Applications of Algebraic Geometry

代数几何研究涉及代数方程组的解集，它为数学领域的非线性问题提供了工具和模型。它现在在科学和工程中有许多应用。该项目针对四个具体方向：光谱面体和轨道、代数统计、热带几何和高维品种的隐式化。谱面体和轨道体是凸代数几何中的基本对象，是一个新兴的交叉学科方向，主要研究具有独特代数表示的凸体。它是由凸优化，特别是半定规划理论中的应用驱动的。该项目将巩固凸代数几何的发展，并将导致具有特殊代数结构的非线性优化问题的新的半数值算法。这类问题在代数统计中经常出现。该领域关注的是可以用模型参数中的多项式表示的统计模型。其中包括高斯和离散随机变量的图形模型。该项目将解决有关这些模型的基本问题，并有望为统计推断提供新的计算工具。热带几何是代数几何的分段线性版本，是为建模应用问题定制的。该项目旨在解决该领域组合方面的关键开放性问题。高维隐化是计算机代数的新范式，它将为寻找参数化代数变量的定义方程提供新的工具。

Research in algebraic geometry is concerned with the solution sets to systems of algebraic equations, and it offers tools and models for non-linear problems across the mathematical spectrum. It now has numerous applications in the sciences and in engineering. This project is aimed at four specific directions: Spectrahedra and Orbitopes, Algebraic Statistics, Tropical Geometry, and Implicitization of Higher-Dimensional Varieties.Spectrahedra and Orbitopes are fundamental objects in convex algebraic geometry, an emerging interdisciplinary direction that focuses on convex bodies with a distinguished algebraic representation.It is driven by applications in convex optimization, especially in the theory of semidefinite programming. The project will cement the development of convex algebraic geometry, and it will lead to new semi-numerical algorithms for non-linear optimization problems with a special algebraic structure. Such problems arise frequently in Algebraic Statistics. That area is concerned with statistical models that can be represented by polynomials in the model parameters. These include graphical models for both Gaussian and discrete random variables.The project will resolve fundamental questions concerning these models, and itpromises novel computational tools for statistical inference. Tropical Geometryis a piecewise-linear version of algebraic geometry that is custom-taylored for modeling applied problems. The project is aimed at key open problems on the combinatorial side of the field. Implicitization in Higher Dimensions isa new paradigm for computer algebra, and it will lead to new tools for findingthe defining equations of parametrically specified algebraic varieties.