Computational Algebraic Geometry

计算代数几何

• 批准号: 0456960
• 负责人: Bernd Sturmfels
• 金额: --
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0456960&HistoricalAwards=false
• 关键词: Computational; Algebraic; Geometry

Tropical geometry, the study of piecewise-linear spaces whichrepresent algebraic varieties over a field with a non-archimedean valuation; phylogenetic algebraic geometry, the study of unirational algebraic varieties derived from statistical models on phylogenetic trees; maximum likelihood degree, which is a new invariant of a variety in projective n-space with a distinguished divisor defined by the intersection with n+2 hyperplanes;toric varieties arising from matroids, which is aimed at resolving basicopen questions at the interface of combinatorics and commutative algebra.The field of computational algebraic geometry is concerned with thealgorithmic study of solution sets to systems of algebraic equations,and with the development of polynomial models in science and engineering.For instance, many widely used statistical models in biological sequenceanalysis can be specified by polynomial constraints. Phylogenetic algebraic geometry concerns this algebraic representation and its implications for the construction of maximum likelihood trees. The ensuing interaction between algebraic geometry and phylogenetics is a healthy two-way street: computational biologists may benefit from new algebraic tools, while algebraic geometers find a rich source of new problems concerning objects familiar from classical projective geometry.The maximum likelihood degree of a statistical model quantitiesthe algebraic complexity of a fundamental problem in estimation,namely, to identify the parameters of a model which best explain the observed data. The work in tropical geometry will solidify the foundations of this new field, and it will lead to new methodsfor solving systems of algebraic equations numerically.

热带几何，研究分段线性空间，它表示具有非阿基米德赋值的域上的代数簇;系统发生代数几何，研究从系统发生树的统计模型导出的单有理代数簇;最大似然度，它是投影n空间中的一个新的不变量，具有由与n+2超平面相交定义的杰出因子;由拟阵产生的环面簇，其目的是解决组合学和交换代数接口上的基本开放问题。计算代数几何领域涉及代数方程组解集的算法研究，以及科学和工程中多项式模型的发展。例如，在生物序列分析中，许多广泛使用的统计模型可以由多项式约束来指定。系统发生代数几何关注这种代数表示及其对最大似然树的构建的影响。代数几何学和遗传学之间的相互作用是一条健康的双行道：计算生物学家可能受益于新的代数工具，而代数几何学家发现了一个丰富的来源，新的问题，有关对象熟悉的经典射影几何。最大似然度的统计模型quantitesthe代数复杂性的一个基本问题的估计，即，确定最能解释观测数据的模型参数。热带几何的工作将巩固这一新领域的基础，并将导致数值求解代数方程组的新方法。