Two polygons are scissors congruent if one of them can be cut up into polygon pieces and put back together to form the other. Clearly if two polygons are scissors congrunet, then they have the same area. The converse, i.e. if two polygons have the same area then they are scissor congruent, is also true. Hilbert's 3^rd Problem asks the analogous question in three dimensions: If two polyhedra have the same volume then are they scissors congruent? Hilbert's student Max Dehn provided a counter example to show that equal volume does not imply scissors congruence in Euclidean geometry. Hilbert's 3^rd Problem in hyperbolic geometry, however, has yet to be solved.

Let k, l < 3 be integers. Consider two polygons in a plane with k and l vertices respectively. We are interested in the problem of determining the maximum possible number of intersections of their boundaries for given k and l. Denote this maximum by f(k,l). When k and l are both even, it is easy to show that f(k,l) = kl. If k is even and l is odd then f(k,l)=k(l - 1). However, if k and l are both odd the problem seems extremely difficult and the exact value of f(k,l) is unknown. In this last case, if k < 1 then (k - 1)(l - 1) < f(k,l) < (k - 1)l. The hypothesis is that the lower bound is tight. This hypothesis is proved for the cases k = 3 and k = 5. In this talk we present the methods used to improved the upper bound to obtain f(k,l) < kl - 1 - (k - 3)/2.

A planar graph is a graph that can be drawn in the plane such that no two edges intersect. The Planarity Problem is to determine if a given graph is planar and if so to provide a planar embedding of the graph. An efficient parallel algorithm for solving the Planarity Problem will be mentioned and the project goals of the student.