Browsing by Author "Dawson, Erin, author"
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Item Open Access Graduate students' representational fluency in elliptic curves(Colorado State University. Libraries, 2023) Dawson, Erin, author; Cavalieri, Renzo, advisor; Ellis Hagman, Jessica, advisor; Zarestky, Jill, committee memberElliptic 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 P2 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.Item Open Access Tropical Tevelev degrees(Colorado State University. Libraries, 2025) Dawson, Erin, author; Cavalieri, Renzo, advisor; Gillespie, Maria, committee member; Miranda, Rick, committee member; Canetto, Silvia, committee memberTropical Hurwitz spaces parameterize genus g, degree d covers of a tropical rational curve with fixed branch profiles. Since tropical curves are metric graphs, this gives us a combinatorial way to study Hurwitz spaces. Tevelev degrees are the degrees of a natural finite map from the Hurwitz space to a product Mgnbar{g,n} cross Mgnbar{0,n}. In 2021, Cela, Pandharipande and Schmitt presented this interpretation of Tevelev degrees in terms of moduli spaces of Hurwitz covers. We define the tropical Tevelev degrees, Tev_g^trop in analogy to the algebraic case. We develop an explicit combinatorial construction that computes Tev_g^trop = 2^g. We prove that these tropical enumerative invariants agree with their algebraic counterparts, giving an independent tropical computation of the algebraic degrees Tev_g. We finally generalize tropical Tevelev degrees to more cases and construct computations of these invariants.