Computational Algebraic Geometry

热带几何，在非阿基米德值域上表示代数变异的分段线性空间的研究；系统发生代数几何，研究由系统发生树统计模型衍生的单国家代数品种；极大似然度是一种新的不变量，它在射影n空间中具有一个由n+2个超平面的交定义的可分辨因子；由拟阵引起的环面变分，旨在解决组合学与交换代数交叉领域的基本开放问题。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通过多项式约束来指定。系统发育代数几何关注这种代数表示及其对最大似然树构造的影响。代数几何和系统发育之间的相互作用是一条健康的双向道路：计算生物学家可以从新的代数工具中受益，而代数几何学者则可以从经典投影几何中熟悉的对象中找到丰富的新问题来源。统计模型的最大似然度量化了估计中一个基本问题的代数复杂性，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，并将导致用数值方法求解代数方程组的新方法。

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.