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 places my own training within a lineage of mentors and mentees spanning nearly a millennium. It is a meaningful reminder of how deeply mathematics depends on the work of others, and a tribute to the many mathematicians whose ideas, teaching, and mentorship have helped bring me to where I am today; in a more modest sense, my Erdős number of 3 offers another small reflection of this collaborative tradition. This same interest also appears in some of the lecture notes, selected course materials, and older notes I typed while preparing talks or learning new topics, which you can find in the Exposition tab.

As an undergraduate, I benefited tremendously from research experiences, and that has stayed with me as a teacher and mentor. In particular, it was through the Mathematical Sciences Research Institute Undergraduate Program (MSRI-UP) that I was first introduced to research in elliptic curves and the abc conjecture, under the guidance of Edray Goins, who would later become my PhD advisor. Experiences such as these have strongly influenced my approach to mentoring and have galvanized me to create opportunities for students to engage in research in algebraic geometry and number theory, and to discover the beauty and depth of these subjects. I am one of the directors of Pomona Research in Mathematics Experience (PRiME), a REU and summer learning community designed to introduce undergraduates to research in these areas; it is currently on hiatus, with plans to resume in the near future. For a list of undergraduate projects I have directed, see the Undergraduate Research tab.

Outside of mathematics, I enjoy cooking, flight simulation, 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

Invited lecturer for the Jumpstart Math Workshop

AMS Research Enhancement Grant ($10,800 — PI)

University of St. Thomas Research Grant ($13,840 — PI)

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)

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

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