UC Davis Mathematics

Summer Research Opportunities

Summer undergraduate research positions provide a unique opportunity to join an active research group in the department and work on a project. In some cases this may lead to employment as an Undergraduate Research Assistant during the Academic Year and/or the following summer.

Every year there are a limited number of paid summer research positions. Please note that if you are planning to do a senior thesis, the optimal time to start is the summer of your sophomore or junior year.


Apply for Summer 2019 Research Projects

The application for Spring 2019 research opportunities is available online.  Please note that you will need the Application Code (from the section below) for the position(s) you are interested in. If you are interested in research opportunities with multiple faculty members, submit a separate application for each faculty member. 

Link to application: https://forms.gle/Tc7Bss83NFTaKSZh8

For full consideration, please apply before Sunday, May 12.

If you are interested in working on an area of research not represented on this list, you are encouraged to contact faculty directly who are doing work in that area. A list of math faculty, including their research areas, is available here.

Summer 2019 Research Projects:


Principal Investigator (PI): Prof. Woodruff

Description: Modify and extend the grid operations scenario maker (gosm) software to produce inputs for extreme events such as hurricanes. The current software deals primarily with energy scenarios; it will be extended to include line and load outages due to extreme events.

Requirements: Proficiency as a Python programmer. Coursework in probability theory and statistics. Optimization coursework is a plus, but not required.

Application Code: woodruff



Principal Investigator (PI): Prof. Bob Guy
I am interested in mentoring students on a variety of projects related swimming micro-organisms. From single cells to complex multicellular organisms, locomotion requires coordination of different parts of the body. In cells such as sperm, this coordination emerges from mechanochemical coupling (i.e. chemical reactions influenced by mechanical feedback). In worms such as C. elegans, the coordination is thought to arise through neuromechanical coupling (i.e. neuron interactions influenced by mechanical feedback). In both systems, the gait emerges from coupled dynamics of the body, muscles/molecular motors, and the surrounding fluid. Below I list several possible projects related to these systems.
Requirements for projects below: Programming and ODEs (MAT 119A)

Project 1A: Models of active filaments: elastic fluids.
Simple models of so called "active filaments" have been used to study the emergence of the beating pattern of flagella. These systems have been studied only in viscous fluids (e.g. water), but many cells function in viscoelastic fluids. We are interested in investigating how fluid elasticity affects the emergence of the beat. This project involves performing numerical simulations of ODEs and PDEs as well as analyzing the bifurcation at which the system breaks symmetry and begins to beat. Application Code: guy1A

Project 1B: Models of active filaments: synchronization.
Using the same type of models as in project 1A, we will investigate whether nearby filaments synchronize their beats. Nearby filaments affect the beat of one another though the motion of the common surrounding fluids. Do interactions though the fluid cause the filaments to beat together or with some other rhythm? How do elastic forces in the fluid affect the rhythms that emerge? Application Code: guy1B

Project 2: Optimal gaits under constraints.
What body shape should a worm use given an energy budget in a given environment? The worm C. elegans swims with larger head movements compared to the tail, while sperm swim with larger tail movements compared to the head. The organisms swim in different environments and have different body design. Did these organisms evolve to optimize their swimming performance? Using computational models of the fluid and body mechanics, we can investigate how the optimal swimming shape depends on the external environment and body design. Application Code: guy2



Principal Investigator (PI): Prof. Javier Arsuaga
Description: In certain viruses DNA is highly condensed and it is believed that the DNA molecule is found in the form of a liquid crystal. Motivated by these experimental observations we are interested in developing a model of liquid crystal for DNA under confinement. The project is open to the computer implementation of continuous or discrete models of liquid crystalline phases.
Requirements: Some background in PDEs or Physics is desirable. Programming experience, preferably C++ and/or Python. Ability to conduct numerical simulations with care and interest in the underlying biological motivation.
Application Code: arsuaga


Principal Investigator (PI): Prof. Tim Lewis
Description: Topics include:
  - neural and mechanical mechanisms of locomotion in "model" systems
  - autonomic (neural) regulation of cardiac activity
  - effects of pharmacological drugs on electrical activity in the heart. 
Requirements: MAT 22AB necessary; MAT 119A and/or MAT 124 highly preferred. Some experience in computer programming is required, and a willingness to learn mathematical modeling and biology is essential.  
Application Code: lewis

2018 Summer Research Projects

Title: Shock formation in compressible gas dynamics

Principle Investigator (PI): Prof. Qingtian Zhang

Description: The dynamics of air flow is described by the compressible Euler equation. When the air flows very fast, shock waves may appear. To understand how the shock waves form, and evolve, is very important. This is also a very popular research direction in recent years.

For this project, we mainly read this paper: https://arxiv.org/abs/1212.2867

As an undergraduate student, you may obtain from this project:
  1. Get a better understanding of the knowledge you've learnt from different courses
  2. Learn the current research status in the related fields
  3. It will be a great help if you will continue doing research in the related fields
  4. If you are interested, we can also extend this project to next quarter

3.6 GPA or higher

You've taken these courses: MAT 21ABCD, 22AB, 116*, 118, 119, 125

If you've taken the course of general relativity, it will be a great help.

