New Directions in Computational Algebraic Geometry

这个项目为计算代数几何开发了新的算法和新的应用。这项研究主要集中在以下四个主题：计算变形器、图模型的环面代数、半定规划和实零点化。它涉及组合学、交换代数和符号计算的工具，特别是Grobner基的方法。研究人员还写了两本书，一本是关于求解多项式方程组的，另一本是关于组合交换代数的。这个项目开发了计算代数的新算法和新的应用。本项目的研究部分是与其他领域的研究人员合作的，它在统计学、生物信息学和计算机科学中都有应用。在美国国家统计科学研究所提供的美国人口普查数据分析中，代数方法的使用发挥着越来越重要的作用，本项目为这一应用提供了基础研究。在计算机科学中，这些技术被用于机器学习和人工智能。

This project develops new algorithms and new applications forcomputational algebraic geometry. If focusses on the followingfour specific topics: Computing amoebas, toric algebra of graphicalmodels, semi-definite programming and the real Nullstellensatz.It involves tools from combinatorics, commutative algebra andsymbolic computation, specifically the method of Grobner bases.The investigator also writes two books, one on Solving Systems of Polynomial Equations, and the other on Combinatorial Commutative Algebra.This project develops new algorithms and new applications forcomputational algebra. The research in this project is partlycollaborative with researchersin other fields, and it has applications to Statistics, Bioinformaticsand Computer Science. The use of algebraic methods plays an increasingly important role for the analysis of the U.S. census data provided by theNational Institute for Statistical Sciences, and this project providesfoundational research for this application. In Computer Science, thesetechniques are used in machine learning and artificial intelligence.