What is an REU?

What is it like to do research?

As in a course, you are expected to work hard and do your "homework," with the big difference that now you are working to create new knowledge! This require working a minimum of 30 hours per week. In a typical day you will be trying to solve a problem or helping to test a conjecture. You will be either quietly thinking with paper and pencil about a proof, writing computer programs to test a conjecture, gathering data, or displaying mathematical objects in a computer screen. You may often find yourself presenting a recent discovery on the chalkboard, on a computer, or using an overhead projector, to a group of fellow REU members and faculty mentors. Like a real mathematician, you would typically perform not just one of the above examples, but many of them.

You will have the opportunity to work closely with your mentor, who will teach you about his/her own research and what it is like to conduct original research. Some projects will be for a small group of students working together under the supervision of professors and graduate students. If you are working with a large group of students, you will learn how to work with a team, to divide up duties amongst yourselves, and to report your results back to the group. While your mentor will guide your project and suggest what to read, what computer "experiments" to conduct, what calculations to make, s/he will not give formal classroom lectures or assignments. You will practice looking for patterns in your data or computations and making conjectures of what theorems hold or what direction to move in next. You will be expected to present your work, and to write it up in clean detail, in the process catching mistakes and reflecting on the "big picture." You may spend a lot of time writing and correcting a paper reporting the final results. One thing is for sure: You will be expected to write a report or paper on your work at the end of the quarter and to give a presentation at our local undergraduate research conference. Some of our hard-working students have even managed to publish their results in international mathematics journals!

What are examples of research projects done in the past?

Below we present a sample of projects that are within reach of juniors and, in many cases, sophomores students; specially 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. For questions please contact professor De Loera.

• 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, Prof. Anne Schilling (anne@math.ucdavis.edu) 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, Prof. Jesus De Loera (deloera@math.ucdavis.edu), 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
  

  Prof. Janko Gravner (gravner@math.ucdavis.edu) 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.

• 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 Prof. Albert Fannjiang (fannjiang@math.ucdavis.edu). 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. Prof. Andrew Waldron (wally@math.ucdavis.edu) 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 Modeling.
  

  Mathematical Optimization is a rapidly evolving field in applied mathematics. For example in logistic and transportation planning and in financial models one tries to find an optimal solution that would minimize cost. Prof. Matthias Koeppe (mkoeppe@math.ucdavis.edu) is a leader in the area. 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.