*MAT 116 is a very important requirement. If you didn't take this course, but you have learnt it by yourself, it's also fine. You can list what you have learnt when you apply for the project.

Application Code: zhang


Title: Projects on applied and computational discrete mathematics 

Principal Investigator (PI): Prof. Jesus DeLoera

Description: Seeking diligent hardworking students with strong interests and background in at least two of the following topics:
Discrete Math, Algorithms, Optimization,  Computational Geometry/topology, Operations Research and Data Sciences, 
Machine learning. One or two projects in the intersection of these mathematical areas open starting Spring 2018. 

Requirements: GPA 3.6 or higher. Very strong programming experience (e.g., at least ECS 60 with an A), A or higher grade in at least two of the following courses: Math 145, 146, 148, 168, 160, 167, 128, 150, or 135. Preferably junior level as the projects could be extended to last for a year. Honors senior thesis possible.

Application Code: deloera


Title: The Riemann Ellipsoids; Hamiltonian dynamics and visualization

Principal Investigator (PI): Prof. Joseph Biello


The equations of fluid dynamics are a  system of partial differential equations whose multitude of solutions give us a wide range of  physical phenomena we see in the world, from stars to our atmosphere and oceans, from swimming in a pool, to bacteria swimming in a biofluid.

However, in the absence of viscosity, there is a an exact reduction of the fluid equations to a low order (15 degree of freedom) system of ordinary differential equations.  These are the Riemann ellipsoids.   This equations have a wonderful, non-canonical Hamiltonian structure, as well as conserved quantities such as total energy and total circulation.  They were the original dynamical models of self-gravitating bodies, and have been studied by eminent mathematicians such as Maclaurin, Jacobi, Dedikind, Riemann, Chandrasekhar, and Lebovitz.   

In this REU, students will move beyond the study of equilibria and bifurcations of the Riemann ellipsoids to study the dynamics in a variety of setting.   The idea will be to visualize the dynamics of the ellipsoidal surfaces as physical objects - in space.   Students will track solutions from equilibria, to periodic orbits, homoclinic orbits and, ultimately, chaotic orbits in the phase space and show what these look like from the perspective of the ellipsoids.

Integrations will be carried out using matlab.   The results will be presented as curated videos published to the UC Davis website - and possibly to youtube.   

Requirements: Students will need experience in dynamical systems (119A/B or equivalent classes from physics) as well as Matlab (or python) experience. Students will be making videos to post to to share on the internet, and some willingness to learn video editing software is also necessary.

Application Code: biello


Title: Reduction algorithms for meromorphic differential equations

Principal Investigator (PI): Prof. Martin Luu

Description: In close analogy with the theory of Jordan normal forms, there exist normal forms for some classes of differential equations (associated to suitable Lie algebras). Furthermore, there exist explicit algorithms to reduce to the normal form, but they are quite involved and computer implementation is very useful. The aim of this project is (starting with sl_n) to write a program calculating these normal forms in an efficient manner.

Requirements: Most important is a good grasp of linear algebra as covered in MAT 67 as well as a strong background in programming (particular languages are not so important).

Application Code: luu


Title: Unit testing and parallelization of existing Monte Carlo software (C++) used to sample knotted self avoiding polygons

Principal Investigator (PI): Prof. Mariel Vazquez

Description: Seeking student with strong C++ background to write unit tests for existing software. Student would gain exposure to statistical sampling techniques in the context of self-avoiding polygons. Additionally, student would have the opportunity to design and implement a multi-threaded approach to an existing serial sampling algorithm. 

Requirements: Strong C++ background. Ability to use git versioning software.

Application Code: vazquez1


Title: Monte Carlo sampling to model chromosome packing in confinement

Principal Investigator (PI): Prof. Mariel Vazquez

Description: Student will learn to use existing software and tool suites for sampling knotted polygons in confinement. Student will produce simulations incorporating methods for topological change as guided by the action of enzymes on circular DNA, and will become familiar with specific experimental (numerical) procedures for sampling and introducing topological changes.

Requirements: Programming background, preferably C++ and Python. Ability to conduct numerical simulations with care, learn existing tool suites, and understand underlying biological motivation.

Application Code: vazquez2


Title: Mathematical modeling projects in neurobiology and cardiac electrophysiology

Principal Investigator (PI): Prof. Tim Lewis
Description: Topics include:
  - neural and mechanical mechanisms of locomotion in "model" systems
  - autonomic (neural) regulation of cardiac activity
  - effects of pharmacological drugs on electrical activity in the heart. 
Requirements: MAT 22AB necessary; MAT 119A and/or MAT 124 highly preferred. Some experience in computer programming is required, and a willingness to learn mathematical modeling and biology is essential.  
Application Code: lewis


The application for Summer 2018 research opportunities is online here.  Please note that you will need the Application Code (from the section above) for the position(s) you are interested in. 

The deadline to apply for Summer 2018 research opportunities has been extended to noon on May 16.