Approximation Theory and Elementary Submodels

解决困难的数学问题的常见方法是尝试一个简单的问题，看看简单问题的解决方案是否能给困难问题带来启示。类似地，简单的概念经常被用来近似、阐明甚至定义更复杂的概念。例如，微积分使用极其简单的直线和矩形概念来近似非常复杂的函数和区域；当一个函数不能用直线很好地近似时，次简单的函数（多项式）通常可以用来做近似。本项目将为研究生提供研究训练机会。代数中也有很多近似问题，它们通常有这样的形式：假设C是一个复杂类别K的子类别，C可以用来近似或提取周围类别K的任意成员的信息吗？近似问题以多种形式出现在同调代数和同伦理论中，这些问题的肯定解可以为k提供有用的新不变量。从20世纪70年代希拉在Whitehead问题上的工作开始，集合论方法在处理这类问题方面令人惊讶地有效，这些问题通常被证明是独立于数学公理的。该项目将采用集合论和代数方法的结合来解决这个领域的一些开放问题。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。

A common approach to a difficult mathematical problem is to try an easier version of the problem, and see if solutions to the easier problem can give any insights for the difficult one. Similarly, simple concepts are often used to approximate, illuminate, or even define more complicated concepts. For example, calculus uses the extremely simple concepts of lines and rectangles to approximate very complicated functions and regions; and when a function cannot be nicely approximated by a line, the next-simplest kind of function (a polynomial) can often be used to do the approximation. This project will provide research training opportunities for graduate students.Approximation Problems abound in algebra too, where they often have the form: suppose C is a subcategory of a complicated category K. Can C be used to approximate, or extract information about, arbitrary members of the ambient category K? Approximation Problems appear in many guises in homological algebra and homotopy theory, and affirmative solutions to such problems can provide useful new invariants for K. Starting with Shelah's work on the Whitehead Problem in the 1970s, set-theoretic methods have been surprisingly effective in dealing with such questions, which often turn out to be independent of the axioms of mathematics. The project will employ a combination of set-theoretic and algebraic methods to address some open problems in this area.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.