Undergraduate research opportunities


"I do not know what I may appear to the world, but to myself I seem to have been only a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."   Isaac Newton

"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."   G.H.Hardy



My research areas are partial differential equations, dynamical systems and applications of biology, chemistry, physics. Partial differential equations (PDE) are traditional mathematical models for physics, chemistry and engineering problems based on the differential and integral calculus developed by Issac Newton . In a PDE model, physical quantities are assumed to be continuous and differentiable functions depending on several variables such as the time, spatial coordinates. Examples of such functions are concentration and density functions of chemical or biological organisms. The rates of change with respect to various variables and physical laws yield partial differential equations. In early years, PDEs are mostly used to model physical phenomena, such as heat equation, wave equation, Navier-Stokes equations in fluid mechanics (which is one of the Millennium Prize Problems). However the applications of PDEs now include biological sciences, economics, finances, and many other subjects requiring qualitative analysis.

My current interests are on the spatial-temporal models of biological and chemical systems. An important class of models is the reaction-diffusion equations based on the diffusion mechanism and reaction/interacting of different species. The famous scientist Alan Turing proposed to use reaction-diffusion equations to study the chemical mechanism of morphogenesis, which becomes one of important foundation of nonlinear sciences today. (The graph on upper right is a computer simulation of the pattern generated by Turing type equation.) Reaction-diffusion equations nowdays have wide applications from the pattern of the seashell (upper left graph), mixing of river water (upper middle), the wave pattern in chemical reaction (lower right), to the stripes on the body of zebra. With reaction-diffusion equations and computer, you can design your favorite patterns by yourself (see lower left).

Possible undergraduate research topics:

  1. Numerical calculation of radial solutions of coupled Schrodinger equations background material
  2. Numerical simulation of nonlocal reaction-diffusion logistic equation
  3. Bistability in self-organized patchiness in ecosystems, such as lakes, coral reef, woodlands, deserts and oceans.
  4. Spatial-temporal patterns in isothemal autocatalytic chemical reactions.
  5. Bistability of matrix population models.
  6. Optimal harvesting strategy for renewable natural resource such as fishrey, forestry.
  7. Sustainable natural resource management with spatial consideration.
  8. Analysis of ODE systems of dynamics of periphyton and phosphorus in Everglades.
  9. Numerical studies of spatial optimization problems related to design of marine reserve.
  10. Numerical calculation of bifurcation diagram of steady state solutions of reaction-diffusion systems.
Student research will be partially supported by National Science Foundation grants DMS-0703532(CSUMS: Computational Science Training for Undergraduates in the Mathematical Sciences), and  EF-0436318 (UBM: Undergraduate Research in Metapopulation Ecology). Please contact me for applications of the support (or Dr. Li for CSUMS support, or Dr. Dan Cristo for UBM support). You can work on some problems from topics above, or work on some topic of your own choice, such as developing and analysing a mathematical model in your area of interest. For the mathematical background of the research, you should have completed either Math 345 Mathematical Models in Biology or Math 302 Differential Equations when you start the project. Fo some projects, you can take Math 490 PDE and Mathematical Biology  as a training course on the basic knowledge on reaction-diffusion equations, and Math 413/414 Numerical Analysis for numerical helps.

The courses I taught on applied mathematics

Math 490 PDE and Mathematical Biology    (Spring 2002, Spring 2004, Spring 2006) Lecture notes of Math 490
Math 441 Applied Mathematics (Fall 2002)
Math 345 Mathematical Models in Biology  (Fall 2006) (or Math 410 before 2004)  Mathematical Models in Biology    (Fall 2001)

My publications

The powerpoint file of an undergraduate leecture on Pattern formation in mathematical biology (partly in Chinese)

Some population science books related to mathematical biology and differential equations

Life's Other Secret: The New Mathematics of the Living World (Paperback)
by Ian Stewart

Mathematics in Nature : Modeling Patterns in the Natural World (Hardcover)
by John A. Adam

Celestial Encounters (Paperback)
by Florin Diacu and Philip Holmes

Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life (Paperback)
by Steven Strogatz

Linked: How Everything Is Connected to Everything Else and What It Means (Paperback)
by Albert-Laszlo Barabasi

Previous undergraduate projects

 Bifurcation Diagrams of Population Models with Nonlinear Diffusion (Young He Lee, Lena Sherbakov, Jackie Taber, Junping Shi)
   (Journal of Computational and Applied Mathematics, Vol. 194, no. 2, 357--367, 2006.)