Geometry, Syzygies and Combinatorics of Algebraic Varieties

主要研究人员和他的合作者将解决代数几何、组合学和计算代数几何领域的几组问题，重点是合子及其在几何和组合学中的相关性。代数几何主要研究用多项式方程系统描述的几何对象。这个领域是数学的中心，因为这样的几何对象包括在最不同的数学学科中的某些基本例子。它在数学应用中的重要性来自于这样一个事实，即这样的代数几何对象提供了在通过方程和在计算机上表示基本几何形式时经常使用的模型。研究者将研究与其复杂性密切相关的多项式理想的几何和代数，将研究与他提出的复杂性界限有关的几个组合方面，并将外部代数方法的使用扩展到射影几何。主要的研究活动还将包括计算代数几何中各种基本算法的发展及其在Grayson和Stillman编写的Macaulay2软件包中的实现。

The principal investigator and his collaborators will addressseveral groups of problems in areas of algebraic geometry,combinatorics, and computational algebraic geometry, with emphasison syzygies and their relevance in geometry and combinatorics.Algebraic geometry at large deals with geometric objectsdescribed by systems of polynomial equations. The area is central tomathematics since such geometric objects include certainfundamental examples in the most diverse disciplinesof mathematics.Its importance in the applicationsof mathematics comes from the fact that such algebro-geometricobjects provide models often used when representing fundamentalgeometric forms via equations and also on a computer.The investigator will study the geometry and algebraof polynomial ideals in close relation to their complexity,will investigate several combinatorial aspects related tohis proposed complexity bounds, and will extend the use ofexterior algebra methods to projective geometry.The principal investigator's activity will also include thedevelopment of various fundamental algorithms in computationalalgebraic geometry and their implementation in the Macaulay2package written by Grayson and Stillman.