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.
2018 Summer Research Projects
Note: 2019 Summer Research Projects will be posted later in the 2018-19 academic year.
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.
As an undergraduate student, you may obtain from this project:
- Get a better understanding of the knowledge you've learnt from different courses
- Learn the current research status in the related fields
- It will be a great help if you will continue doing research in the related fields
- 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