DIMACS DCI 2002 July 18, 2002   REU Abstracts   Students Jakub Cerny, Charles University, Czech Republic jcerny@dimacs.rutgers.edu Zdenek Dvorak, Charles University, Czech Republic dvorak@dimacs.rutgers.edu Vit Jelinek, Charles University, Czech Republic Pavel Podbrdsky, Charles University, Czech Republic Mentors Janos Komlos, Department of Mathematics Martin Mares, Charles University On the Polygon Crossing Problem 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 ≤l then (k −1)(l −1) + 2 ≤ 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 − l −(k −3)/2.   Student Samuel Stechmann, University of ST. Thomas, MN Mentor Mathew Leingang, Department of Mahtematics Hilbert’s 3rd Problem and Hyperbolic Geometry   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 congruent, 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 3rd Problem asks the analogous question in three dimensions:  If two polyhedra have the same volume then are they scissors congruet?  Hilbert’s student Max Dehn provided a counterexample to show that equal volume does not imply scissors congruence in Euclidean geometry.  Hilbert’s 3rd Problem in hyperbolic geometry, however, has yet to be solved.     Student Yuki Saka, University of California at Berkeley, CA Mentors Leonid Khachiyan, Department of Computer Science A Generalization of Spanning Trees and Cuts     Student Paul Gross, Rose-Hulman Institute of Technology, IN Mentors Eric Allender, Computer Science Department Extracting a Sequential Algorithm for Solving the Planarity Problem   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.