Introduction
Dr. Barrios' research focuses on bridging the gap between the theoretical understanding of elliptic curves and the explicit construction of examples and the calculation of data pertaining to elliptic curves. Elliptic curves have provided the mathematical bridge to solving intractable problems in number theory, such as Fermat's Last Theorem. His recent works include the explicit classification of rational elliptic curves with non-trivial isogeny and the classification of minimal discriminants and local data (joint work with Manami Roy) of rational elliptic curves with a non-trivial torsion subgroup. These works have allowed him to further study the abc conjecture, which has an equivalent formulation in the language of elliptic curves.
Dr. Barrios is a number theorist specializing in Diophantine geometry, a branch of mathematics that focuses on solving number-theoretic questions through techniques in algebra and geometry. Since 2019, he has co-directed the NSF-funded Pomona Research in Mathematics Experience (PRiME), an 8-week summer residential program that introduces undergraduates to algebraic geometry and number theory research. Several of his undergraduate research groups have received presentation awards for their research at national conferences.