenic exposure may be a causal agent in the development of bladder, lung, kidney, and skin cancers. Furthermore, arsenic is known to vary across geographic locations. GIS technology has made it more feasible to link multiple sources of descriptive attribute information for various geographic levels with health outcome data.
Several geographically delineated data sets exist in Arizona that allow for exploration of the relationship between arsenic exposure and cancer occurrence. Geocoded cancer incidence and mortality data are available from the Arizona Cancer Registry for bladder, kidney, and lung cancers. Skin cancer data are available from a completed population-based case-control study. Arsenic concentrations are available from a multimedia, multipathway survey conducted in Arizona.
The researchers will examine cumulative, aggregate, and cumulative-aggregate arsenic exposures, as potential doses, with the incidence of specific cancers. They will determine the homogeneity of the associations across various geographic scales.
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Mahler measure of polynomials of small degree
Advisor: Dan Madden.
Prerequisites: None.
Summary: Attached to a polynomial with integer coefficients is a certain number, called the Mahler measure, which is computed in terms of the roots of the polynomial. There are also other ways to define this measure. Mahler measure is not very well understood, and the idea of this project is to explore it by re-establishing some known results using new techniques. To learn a little more, see our detailed project description.
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Simulating self-avoiding random walks
Advisor: Tom Kennedy
Prerequisites: Basic probability concepts, such as the distribution function of a random variable. Familiarity with some topics from MATH 468, in particular Markov chains, would be useful, but is not essential. Since the project will consist of computer simulations, programming skill is needed. However, programs to simulate these self-avoiding walks can be surprisingly short.
Summary: To construct a random walk, each time you take a step you randomly choose one of the four directions north, east, south or west. In a self-avoiding random walk you are not allowed to visit the same place more than once. While very little has been proved about these walks, they can be simulated rather easily on a computer, and so can be studied numerically. Here are a couple of pictures of computer generated self avoiding walks:
If you run an ordinary random walk for a long time and then look at it from far away, it looks like a stochastic process called Brownian motion. Now suppose you do the same thing for self-avoiding random walks. Does it look like a stochastic process, and if so what can you say about the process? The goal of the project is to study this question empirically, i.e., by simulating lots of self-avoiding walks and seeing what we can say.
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Modeling microvascular networks
Advisor: Timothy Secomb
Prerequisites: Ordinary differential equations, linear algebra.
Summary: The microcirculation is an intricate network of tiny blood vessels that carries nutrients to every part of a living tissue. Current projects are aimed at developing mathematical models for several aspects of the system's function. For example, networks continually adjust their structures in response to local signals, during growth and when metabolic requirements change. This process can be modeled using a system of ordinary differential equations to describe changes in vessel diameters. The problem is to determine what types of responses can lead to stable, adequate network structures consistent with experimental observations.
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Mathematical modeling
Advisor: David Lomen
Prerequisites: None.
Summary: Topics chosen by the student in areas of mathematical modeling dealing with fluid motion (soil physics, physiology, engineering), with differential equations (pharmacokinetics, biological, chemical, or physical processes, ...) or with different ways to teach mathematics or the development of innovative teaching materials.
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Computation for optical pulse propagation in dispersive, active and nonlinear media
Advisor: Moysey Brio
Prerequisites: MATH 223, 215, and 355 useful.
Summary: Code development, grid generation and optimization of finite element and finite differencing with applications to computation of guided modes in photonic structures, pulse propagation dispersive, active and nonlinear media, phase locked mode propagation in semiconductor lasers.
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Integrable and non-integrable billiards
Advisor: Marek Rychlik
Prerequisites: Basic calculus sequence required, proficiency in a programming language required, Fourier analysis helpful, probability helpful, linear algebra helpful.
Summary: Study the motion of a billiard ball on a table of an arbitrary shape. Only elliptic tables are known to be integrable. Other tables are conjectured to be chaotic. In the course of this project the student would perform simple numerical experiments to measure the chaotic behavior of tables of various shapes.
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Speech and attractors
Advisor: Marek Rychlik
Prerequisites: Calculus and linear algebra necessary, Fourier analysis helpful.
Summary: The student would collect samples of speech, using a computer with a sound card and a microphone. Subsequently, these samples would be visualized, using software like MATLAB. The individual sounds, like the ones produced by saying “aaaah” or “oooh” produce patterns which are referred to as attractors. These patterns will be subjected to a mathematical analysis to explore the possibility of distinguishing between them.
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Neural networks and genetic algorithms
Advisor: Marek Rychlik
Prerequisites: Discrete math, programming language.
Summary: A neural network is a system that will produce “correct” responses to a range of inputs. The traditional method of training of neural networks is the back propagation algorithm and it is known to be very slow. The student will examine a new method of genetic breeding and compare it to the old method by performing numerical experiments.
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Population biology — modeling and education
Advisor: Joseph Watkins
Prerequisites: Varied.
Summary: My recent work involves interaction between the mathematics and biology communities. Under the auspices of the Southwest Regional Institute in the Mathematical Sciences, we are developing ways to communicate research ideas in population biology. This is now being pursued on the biology of bees and soon we will start investigating questions on HIV both its immunology and its epidemiology. Undergraduate projects here come in two types - one is to work with scientists and teachers to develop strategies for communicating these ideas to broad community.
At this moment, we are just starting to write software on bee population models. These models will become tools for high school students, beekeepers and for bee scientists. These groups are concerned about the Africanization of the European honey bee population and mite infestations of bee hives. We hope that these tools will help understand the nature of these events and suggest methods for remediation.
The HIV models will also require some software development. The issues in the immunology are optimal drug protocols for improving the lives of those with HIV. The epidemiological issue is to investigate what changes in behavior will have the biggest impact in the spread of the virus.
Separated from my activities with the Regional Institute, I am also looking into issues involving enzyme kinetics. When the number of enzyme molecules and substrate molecules was high, mass action equations were an adequate mathematical model of the dynamics of the enzyme reactions. As biochemistry is learning more about enzymatic reaction in which the number of molecules is smaller, the stochastic or random effects are playing a more important role. Here we will be writing programs that find the importance of these effects and looking into the experimental data for verification.