Robin Zhang
AMSI-MSRI Winter School 2022
In this interview, Robin Zhang shares how a casual dining hall conversation at the 2022 AMSI-MSRI Winter-Summer School in Hawaiʻi unexpectedly launched a new research direction and international collaboration. Robin reflects on how AMSI Winter School fostered the kind of cross-disciplinary, cross-cultural exchange that led to his recent publication.
Your recent paper tackles a problem that you first encountered during a conversation with a fellow student at the 2022 AMSI-MSRI Winter-Summer School. Can you take us back to that moment? What was it about the problem that intrigued you?
I believe it was in the dining hall at the University of Hawaiʻi at Hilo when I first encountered this problem. It was early in the 2-week program, and I had just met Antoine de Saint Germain, a PhD student at the University of Hong Kong at the time. Over dinner, I mentioned that while I was interested in representation theory (the main focus of the Winter-Summer School), my background and primary research interests were in number theory. Antoine immediately lit up and said he had a problem that could use a number theorist’s perspective. That problem turned out to be the Fontaine–Plamondon conjecture on counting friezes, which are combinatorial patterns connected to representation theory and algebraic geometry through cluster algebras. At the time, I had never heard of friezes and didn’t even know what a cluster algebra was! Antoine was incredibly patient and presented the problem in its elementary formulation in terms of arrays of integers. When he mentioned that the conjecture could be framed as a question about finding positive integer solutions to certain systems of equations, this deeply resonated with my number-theoretic background.
It was an exciting moment: here was a problem from a different area of mathematics that I felt could be tackled using number theory. Looking back, I feel incredibly fortunate to have met Antoine at the Winter-Summer School. In particular, this chance encounter shaped an entirely new research direction for me.For context, international borders were still just opening in mid-2022 so all of the MSRI participants (i.e. the non-Australian contingent) were based at the University of Hawaiʻi Hilo campus during the Winter-Summer school.
How did you and the other student initially attempt to solve the problem together? How did your partnership evolve from that first discussion?
In fact, Antoine’s then-in-progress thesis had already done some of the heavy-lifting before our conversation even started. It’s a consequence of his work with Min Huang and Jiang-Hua Lu that the Fontaine–Plamondon conjecture can be moved from the world of combinatorics & cluster algebras to the world of number theory. In a way, they give a dictionary between these two otherwise unrelated languages.
I spent a few nights in my dorm at the Winter-Summer School working on “baby versions” of the conjecture using elementary techniques in Diophantine geometry. Each day, over lunch, I would share my latest discoveries (or lack thereof) with Antoine. No matter how promising an approach seemed at first, there was always a missing step here or an incomplete argument there. By the end of our two weeks in Hawaiʻi, I couldn’t say that I had made tangible progress on the problem, but I left with a strong conviction that the problem was solvable by number theoretic methods.Antoine and I kept in touch over the next three years, in part because we became good friends during the program and in part because I kept emailing him naïve questions! At one point, he generously set aside two hours for a Zoom session to walk me through the fundamentals of the “classical” view of frieze patterns. His explanations helped me see how my arithmetic techniques fit into the broader cluster algebra framework. That masterclass was pretty essential to my understanding of the problem.
Were there any concepts or techniques from the Winter-Summer School mini-courses that helped shape your approach to solving this problem?
While the problem is closely tied to representation theory, and therefore related to the mini-courses in a broad sense, I was very happy to take Antoine’s dictionary and translate the problem to the language of number theory. In the end, my proof really relied on applying number theoretic perspectives and techniques rather than representation theoretic ones.
Your connection with Professor Uri Onn also stemmed from the Winter-Summer School. What was your first impression of his mini-course, and how did you come to interact with him during the program?
I really enjoyed Professor Uri Onn’s minicourse on “Representation Zeta Functions”. The topic was especially exciting to me because of its close connection to automorphic forms, an area at the intersection of number theory and representation theory that I was actively studying as a graduate student. I remember introducing myself and taking the opportunity to ask a few specific questions during his office hours. As a student, I rarely attended office hours in university courses, but I really appreciated that they were built into the Winter-Summer School schedule. Uri and the other instructors were very helpful and open to chatting during these office hours.
When you reconnected with Professor Onn during your postdoc applications, did you anticipate that it would lead to a research collaboration? How did the opportunity to visit ANU in 2024 come about?
No, I didn’t anticipate a research collaboration when I first reached out! I was finishing my PhD at Columbia University and considering a move to Australia, so I sent an email to Uri to ask a few questions about the postdoctoral environment in Australian universities.
I ended up taking a postdoctoral position at MIT, but we kept in touch. At some point, Uri suggested we explore a few research problems related to his mini-course and mentioned that the Australian National University (ANU) had a research visitor scheme that would make an in-person collaboration possible. This sounded like a great opportunity, so I applied to their Early Career Researcher Visitor Program. They provided funding for a 2-week visit to ANU in March-April 2024, when our collaboration began in earnest.
Without giving too much away, what research directions are you currently exploring with Professor Onn? How does this collaboration build on the foundation of your earlier work?
We’re working on a small aspect of this broad web of conjectures known as the Langlands program. Roughly speaking, it relates some of the most important objects in number theory, representation theory, and algebraic geometry to each other. One of these conjectures is the existence of the local Langlands correspondence, which was proved by Michael Harris (my doctoral advisor) and Richard Taylor for general linear groups at the turn of the century.
One aspect of the local Langlands correspondence is having a very good understanding of the representation theory of general linear groups over local fields. Uri and I are looking at constructing similar correspondences in settings where the representation theory is currently not as well-understood.
Your experiences highlight how participation in AMSI programs can lead to long-term academic and professional connections. What advice would you give to future students on making the most of opportunities like the Winter-Summer School?
I’d really encourage students to take part in programs like the Winter-Summer School, even if the focus isn’t directly aligned with their main area of interest. Some of the most valuable insights come from the cross-pollination of ideas between different fields. Often, mathematicians from seemingly unrelated areas are thinking about the same objects or problems—just from different angles.
Another benefit to stepping outside of your comfort zone is that when you’re new to a topic, it can feel easier to ask questions that others might hesitate to voice. You have the perfect excuse – no one expects you to know anything yet! Embracing that beginner mindset can lead to fresh perspectives and interesting conversations.
