Alex Barrios

About Me

I am a number theorist at the University of St. Thomas, working in algebraic number theory, arithmetic geometry, class field theory, and Galois representations. A central focus of my research program is on bridging the gap between our theoretical understanding of elliptic curves and the explicit construction of examples, as well as the calculation of arithmetic data attached to them. In particular, I study how arithmetic phenomena vary in families of elliptic curves, with an emphasis on making these questions explicit and computable. While many of these problems are naturally studied over the field of rational numbers, I am also interested in their behavior over more general global and local fields. A guiding motivation for my research is to better understand deep conjectures, such as the Birch and Swinnerton-Dyer conjecture and the abc/Szpiro conjectures, through explicit and computable methods. 

For a list of my publications, see the Publications tab. Preprints of my works are also available on arXiv, and you can find me on Google Scholar and ORCID. 

Beyond my research, I have a strong appreciation for the history and philosophy of mathematics. In my teaching, I incorporate the historical development of the subjects I teach to emphasize that mathematics is a collaborative discipline, best understood as a continuing conversation across generations rather than the product of isolated geniuses. In this spirit, you can find my academic genealogy here, which reflects the lineage of ideas informing my work. For lecture notes, selected course materials, and older notes I typed while preparing talks or learning new topics, see the Exposition tab.

As an undergraduate, I benefited tremendously from research experiences, and that has stayed with me as a teacher and mentor. I enjoy leading undergraduate research projects in algebraic geometry and number theory and introducing students to the beauty and depth of these subjects. For a list of undergraduate projects I have previously directed, see the Undergraduate Research tab.

Outside of mathematics, I enjoy cooking, reading, running, and traveling. I am currently working my way through the Hugo and Nebula Award winners.

Education History

Purdue University, MS, PhD

Brown University, ScB

Miami-Dade College, AA

Highlights & Recent Activity

National Science Foundation DMS-2228858 ($64,920 — Co-PI) 

National Science Foundation DMS-2113782 ($548,786 — Co-PI)

Co-director of the Pomona Research in Mathematics Experience (PRiME) (2019-2014)

Project leader (joint with Manami Roy) for the Rethinking Number Theory 2 Workshop during Summer 2021

Invited lecturer for the 2021 Arizona Winter School

Organizations & Scholarly Affiliations

• American Mathematical Society

• Association for Women in Mathematics

• Math Alliance

• Mathematical Association of America

• National Association of Mathematicians