Advisor: Dr. Renzo CavalieriCo-Advisor:Jess Hagman

Committee:  Dr. Jill Zarestky

Title: Graduate students’ representational fluency in elliptic curves

Abstract: Elliptic curves are an important concept in several areas of mathematics, including number theory and algebraic geometry. Within these fields, three mathematical objects have each been referred to as an elliptic curve: a complex torus, a smooth projective curve of degree 3 in $\PP^2$ with a chosen point, and a Riemann surface of genus 1 with a chosen point. In number theory and algebraic geometry, it can be beneficial to use different representations of an elliptic curve in different situations. This skill of being able to connect and translate between mathematical objects is called representational fluency. My work explores graduate students’ representational fluency in elliptic curves and investigates the importance of representational fluency as a skill for graduate students. Through interviews with graduate students and experts in the field, I conclude 3 things. First, some of the connections between the above representations are made more easily by graduate students than other connections. Second, students studying number theory have higher representational fluency in elliptic curves. Third, there are numerous benefits of representational fluency for graduate students.

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