Computational Algebraic Geometry

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Tropical geometry, the study of piecewise-linear spaces which represent 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 basic open questions at the interface of combinatorics and commutative algebra. The field of computational algebraic geometry is concerned with the algorithmic 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 sequence analysis 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 quantities the 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 methods for solving systems of algebraic equations numerically.