Modeling in Ecology
Professor Glenn Ledder,
Prerequisites: a course in linear algebra, a lot of intellectual curiosity, and some background in one of the following: advanced calculus, real analysis, ecology, and/or programming with Maple, Matlab, or Mathematica. (It would be helpful to have an advanced calculus course and some experience with computational software such as Maple, Matlab, or Mathematica. Some background in ecology would also be helpful.)
Project Description: The simplest discrete model for growth of a population is the exponential growth model, in which the population at time t+1 is taken to be the product of the population at time t with a constant growth factor. Such a model can be made more realistic by making the growth factor decrease as the population becomes more dense. This makes the model nonlinear, however, which makes it more difficult to analyze.
A more realistic discrete model arises from the division of a population into age classes, with separate linear growth equations for each class. The resulting model is the vector equivalent of the scalar exponential growth model. One is again motivated to make the model more realistic by making one or more of the parameters dependent on the population density. As in the scalar case, this also makes the model nonlinear. We will study models of this general class and identify a research problem involving such a model.
Differential/Difference Equations on Time Scales
Professor Allan Peterson,
We will be concerned with problems that come up in Differential Equations and Difference Equations. One of our interests will be to see how these two theories can be unified. For example, calculus is very useful in Differential Equations and when one studies Difference Equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general calculus. Consequently, we get a generalization of difference equations and differential equations. We study dynamic equations, where differential equations and difference equations are special cases. This leads to many interesting open questions which we will consider. Any student interested in Differential Equations should enjoy this experience. About a fourth of our time will be spent studying Difference Equations so knowledge of Difference Equations is not a prerequisite. Details
Graph Theory
Professor Jamie Radcliffe,
Prerequisites: There aren't many prerequisites in terms of material studied, but it would be very beneficial to have had a proof based course. Background in linear algebra and discrete mathematics will be beneficial.
Project Description: Graph theory is the mathematical study of networks. This is an extremely broad area which has applications in a wide variety of networks. We will work on studying some graph polynomials. A graph polynomial is a polynomial that we associate with a graph that captures some important information about its structure. In particular, we will think about the interlace polynomial, a relative newcomer to the world of graph polynomials. There are several interesting open questions concerning the interlace polynomial that we will work on.