Approximation Theory and Elementary Submodels

解决复杂数学问题的一种常见方法是尝试较简单版本的问题，并看看较容易问题的解决方案是否能为困难的问题提供一些见解。同样，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，微积分使用非常简单的直线和矩形的概念来逼近非常复杂的函数和区域；当一个函数不能用直线很好地逼近时，通常可以使用次简单的函数(多项式)来进行逼近。这个项目将为研究生提供研究培训的机会。接近问题在代数中也很常见，它们的形式通常是：假设C是复杂范畴K的一个子范畴，C可以用来逼近或提取关于环境范畴K的任意成员的信息吗？在同调代数和同伦理论中，逼近问题以多种形式出现，对这类问题的肯定解可以为K提供有用的新的不变量。从20世纪70年代谢拉关于怀特黑德问题的工作开始，集合论方法在处理这类问题时出人意料地有效，这些问题往往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相结合的方法来解决这一领域的一些公开问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.