The P.I. will study his newly developed techniques relating group presentations to certain graph theoretical data, allowing him to extend earlier results concerning s-transitive graphs, generalized polygons, the sporadic group and distance-transitive graphs. He will continue his efforts to generalize these techniques and to apply them to a variety of problems related to sporadic groups, finite geometries and amalgams. He also will continue his investigation into the theory of locally s- transitive graphs based on various innovations of Delgado, Stellmacher and the P.I. which have recently yielded a dramatically simplified proof of Goldschmidt's theorem on trivalent graphs with an edge-transitive automorphism group. This research will focus on the area of mathematics which bridges graph theory and group theory. This blending of combinatorics and algebra has been a very fruitful endeavor enhancing both subjects. Weiss is a major worker in this area and his results will surely prove very important.
