Homology of Monomial and Toric Ideals

PeevaDMS-9970334 DR.Peeva将致力于交换代数一般领域的项目。她的项目涉及一元思想和环面思想的最小自由分解的结构。我们将应用交换代数、组合学和子空间排列的拓扑理论中的思想和方法来研究这种归结。以前用于描述复杂的子空间排列的上同调的结构将被修改以产生最小自由分辨率。目标是建立高度结构化的解和解子复形，并得到关于Betti数和正则性的界。交换代数是研究数学的基本结构的学科。交换代数的结果在通信、安全和机器人学中得到了应用。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。现代几何学中被称为代数几何的重要领域是这一攻击线上最有力和最富有成效的数学例子。代数对象取代了几何对象，并应用了交换代数强大而抽象的方法来加深我们对几何的理解。皮瓦博士的研究应该会对几何学和组合学产生直接和直接的影响。

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.