The UCDavis VIGRE project

Sample Research Projects

Below we present a sample of projects that are within reach of junior and, in many cases, sophomore students, especially those projects where writing a computer program is useful. Keep in mind this is just a sample, feel free to ask other faculty member not listed here about other opportunities. They will be happy to hear your are curious about their research! Are you confused about where to start your search for a summer internship? You don't know what the right prerequisites are to join a project? Feel free to contact the Undergraduate Research Advisor who can provide you with useful tips.

• Mathematical Physics and Combinatorics
  There is an exciting new relation between combinatorics and mathematical physics. Several members of our department participate with enthusiasm in this area and expect to have projects suitable for undergraduates. For example, Professor Anne Schilling has had students help her write a Mathematica program which encodes a conjectured bijection between two combinatorial sets, crystal paths and rigged configurations. These are notions of importance in the representation theory of quantum algebras. This program was useful to confirm certain properties of the map and check the conjecture for big examples.

• Algebraic Methods in Statistics
  Statistical data comes often in the form of multi-way tables. For analysis of independence and model-fitting there is often the need to compute or approximate the number of tables filled with non-negative integer numbers and with fixed set of marginal sums (add the entries of the table to give a count in some way). It is known that it is possible to use Markov chains to do approximations. Undergraduate students could help the leading faculty, Professor Jesus De Loera, to calculate a set of Markov basis of using Grobner bases. Grobner bases are generalizations of linear algebra bases for high degree polynomials. Students can help with carrying calculations in some computer algebra package and then analyzing the results. This has also strong connections to disclosure limitation of table information under marginal release.

• Cellular Automata and Stochastic Processes
  Professor Janko Gravner studies cellular automata theory with emphasis on probabilistic problems. He has analyzed various cellular automata with random initial conditions or random choices in the transition rules. In this area undergraduates can help with research in spatial stochastic processes writing efficient simulation programs. They learn the basics of the theory while improving the implementation. The long term fate of many dynamical processes (e.g., artificial life models) depends on existence of certain finite objects so students would be exposed to combinatorial analysis. They could also help with the computation of multidimensional integrals, solving a large system of equations or estimating coefficients in a series, all useful steps in in the analysis, that would require students to exercise things they studied in their courses.

• Computational Geometry and Topology
  Our department has a strong involvement on computational questions related to topology and geometry. The geometry-topology group will provide several opportunities for undergraduates. For example, Professor Mikhail Khovanov constructed an unusual cohomology theory of knots, which associates bigraded abelian groups to a knot, such that their graded Euler characteristic is the Jones polynomial. In a long-term project, which would involve several undergraduates, it would be desirable to write computer programs for computing ranks, module structure, torsion, and other invariants of these groups. Students would learn about knots, Jones polynomial, homology theories, etc along the way.

• Dynamical Systems
  For students who have some knowledge in dynamical systems, say at the level of Math 119A, there are two directions of research that are suitable for undergraduate research (both will focus on one and two dimensional systems). This will be coordinated by Professor Albert Fannjiang. First is the study of iterated systems with random perturbations noises. Such pertubations can induce dissipation and irreversibility in short times or can induce transport over long distances. These effects are usually absent when there is no noise. Another interesting case are quantized iterated systems where one considers an additional quantum effect which can be thought of as a certain kind of noise (uncertainty). But the quantum "noise" can have very different effects on the dynamics. The research activities will involve explicit calculations that are very instructional and can be pursued theoretically or numerically.

• Topics in Theoretical Physics
  For many students, the decision whether to pursue ongoing studies in pure mathematics or theoretical physics in a difficult one. In principle the two alternatives need not be mutually exclusive, as the tightly interwoven history of the two disciplines shows. A fun possibility to investigate what is like to do research in the area is to participate in a "laid-back" reading seminar for undergraduates, using their recently acquired mathematical skills to understand topics in theoretical physics. Obvious direct links are vector calculus and classical mechanics, elementary differential geometry and general relativity, functional analysis and quantum mechanics etc. Professor Andrew Waldron is already conducting such a seminar on Lagrangian mechanics. The format is to read the material together for an hour each week, she then writes a short synopsis of the main material ready to read further the following week.

• Applied Optimization and Financial Modeling.
  Mathematical Finance is a rapidly evolving field in applied mathematics. Financial models try to predict market behavior under uncertain situations. Often probability and multiple scenarios need to be consider and to find an optimal solution. Professor Roger Wets is a leader in the area for many years, he also takes an active role in applications ranging from environmental questions related to lake pollution to problems in finance concerning asset/liability management. There are several interesting projects that a student who has taken Math 168 can appreciate. For example, the approximation theory for stochastic optimization problems basically asks the question: can I replace a 'large/complicated'-problem by a simpler (small) problem that yields the same solution (or one that's nearly optimal). The best approximating problems are linear programs (they certainly are easy to solve). Students would be able to help with implementing approximation schemes and learning about modeling.