URA Research Project Ideas
What follows is a list of some of the project topics that faculty members in the department of mathematics have suggested as suitable for undergraduate research projects. 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. Students are encouraged to contact the URA Program Coordinator for help finding a suitable faculty research mentor.
Students participating in undergraduate research for credit must submit a proposal form through the math academic office. Stop by the window at Math 108 once you have lined up your project advisor and topic.
Project ideas list is not exhaustive  there are additional faculty who are interested in working with undergraduates.
Name  Research Area(s)  Prerequisites  Honors Thesis?^{*}  URA for Credit?  URA for Pay?**  Last Updated 

Numerical Simulation of Waves in Optics, fluids and solids; 
introductory numerical analysis, basic physics/optics and computer programming. 
Yes 
Yes 
Yes 
6/13/2013 

Vegetation pattern formation in dryland ecosystems: In waterlimited regions (think subSaharan Africa, Australia, southwestern US), competition for water resources results in the formation of vegetation patterns. Interestingly, the type of pattern appears to depend on the topography; on flat ground, vegetation arranges in disorganized spot or labyrinth patterns, while on sloped terrain, the vegetation aligns in stripes or arcs. The dynamics of these patterns can be modeled by PDEs describing the interplay of vegetation and water resources. The goal of this project is to explore these models and study the effect of terrain and climate on vegetation patterns, and in particular the effect of terrain curvature on the arcing of vegetation stripes on ridges and valleys. 
ODEs, Linear Algebra, experience with MATLAB. Preferably one (or more) of: ODEs and Stability Theory (Math 454), PDEs (Math 456), Numerical Methods (Math 475) 
Yes 
Yes 
Ask 
8/27/18 

Andrew Gillette  Derive formulas that will be used in "maximally efficient" code for the numerical simulation of physical phenomena. The specifics of the project will be determined later but may rely on computational geometry, combinatorics, and numerical analysis.  Vector calculus (223) and linear algebra (313 or higher). Computer science and programming background is preferred. Interested students should contact me in November or December 2017 to start research in Spring 2018.  Yes  Yes  Maybe 
9/6/2017 
Karl Glasner  Pattern formation on graphs; view description.  ODEs, Linear Algebra, experience with MATLAB. Ideally one or more of: Graph Theory (math 443), Dynamical Systems (math454), PDEs (math456), Numerical Methods (math475)  Yes  Yes  Ask 
1/17/2017 
Karl Glasner  Dynamics of self assembly at the nanoscale; view description.  ODEs, some experience with MATLAB. Ideally one or more of: Dynamical Systems (math454), PDEs (math456), Numerical Methods (math475).  Yes  Yes  Ask 
1/17/2017 
Developing computer software to visualize abstract geometries and polyhedral geometries (like the dodecahedron). Study of differential equations that deform arbitrary embeddings of graphs into "nice" embeddings for graphs. 
Basic linear algebra, differential equations. Topology can be a plus, but not necessary. General mathematical sophistication. Some computer science/programming background is a plus. 
Yes 
Yes 
Yes 
9/17/2012 

Geometry: Study the space of three and four point configurations on the (projective) plane.

good command of Linear Algebra 
Yes 
Yes 
Ask 
1/3/2017 

Unifying principles for vectorborne disease spread
Join our team working on the spread of mosquitoborne diseases. This undergraduate research assistantship, provided through support from the UA’s Improving Health/BIO5 Institute Graduate/Undergraduate Interdisciplinary “Link” Program, will allow you to work closely with an interdisciplinary, verticallyintegrated team of professors and students (graduate and undergraduate) from Geography, Mathematics and Public Health on a current hot topic in the news: developing unifying principles for vectorborne disease spread. As a member of this team, you will work with a PhD student in Geography and three professors (in Mathematics, Geography, and Public Health) on collecting, maintaining, cleaning and analyzing data on mosquito presence, network information, and models of disease spread.
Please submit a letter of interest, transcript (unofficial is fine), and your CV to Dr. Joceline Lega (lega@math.arizona.edu) by September 1st, 2016. The upto 20 hr/week position (at $10 per hour or more depending on experience) begins this fall and can go through the spring 2017.

Preferred experience: math major core courses and some programming experience. 
Ask 
Ask 
Ask 
9/29/2016 

Internal and external structure of deep neural networks; Description: Deep neural networks (DNNs) are a class of statistical methods that have received substantial attention in recent years, mainly due to their performance on certain computational tasks which have proved difficult to tackle by other means. The project aims to address the following question: when a DNN is trained on structured data, is the structure reflected in the trained network, and if so how? The project requires applying standard mathematical ideas to some notsostandard problems, willingness to generate and shoot down new ideas, and lots of computer experiments. It may be suitable for an Honors Thesis. References: 
Linear algebra (313, 310, or higher); ability (or willingness to learn) programming in Python and/or C/C++; insatiable intellectual curiosity. Helpful to have at least one of the following, but not required: concurrent enrollment in Abstract Algebra (415A); Probability (464); knowledge of Fourier series and transforms, for example at the level of 322, 422, 456, or an equivalent course.  Yes  Yes  Ask  4/11/18  

Evolution of gene circuits controlling segmentation;Description: A basic question in evolutionary developmental ("evodevo") biology is how the genetic circuitry controlling development have evolved to produce more and more complex organisms. Motivated by this question, Eric Siggia and collaborators have proposed a highly idealized model that tries to capture some of the basic features of the evolution of segmented bodies. Though extremely simplified, their models have yielded some insights into the origin and structure of gene circuits controlling development, and also leads to some interesting mathematical questions involving e.g. bifurcations. This project proposes to implement and study the model. Potential directions include bifurcation analysis and modifying the model to address features other than segmentation. 
Probability (464); programming (Python, Matlab, Java, or C/C++); stochastic processes (468) and/or nonlinear dynamics (454) helpful.  Yes  Yes  Maybe  3/1/2016 
Computational Group Theory; 
413 Linear Algebra or 
Yes 
Yes 
Maybe 
9/18/2012 

Douglas Pickrell  power series identities; conformal mappings  linear algebra, complex variables.  Yes  Yes  Maybe  9/23/2012 
Shankar Venkataramani  Differential equations and modeling physical phenomena; Geometry and applications; Problems in Complex analysis 
Math 254/Math 355 (for Differential equations); Math 323 (for all the problems); MATH 425 (for Complex analysis). 
Yes  Yes  Yes  9/12/2014 
Calvin Zhang  Research in Mathematical Biology: use methods from mathematics and computing to study operating mechanisms of living systems. In particular, I am interested in synaptic transmission and sensorimotor system. Topics of recent interest are stochastic synaptic vesicle release and rhythmic motor pattern generation in crustacean swimming.  Familiarity with differential equations and probability as used in applications.  Yes  Yes  Ask  9/29/2016 
*Honors Thesis MATH 498H credit available to students in the Honors College.
**Restrictions may apply. Ask the individual faculty member for details.