A study of hypersurface singularities

该项目将研究多项式方程系统的微妙几何方面，同时培养学术界和工业界的下一代科学家。多项式是可以用加法、减法和乘法等基本运算形成的任何方程，而求解多项式方程系统意味着同时求解多个多项式方程。多项式系统虽然定义简单，但却足够丰富，可以描述或建模许多有趣的现象。它们在许多不同的科学领域中无处不在，包括计算机科学、生物学、化学、物理学和工程学，并在这些领域的工业应用中发挥着关键作用。多项式系统可能涉及许多不同的未知数或变量，并且在应用中，这些变量的数量通常相当大（例如，在系统中有数千个变量并不罕见）。不幸的是，当我们处理4个或更多变量时，使我们能够可视化和研究多项式方程的绘图技术就不再适用了。这个项目的重点是克服这一障碍，通过开发系统的方法来描述和精确测量多项式系统的复杂性，无论变量的数量。此外，这将以代数的、离散的方式完成；也就是说，使用可以翻译成计算机语言的方法。在这个过程中，PI将培训、指导和支持学生。他将培养研究生进行技术数学研究，使用开源计算语言实现他们的结果，并有效地将他们的结果传达给广泛的受众。这将为这些学生在学术研究和教学以及工业领域的职业生涯做好准备。在本科阶段，我们将指导学生，特别是那些来自代表性不足的群体的学生，为他们在工业领域的成功职业生涯做好准备，也为STEM或邻近领域的研究生院做好准备。更准确地说，PI将通过理解与其相关的不变量来研究代数变量或多项式方程系统的奇异性。不变量是以一致的方式从一个变量衍生而来的对象（如数字、环中的理想或多面体形状），并且可以有意义地、精确地量化该变量的细微特性。所研究的不变量既有离散的（例如，用素数特征的Frobenius态射定义），也有连续的（例如，用可积性条件或奇点解析定义）。PI将采用许多具体的技术，包括一些基于凸几何，多面体几何和线性优化问题的变体，并将产生有效的算法来显式计算许多有趣的奇点不变量。这些算法将在诸如Macaulay2之类的开源计算机代数系统中实现。该项目由代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will study subtle geometric aspects of systems of polynomial equations, while at the same time training the next generation of scientists, both in academics and industry. A polynomial is any equation that can be formed using the fundamental operations of addition, subtraction, and multiplication, and solving a system of polynomial equations means solving multiple polynomial equations simultaneously. Polynomial systems, though simple to define, are rich enough to describe, or model, many interesting phenomena. They are ubiquitous in many different areas of science, including computer science, biology, chemistry, physics, and engineering, and play a critical role in the applications of these areas to industry. A polynomial system can involve many different unknowns, or variables, and in applications, the number of these variables is often quite large (e.g., having thousands of variables in a system is not rare). Unfortunately, the technique of graphing, which allows us to visualize and study polynomial equations, is no longer available to us once we are dealing with four or more variables. This project is focused on overcoming this obstacle by developing systematic ways to describe and precisely measure the complexity of polynomial systems, no matter the number of variables. Furthermore, this will done in an algebraic, discrete manner; that is, using methods that can be translated into the language of a computer. In this pursuit, the PI will train, mentor, and support students. He will train graduate students to conduct technical mathematical research, to implement their results using open-sourced computing languages, and to effectively communicate their results to a wide audience. This will prepare such students for careers in academic research and instruction, as well as in industry. At the undergraduate level, we will mentor students, especially those from under-represented groups, to prepare them for successful careers in industry, and also for graduate school in a STEM, or adjacent, field.More precisely, The PI will study the singularities of algebraic varieties, or systems of polynomial equations, through understanding invariants associated to them. Invariants are objects (such as a number, or an ideal in a ring, or a polyhedral shape) derived from a variety in a consistent manner, and that can meaningfully, precisely, quantify subtle properties of that variety. The invariants studied are both discrete (e.g., defined using the Frobenius morphism in prime characteristic), and continuous (e.g., defined in terms of integrability conditions, or resolution of singularities). The PI will employ many concrete techniques, including some based on convex geometry, polyhedral geometry and variants of linear optimization problems, and will produce effective algorithms to explicitly compute many interesting invariants of singularities. These algorithms will be implemented in open-source computer algebra systems such as Macaulay2.This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.