This project studies rational functions. Rational functions arise in modeling and computation in complex systems and provide a flexible yet numerically tractable means of describing a wide range of phenomena. Their study is key to advancing knowledge in areas such as signal processing, dynamical systems, and scientific computing. This project seeks to contribute to broader progress in the mathematical sciences by developing new understanding of the structure of the space of rational functions, with an eye towards applications as well as advancing foundational knowledge in analysis and dynamics. As part of the project, future generations of mathematicians are trained through the establishment of a yearly undergraduate summer research program and through engagement in outreach activities involving the broader community. Rational functions are usually described using zeros and poles as parameters, or by coefficients in a representation as the quotient of two polynomials. These descriptions do not capture important aspects of the structure of this space. The investigator plans to use the theory of quasiconformal mappings in one complex variable to parametrize spaces of rational functions using critical points as parameters, together with combinatorially defined objects needed to distinguish between functions sharing a common set of critical points. This approach is expected to lead to new understanding of the space of rational functions, with potential applications to analysis, dynamical systems, and topology. 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.