Computational Algebraic Geometry

点击翻译按钮获取中文摘要

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.