Álvaro Lozano-Robledo, Ph.D.
Professor University of Connecticut
- Storrs CT
Mathematics professor specializing in number theory, arithmetic geometry, and elliptic curves.
Biography
His work focuses on rational points on abelian varieties, including elliptic curves and modular curves; fields of definition of torsion points; and Galois representations associated with abelian varieties. His research has appeared in journals including Mathematische Annalen, Mathematics of Computation, Algebra & Number Theory, the Journal of Number Theory and Transactions of the London Mathematical Society.
Lozano-Robledo also develops expository and instructional mathematics materials. He is the author of books on elliptic curves, modular forms, L-functions, number theory and arithmetic geometry, and creates online lectures and courses covering topics in mathematics. His public-facing work includes explanations of advanced mathematical concepts for students and broader audiences.
He earned a Ph.D. in mathematics from Boston University and a mathematics degree from Universidad Autónoma de Madrid, with additional mathematical study at Imperial College London. Before joining the University of Connecticut, he held academic positions at Cornell University and Colby College.
Areas of Expertise
Education
Boston University
Ph.D.
Mathematics
2004
Universidad Autónoma de Madrid
Licenciado en Matemáticas
Mathematics
2000
Imperial College of Science, Technology and Medicine
International Diploma in Mathematics
Mathematics
1999
Languages
- Spanish
- English
Accomplishments
AMS 2027 Einstein Public Lecture in Mathematics
The AMS Einstein Public Lecture in Mathematics is an annual invited address given by a mathematician about a compelling mathematical topic of broad appeal. These lectures began in 2005 as a centenary celebration of Albert Einstein's annus mirabilis, when he published four papers that revolutionized humanity's fundamental understanding of physics.
Visiting Research Professor, Harvard University
August 2022 to December 2022
Mathematics Junior and Visiting Faculty Teaching Award
Cornell University - November 2007
Dean’s Fellowship
Sept 2000 - June 2001, Boston University
Premio Extraordinario de Matematicas
Best math major graduating in 2000, out of 400 matriculating students.
June 2000, Universidad Autónoma de Madrid, Spain.
Links
Social
Media
Media Appearances
AI’s mathematics blitz pushes science into strange new territory
The Economist online
2026-10-12
In early September OpenAI, an American firm, claimed one of its internal AI models had solved a Millennium Prize problem: the Navier-Stokes existence and smoothness problem, which relates to the motion of fluids. It was only the second of the lot to be conquered and the first to fall to a machine. Mathematicians had been wrestling with this one for at least a century. The AI solved it in 88 hours.
Then came what Álvaro Lozano-Robledo, a mathematician at the University of Connecticut, has called “possibly the single most important day in the history of mathematics thus far”. On October 6th OpenAI released a torrent of 719 papers detailing solutions the same model had found to at least 372 other long-standing open problems. No field of research has ever experienced an event like it.
‘Pure insanity’: Mathematicians will need years to make sense of OpenAI’s latest drop
The Verge online
2026-10-09
The sheer volume of work makes even a preliminary assessment as to exactly what the company has released difficult. In the hours and days following the drop, most mathematicians The Verge spoke with said they were still struggling to digest everything; several said that simply working through the roughly 40-page table of contents and abstracts took them the better part of an hour. “Just going over the entire list of abstracts is overwhelming,” said Álvaro Lozano-Robledo, a professor of mathematics at the University of Connecticut.
OpenAI posts 700 maths preprints online: mathematicians are up in arms
Nature online
2026-10-07
Although some mathematicians celebrated the solution of long-standing problems, others took to social media to complain about being scooped. Some were incensed at what one mathematician called a slopocalypse, even if the mathematical content could end up being formally correct.
The sheer number of papers posted at the same time is “staggering”, wrote Alvaro Lozano-Robledo, a number theorist at the University of Connecticut in Storrs, on the social-media platform X. “Why is OpenAI trying to solve hundreds of math problems to be released all at once?”, he added.
OpenAI keeps bulldozing mathematicians
The Verge online
2026-09-28
“This is not how any academic behaves,” said Álvaro Lozano-Robledo, a professor of mathematics at University of Connecticut. “I don’t go around saying, like, ‘Oh, I’ve proved all these things, but I don’t know what to do with them.’”
He sees a similar disconnect in other OpenAI pronouncements, such as when OpenAI’s Laurance Fauconnet told The Verge the company had “made substantial progress” on another Millennium Prize problem. “No mathematician would go out and say that,” Lozano-Robledo said. “You either have solved it, or you’re still trying.”
Inside the generational clash over AI in math: 'We don’t have the luxury of waiting'
Business Insider online
2026-09-14
Math professors, for their part, told Business Insider that despite the alluring headlines, AI's work in math is often quick and dirty and requires hours of human-led research to make sense of the solutions.
"If the machine understands something new, then there is no progress," said Álvaro Lozano-Robledo, a professor at the University of Connecticut. "The actual progress is when human knowledge advances."
Only Geniuses Can Solve This Viral Math Equation—Can You?
Reader's Digest
2025-03-07
“Math is very clear,” says Lozano-Robledo. “But I can imagine people who have been removed from math for a long time might not remember what goes first.”
