Mathematical Sciences: Small Geometry Project

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Eight students supported by this REU site award will be involved in researching several topics of current mathematical interest. Continuing a highly successful pilot project conducted by Williams College in 1988, this award will provide resources for its expansion and for the replacement of seed funds which are no longer available. The term geometry in the title does not cover all the topics to be investigated by the students, although it does provide a focus. Geometric problems include study of area minimizing networks which have immediate and important practical applications. Of particular interest will be an analysis of the structure of singularities of such networks and the structure of directed minimizing networks. Work will also be done in knot theory and on triangulations of hyperbolic 3-manifolds which falls within the province of topology. Questions involve the enumeration of all knots having 14 or fewer crossings (those with 13 or fewer have been catalogued) and the relationship between the new knot and link polynomials and the hyperbolic volume of hyperbolic knot or link complement. Other research will involve visual programming systems for parallel processing, explorations of the relation between sum graphs and product graphs - devising more efficient labeling algorithms than those currently known, and computational geometry problems which arise in trying to asses visibility of portions of the plane in the presence of certain obstacles. The last topic has interesting applications to the design of fortresses. The award will provide support for part of a larger effort involving up to 17 students working with six faculty members.