Syzygies and Geometry

PI将从事代数几何、交换代数、组合学及其计算方面的研究，重点是协同及其与几何的相关性。PI将研究(1)Lawrence环理想的合子及其与图的色数的关系，(2)orliko - solomon代数的合子，秩变化和复杂性，(3)对称束的合子，以及相应的Witt群中对这种束的对称局部自由分辨率的存在性的阻塞类，(4)投影空间中点的构型的自然紧化及其在枚举问题中的应用。代数几何是现代数学中最古老的部分之一，但在过去的25年里，它已经有了革命性的发展。其主要目的是发现由多项式的消失所定义的集合的几何和由这些集合的定义方程所衍生的代数不变量之间的联系，例如syzygies。代数几何和组合学的结果也被用于通信、安全和机器人。

The PI will pursue research in areas of algebraic geometry,commutative algebra, combinatorics, and their computational aspects,with emphasis on syzygies and their relevance to geometry. The PI willstudy (1) Syzygies of Lawrence toric ideals and their relation tochromatic numbers of graphs, (2) Syzygies, the rank variety and thecomplexity of Orlik-Solomon algebras, (3) Syzygies of symmetricsheaves, and corresponding obstruction classes in derived Witt groupsto the existence of symmetric locally free resolutions of suchsheaves, (4) Natural compactifications of configurations ofpoints in projective space and their applications to enumerativeproblems. Algebraic geometry is one of the oldest parts of modern mathematics, butone which has had a revolutionary flowering in the past quarter-century. The main thrust is towards discovering connections between the geometry of sets defined by the vanishing ofpolynomials and the algebraic invariants derived from the definingequations of these sets, for instance syzygies. Results in algebraicgeometry and combinatorics have also found uses in communications, security and robotics.