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Research Experience for Undergraduates 
Project Descriptions - Summer 2009

(1)  Terrence Blows:  Nonlinear Age-structured Population Models

Let N(0,t), N(1,t), N(2,t) .. N(m,t) denote the number of individuals aged 0,1,2,… m respectively at time t, and consider the vector N(t) = (N(0,t), N(1,t), N(2,t) .. N(m,t)). The matrix difference equation N(t+1)=LN(t), due to P.H. Leslie, provides a simple model of population growth. In this model, a viable population settles to a stable growth rate and also settles to a stable age distribution. However the population will become unbounded in time. Nonlinearities may be introduced to put a carrying capacity in place, but the dynamics of such models can be very complex. The project will investigate how the resulting dynamics changes over a range of parameter values for a selection of nonlinearities. Prerequisites: Linear Algebra and Differential Equations.

(2) Michael Falk - Arrangements

In this project students will work with mentor Michael Falk on aspects of  his research on arrangements of  complex hyperplanes. Algebraic, geometric and combinatorial techniques all come into play in understanding the topological structure of the complement of the union of a finite set of codimension-one linear subspaces of a complex vector space. Motivating examples and results come from the theory of the braid group, studies of complex plane curves and singularity theory, and applications to generalized hypergeometric functions and the Knizhnik-Zamolodchikov equation of mathematical physics.

Through the combinatorial notion of matroid, arrangements can be thought of as generalizations of graphs. Many open problems in the theory of arrangements can be approached using only the combinatorics of graphs or matroids and certain vector spaces, rings, or groups associated with them. Some major conjectures in the field can be reduced to elementary problems involving presentations of these groups and rings. Oftentimes methods developed by studying structures associated with graphs can be extended to the context of matroids and arrangements. Also, interesting examples can often be described in terms of graph- or matroid-theoretic constructions. In this way, research problems in this active area of topology are made accessible to undergraduates with background in linear and abstract algebra.

Previous REU projects have resulted in publications that have had a direct impact on research in the field:

M. Falk. Resonance varieties over fields of characteristic p, Int. Math. Research Notices, Vol. 2007, article ID rnm009, 25 pages, doi:10.1093/imrn/rnm009.

M. Falk and S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves, Compositio mathematica, 143 No. 4 (Steenbrink issue) (2007), 1069-1088.

M. Falk and N. Proudfoot. Parallel connections and bundles of arrangements, Topology and Its Applications, 118 (2002), 65-83.

Tutte polynomials and Orlik-Solomon algebras, with C. Eschenbrenner, J. Algebraic Combinatorics, 10 (1999), 189-199.

The latter two papers were co-authored with REU students. The first two were helped along greatly by examples discovered by REU students.

(3) Shafiu Jibrin – Operations Research

Operations research is concerned with decision-making where one chooses decision variables that maximize or minimize an objective function, subject to the requirement that the decision variables satisfy certain constraints. For example, one may wish to find the amount of each product to produce in a factory in order to maximize the profit subject to restriction on labor, raw materials and demand on the products. A relatively new area of operations research that has many applications is called semidefinite programming. In a semidefinite programming problem one optimizes a linear objective function subject to a system of constraints called linear matrix inequalities. One part of the proposed project would develop an interior point method for solving semidefinite programming problems. The method would use a concept of weighted analytic centers and repelling limits for linear matrix inequalities. The second part would develop an efficient algorithm for computing weighted analytic centers for linear matrix inequalities when some of the weights are relatively large compared to the others. One would have an option to do either part of the project.

Prerequisites: A background in Linear Algebra is required. Some experience with a programming language or mathematics software (e.g., C, Maple, Mathematica, or MathLab) is also necessary.

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