A Research Experiences for Undergraduates project involving 12 students will be supported by this award. The students will work for ten weeks under the supervision of seven faculty advisors. Other funds will provide support for additional students bringing the total size of the group to about 20. The grant renews funding of successful projects of similar size carried out during the past three years. Although many of the participants have not yet graduated, at least seven are known to be enrolled in graduate study in mathematics and computer science. The program is structured so that each student works with two subgroups. They focus on several problems initially until one emerges as the most promising. The themes are geometric although the term SMALL in the title is an acronym representing the initials of the first group of advisors and is not used as an adjective. Typical problems areas will include questions involving knot theory. Students will consider how an analysis of the crossings of knots can provide information about their general classification. These ideas are related to others in graph theory in that certain knots can be embedded in graphs. One would like to characterize such graphs. Other topics include the calculation of topological characteristics called Betti numbers, vector-valued Hermitian forms arising in the differential geometry of Cauchy-Riemann structures and work in analytic number theory on the divisor problem for arithmetic progressions. Students will be recruited from the New England area. Efforts will be made see that the research continues after the summer ends. This will be particularly true of the students who are recruited from Williams College. It is expected that several publishable papers will result from the project and that some students will have the opportunity to present their work at professional meetings.
