Approximation Theory and Elementary Submodels

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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.