This REU site award will support work of six undergraduate students investigating problems in the areas of number theory and probability. Students will be immersed in a graduate school type setting involving seminars, individual research and presentation of results. Research topics include cyclotomic function fields; rings of integers and prime factorization; unique factorization domains; Fibonacci sequences; continued fraction; probability bounds and linear programming. In algebraic function fields, the students' projects will deal with analogues to classical results from algebraic number fields. Of particular interest will be efforts to discover analogues of fundamental principles regarding quadratic reciprocity. The properties of the Fibonacci sequence modulo m has been the target of several recent investigations and work will continue on extensions of periodicity relations which have been established for certain special classes of moduli. Work in the area of probability will focus on obtaining information on joint probabilities when only partial information is available on the individual events. Exact solutions are replaced by bounds on the probability. These questions lead naturally into interesting questions of multi-parametric linear programming problems which yield answers to questions of how one can obtain the above bounds in an efficient manner. The project will use resources from the Lehigh Valley Association of Independent Colleges including two faculty advisors. It is expected that work completed under this award will be presented at the fourth meeting of the Annual Moravian Mathematical Student Conference in 1990.
