Research projects suitable for undergraduates
What follows is a sample, provided by members of the faculty, of mathematical research projects where undergraduate students in the honors program in mathematics could participate. Interested students should contact either the faculty members directly, one of the honors advisors: professors Al Novikoff or Steve Childress.
A joint research project of Helmut Hofer and Esteban Tabak studies the behavior of Hamiltonian flows on a prescribed energy surface. Computer experiments using symplectic integrators could give some new insight. Such a project would be ideal for a team of an undergraduate and a graduate student. Codes would be developed and experiments would be conducted, shedding new light on the intriguing dynamics of these flows.
Charles Newman has recently studied zero-temperature stochastic dynamics of Ising models with a quenched (i.e., random) initial configuration. When the Ising models are disordered (e.g., a spin glass), there are a host of open problems in statistical physics which could be profitably investigated via Monte Carlo simulations by students (graduate and undergraduate) without an extensive background in the field. For example, on a two-dimensional square lattice, in the +/- J spin glass model, it is known that some sites flip forever and some don't; what happens in dimension three?
Current experiments in the Applied Mathematics Laboratory (WetLab/VisLab) include one project on dynamics of friction, and another involving the interaction of fluid flow with deformable bodies. Gathering data, mathematical modeling, and data analysis all provide excellent opportunities for undergraduate research experiences. In the friction experiment of Steve Childress, for example, the formulation and numerical solution of simplified models of stick/slip dynamics gives exposure to modern concepts of dynamical systems, computer graphics and analysis, and the mathematics of numerical analysis.
Marco Avellaneda's current research in mathematical finance demands econometric data to establish a basis for mathematical modeling and computation. The collection and analysis of such data could be done by undergraduates. The idea is to get comprehensive historical price data from several sources and perform empirical analysis of the correlation matrices between different price shocks in the same economy. The goal of the project is to map the ``principal components'' of the major markets.
Joel Spencer is studying the enumeration of connected graphs with given numbers of vertices and edges. The approach turns asymptotically into certain questions about Brownian motion. Much of the asymptotic calculation is suitable for undergraduates, while the subtleties of going to the Brownian limit would need a more advanced student.
A joint project of David McLaughlin, Michael Shelley, and Robert Shapley (Professor, Center for Neural Science, NYU) is developing a computer model of the area V1 of the monkey's primary visual cortex. Simplifications of this complex network model can provide projects for advanced undergraduate students, giving excellent exposure to mathematical and computational modeling, as well as to biological experiment and observation.
Peter Lax has carried out many numerical experiments with dispersive systems, and with systems modeling shock waves. The basic theory of these equations is well within the grasp of interested undergraduates, and calculations can reveal new phenomena.
A joint research project of David Holland and Esteban Tabak investigates ocean circulation at regional, basinal and global scales. Their approach is based on a combination of numerical and analytical techniques. There is an opportunity within this framework for undergraduate and graduate students to work together to further develop the simplified analytical and numerical models so as to gain insight into various mechanisms underlying and controlling ocean circulation.
Aspects of Lai-Sang Young's work in dynamical systems, chaos, and fractal geometry are suitable for undergraduate research projects. Simple analytic tools for iterations are accessible to students. Research in this area brings together material the undergraduate student has just learned from his or her classes. With proper guidance, this can be a meaningful scientific experience with the possibility of new discoveries.
David McLaughlin and Jalal Shatah's work on dynamical systems provides opportunities for undergraduate research experiences. For instance, the study of normal forms and resonances can be simplified to require only calculus and linear algebra. Thus undergraduate students can study analytically what is resonant in a given physical system, as well as its concrete consequences on qualitative behavior.
Leslie Greengard and Marsha Berger's work on adaptive computational methods plays an increasingly critical role in scientific computing and simulation. There are a number of opportunities for undergraduate involvement in this research. These range from designing algorithms for parallel computing to using large-scale simulation for the investigation of basic questions in fluid mechanics and materials science.