New Directions in Computational Algebraic Geometry

该项目开发了计算代数几何的新算法和新应用。它集中在以下四个特定的主题：计算变形虫，图形模型的环代数，半确定规划和真正的Nullstellensatz。它涉及到组合学、交换代数和符号计算的工具，特别是格罗布纳基的方法。研究者还写了两本书，一本是关于多项式方程组的求解，另一本是关于组合交换代数的。这个项目为计算代数开发新的算法和新的应用。该项目的部分研究是与其他领域的研究人员合作进行的，并应用于统计学、生物信息学和计算机科学。代数方法的使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.