Homology of Monomial and Toric Ideals

AbSTRACTirena PeevaDMS-9970334Dr. Peeva将致力于交换代数一般领域的项目。 她的项目涉及Monomial和Toric思想的最小自由决议的结构。 从交换代数，组合学和拓扑理论的子空间安排的思想和方法将被应用到研究这样的决议。 以前用于描述复子空间安排的上同调的构造将被修改以产生最小的自由分辨率。目标是建立高度结构化的决议和决议的子复形，并获得Betti数和正则性的界限。 交换代数是研究数学基本结构的学科。交换代数的结果已在通信、安全和机器人技术中得到应用。通过研究抽象的数学对象，数学家对数学的许多其他分支有了更深入的了解。现代几何学中的重要领域--代数几何学，是这条攻击线上最有力、最富有成果的数学例子。 用代数对象代替几何对象，用交换代数的强大而抽象的方法来加深对几何的理解。 Peeva博士的研究应该在几何学和组合学中有直接和直接的影响。

ABSTRACTIrena PeevaDMS-9970334Dr. Peeva will work on projects in the general area of Commutative Algebra. Her project deals with the structure of the minimal free resolutions of Monomial and Toric ideas. Ideas and methods from commutative algebra, combinatorics, and the topological theory of subspace arrangements will be applied to study such resolutions. Constructions previously used to describe the cohomology of complex subspace arrangements will be adapted to yield minimal free resolutions. The goal is to build highly structured resolutions and subcomplexes of resolutions, and to obtain bounds on Betti Numbers and regularity. Commutative algebra is the study of the fundamental structures of mathematics. Results of commutative algebra have found uses in communications, security and robotics. By studying abstract mathematical objects, mathematicians gain deeper insight in many other branches of mathematics. The important area of modern geometry known as algebraic geometry is the strongest and most fruitful mathematical example of this line of attack. Algebraic objects are substituted for geometric objects, and the powerful and abstract methods of commutative algebra are applied to further our understanding of the geometry. Dr. Peeva's research should have immediate and direct implications in geometry and combinatorics.