This research will focus on the relation between vector bundles and the geometry of space curves. In particular, it will consider questions about the Hilbert scheme of smooth space curves and conjectures on the syzygies of the generic canonical curves. Also work will be done on varieties with small dual varieties and their relations to questions about varieties of small codimension in projective space. This research is in the area of algebraic geometry, which is the study of the geometry of sets of simultaneous solutions of systems of polynomials in many variables. This subject is one of the most important and venerable in all of mathematics and has in recent years been instrumental in fundamental advances in many areas of core mathematics as well as applied mathematics and computational mathematics. This research, by an extremely successful and productive mathematician, will have a great impact on the field.