Two teenagers have once again proved an ancient math rule
Science News online
2024-10-28
“Some people have the impression that you have to be in academia for years and years before you can actually produce some new mathematics,” says mathematician Álvaro Lozano-Robledo of the University of Connecticut in Storrs. But, he says, Jackson and Johnson demonstrate that “you can make a splash even as a high school student.”
Using TikTok to conquer math phobia
WILI Radio radio
2024-06-27
UConn Math Professor Álvaro Lozano-Robledo speaks with host Wayne Norman about how he's using TikTok to help conquer math phobia.
New Orleans teens’ Pythagorean proof gains compelling evidence
The Guardian online
2023-04-08
Lozano-Robledo reviewed MathTrain’s reconstruction, broke it down in his own video and concluded that the students had done what they said.
In a follow-up video, he summarized how the proof involved “a fractal of similar triangles” as well as “infinite series” to compute the shapes’ sides.
“It’s so ingenious,” Lozano-Robledo said. “The proof itself is just so beautiful and so elegant.”
Research Focus
Arithmetic geometry
The interaction of algebraic number theory and algebraic geometry
Research focused on arithmetic geometry, including rational points on abelian varieties such as elliptic curves and modular curves.
Elliptic curves and torsion points
Fields of definition and arithmetic properties
Studies torsion points on elliptic curves, division fields, torsion subgroups and the degrees and ramification behavior of fields in which these points are defined.
Galois representations
Representations attached to abelian varieties
Investigates Galois representations associated with elliptic curves and other abelian varieties, including representations related to complex multiplication.
Articles
What should we tell our students?
Terry Tao Blog2026-10-08
TL;DR: Keep calm and carry on studying math.
I would like to give Terry my heartfelt thanks for giving me the opportunity to contribute a post to his blog. After giving much thought to what topic I should write about to maximize impact, I decided to take this opportunity to reach out to the students: particularly to those undergraduate and graduate students who just a few months ago were dreaming of an academic career in mathematics, but their dreams may now seem distant and, for some, apparently impossible to ever become a reality. This post was inspired by a message (quoted below in its entirety, with permission) that I received from a student desperately looking for advice and guidance. This is not the only such message I have received (and I suspect that many of us are receiving many similar requests), but it is perhaps the most heartfelt, and the one that has moved me the most. Please also note the urgency of the message. Students are making decisions now.
Invest in Human Mathematicians!
Proofs and Prompts2026-08-31
In one of the most special and emotional moments I have witnessed during a mathematics conference, Benedict (Dick) Gross’s face unexpectedly appeared on all three of the large screens of the auditorium 10-250 and, after some initial surprised gasps, the audience went completely silent. For the duration of the video (about ten minutes), the number theorists gathered at MIT for the “Gross–Zagier formula, 40 years later” conference watched quietly, and some of us with teary eyes, while Gross explained to us how the formula came to be, in a collaboration that already is a beautiful and critical moment in the history of mathematics.
Teaching Debut
Notices of the American Mathematical SocietyÁlvaro Lozano‐Robledo
2026-08-07
I am in my office in the Mathematics building at 111 Cummington Street, on the Boston University campus, fidgeting while I go over my notes for the -nth time — just a few minutes before my very first class. As a teaching assistant (TA), I am only responsible for four weekly 50-minute discussion sections with about 20 students in each, but I am seriously nervous nonetheless. After all, it is the very first time I will be in front of a classroom.
How to Referee a (Math) Paper by Álvaro Lozano-Robledo
Notices of the American Mathematical SocietyÁlvaro Lozano‐Robledo
2023-01-01
You can run but you can’t hide. Eventually, an editor will find you and send you a referee request.
Now what?
If you do not have any previous experience on what to do next, this article is for you. I would also recommend Arend Bayer’s “Writing, and reading, referee reports,” and Brian Katz’s “What makes a good PRIMUS review.”
The Day She Proved It
The Mathematical IntelligencerÁlvaro Lozano‐Robledo
2022-06-30
Claudia Suárez woke up and immediately knew she had solved it. It had been one of those nights when her subconscious mind kept working on mathematics all night long while her conscious self tried in vain to rest. She had been tossing, turning, tossing and turning, while her neurons, taking no prisoners, kept inexorably marching toward a proof of the conjecture she was obsessed with. A few times she awoke briefly, in a daze, and fell back into a light slumber only to wake up again after what seemed just a few minutes of poor-quality sleep. During the rare occasions when she actually reached a deeper part of a sleep cycle, her dreams confounded plausible reality with abstract concepts from her research in disconcerting ways that triggered spikes in her anxiety levels. In what was a recurring dream, a surgeon had to remove a cohomological theory from her brain to save her life, and with the first incision, she would abruptly wake up still feeling the pain of sharp needles digging into her scalp.
A classification of isogeny‐torsion graphs of Q‐isogeny classes of elliptic curves
Transactions of the London Mathematical SocietyGaren Chiloyan, Álvaro Lozano‐Robledo
2021-01-14
Let ℰ be a ℚ -isogeny class of elliptic curves defined over ℚ. The isogeny graph associated to ℰ is a graph which has a vertex for each elliptic curve in the ℚ-isogeny class ℰ, and an edge for each cyclic ℚ-isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge.
In this paper, we define an isogeny-torsion graph to be an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over ℚ of the corresponding elliptic curve. Then, the main result of the paper is a classification of all the possible isogeny-torsion graphs that occur for ℚ-isogeny classes of elliptic curves defined over the rationals.
