Syzygies and Geometry

PI将继续研究代数几何，交换代数，组合数学及其计算方面的领域，重点是合朔及其与几何的相关性。 PI将研究（1）Lawrence环面理想的合合及其与图的色数的关系，（2）Orlik-Solomon代数的合合，秩簇和复性，（3）Orlik-Solomon层的合合，以及导出的Witt群中相应的阻塞类对这类层的对称局部自由分解的存在性，（4）射影空间中点位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，但在过去的四分之一世纪里，它却有了革命性的发展。主要的推力是对发现之间的连接几何定义的集合消失的多项式和代数不变量来自definingequations这些集，例如syzygies。 代数几何学和组合数学的成果也被用于通信、安全和机器人技术。

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.