Mathematical Sciences: Vector Bundles and Geometries of Space Curves

本文主要研究向量丛与空间曲线几何之间的关系。特别是，它将考虑光滑空间曲线的Hilbert格式的问题和关于一般正则曲线合子的猜想。此外，还将研究射影空间中具有小对偶簇的簇及其与小余维簇问题的关系。这项研究是在代数几何领域进行的，它是研究多元多项式系统的联立解集的几何。这门学科是所有数学中最重要和最古老的学科之一，近年来在核心数学以及应用数学和计算数学的许多领域取得了根本性进展。这项由一位非常成功和富有成效的数学家进行的研究，将对该领域产生重大影响。

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.