Syzygies and Geometry

PI将在代数几何、交换代数、组合学及其计算方面进行研究，重点是合子及其与几何的相关性。PI将研究(1)Lawrence Toric理想的合子及其与图的色数的关系，(2)Orlik-所罗门代数的合子，秩变和复形，(3)对称层的合群，以及导出的Witt群中相应的阻塞类，以证明这种层的对称局部自由分解的存在性，(4)射影空间中点的构形的自然紧化及其在计数问题中的应用。代数几何是现代数学中最古老的部分之一，但它在过去的25年里取得了革命性的成就。主要目的是发现由多项式的消失定义的集合的几何与由这些集合的定义方程(例如合集)得到的代数不变量之间的联系。代数几何和组合学的结果也在通信、安全和机器人中得到了应用。

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.