Computational Polyhedral Geometry: Applications in Algebra, Combinatorics, and Optimization

计算多面体几何：在代数、组合学和优化中的应用

• 批准号: 0309694
• 负责人: Jesus De Loera
• 金额: $ 18.89万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-08-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0309694&HistoricalAwards=false
• 关键词: Computational; Polyhedral; Geometry; Applications; Algebra

The investigator uses methods from commutative algebra,convexity, and analysis to design computer algorithms that allowthe practical fast detection or counting of lattice points insiderational convex polytopes. The main idea is to represent thelattice points using rational functions. This can be done in atleast three different complementary ways. He implements themethods in the computer software LattE. The investigator usesthis software to help investigate theoretical questions incombinatorics (properties of Ehrhart functions), algebra(representation theory and partitions), and optimization (newinteger programming algorithms). The investigator carries out geometric, analytic, andcomputational studies of problems in the convexity andcombinatorics of polyhedra, including development of algorithmsand software. Informally speaking, the project can be thought ofas an attempt to understand how to count the possiblenon-negative integer solutions of a system of linear equations.With two unknowns, this amounts to counting the number ofregularly distributed points -- lattice points -- in a polygon.With more unknowns, the solutions correspond geometrically to thehigher-dimensional analogs to cubes and polygons. Examples ofregularly distributed points are arrangements of atoms orcrystals. Many of the theoretical questions under study aremotivated by problems in data security and computation (in thecontext of RSA encryption, which is used in internettransactions), operations research (via certain techniques forsolving integer programs when levels of uncertainty areexpected), statistics (contingency tables, data analysis), andproblems in abstract algebra (number theory, representationtheory). The training of students is a very important componentof the project.

研究者使用来自交换代数，凸度和分析的方法来设计计算机算法，从而可以实用的快速检测或计数晶格点内部凸音多面体。 主要思想是使用有理功能表示thelattice点。 这可以至少以三种不同的互补方式来完成。 他在计算机软件拿铁中实现了它们。 研究者USESTHIS软件旨在帮助研究理论问题In -In -Inbionatorics（Ehrhart功能的属性），代数（表示理论和分区）和优化（Newinteger编程算法）。 研究人员对Polyhedra的凸度和康复器中的问题进行了几何，分析和计算研究，包括开发算法和软件。 Informally speaking, the project can be thought ofas an attempt to understand how to count the possiblenon-negative integer solutions of a system of linear equations.With two unknowns, this amounts to counting the number ofregularly distributed points -- lattice points -- in a polygon.With more unknowns, the solutions correspond geometrically to thehigher-dimensional analogs to cubes and polygons. 规则分布点的示例是原子orcrystals的排列。 研究安全和计算中的问题（在Internet发行中使用的RSA加密的问题）在研究中引起了许多理论问题，在Internet发行中使用了），操作研究（通过某些技术，当不确定性级别的水平指出），统计数据，统计数据表，数据分析，数据分析），和alge alge Bromection（数字）（数字）（数字分析）（数字）（数字）。 对学生的培训是该项目非常重要的组成部分。