Exciting Mathematics and L-functions: Meet our PIMS PDF, Félix Boudreau, at the University of Lethbridge.
By Lisa Sammoh, Communications and Events Assistant.
Félix Baril Boudreau is a 2022 PIMS Postdoctoral Fellow at the University of Lethbridge. He completed his PhD at the University of Western Ontario, where he studied arithmetic geometry and specialised in elliptic curves over function fields and their L-functions under the supervision of Dr. Chris Hall. Félix’s main research area explores number theoretic questions that can be stated over function fields, and in particular, elliptic curves ( more generally, Abelian varieties — elliptic curves are examples of Abelian varieties) and their L-functions, as well as L-functions defined by characters like Dirichlet L-functions and Artin L-functions. He also obtained Masters degrees at the University of Western Ontario, the Université Pierre et Marie Curie in France, and a Bachelor degree at the Université de Sherbrooke in Québec. We connected with Félix to learn more about his academic journey and the postdoctoral work he does now at the University of Lethbridge.
Tell us about your academic history and how you connected with your current PDF supervisor.
My PhD was at the University of Western Ontario, where with the guidance of my first advisor, I enjoyed working on subjects and questions involving stacks and twisted sheaves. About halfway through, I had to suddenly change my supervisor. I began working with Prof. Chris Hall on a new problem involving elliptic curves over function fields and their L-functions. Then the COVID-19 pandemic happened and everything had to be moved online. This, and also having a small child at home, made it all the more challenging. I was very fortunate to have had the support of my wife during this time and throughout my studies. Moreover, I appreciated Chris’s passion for elliptic curves over function fields and their L-functions. His enthusiasm encouraged me to excel in research and overall helped me develop a general culture in my research area while also keeping a healthy balance with my family life. During this time, I was even able to write my first paper: L-Functions of Elliptic Curves Modulo Integers (now submitted for publication).
Interestingly, I began my undergraduate studies in physics at the Université de Sherbrooke (Québec) before switching to mathematics the following term. I’d quickly realised that what I enjoyed most was actually understanding the underlying mathematics used in describing physical phenomena. A highlight I remember spurring a desire to seriously pursue research was when I took an introductory research semester in France on Optimal Transport, with Fields medallist Cédric Villani at Institut Henri Poincaré and with Étienne Ghys at ENS Lyon. It was Ghys who introduced me to the world of Algebraic Geometry via the Zariski topology. I remember feeling amazed at how one could define a topological space whose points were prime ideals.
After that, I spent a bit more time in France. I did a Master 1 at the Université Pierre et Marie Curie, where I was exposed to more of Algebraic Geometry. I was also accepted for a Master 2 at the University of Paris-Saclay (formerly the Paris-Sud University) and expanded on the opportunity for an even deeper exposure to Algebraic Geometry and Algebraic Number Theory. The experience overall strengthened my resilience and perseverance, and I had this new understanding and broader culture of these parts of mathematics. I then moved back to Canada, married my wife in Mexico, and moved together to London, Ontario, where I completed my Master’s degree at the University of Western Ontario.
When it came time to apply for a postdoctoral position, I learned that the University of Lethbridge had a team of various mathematicians who were working on the analytic aspects of L-functions. I saw it as an opportunity to get a new point of view on something that already interested me. I reached out to Prof. Amir Akbary who, together with Prof. Andrew Fiori, warmly encouraged me to be a candidate for the PIMS fellowship position with them. I applied, and it was very exciting to receive an offer! With Amir, I get to work on a fascinating problem about distributions of some properties of L-functions using probabilistic random models. With Andrew, who is an Algebraic Geometer, we study some theoretic and computational aspects of the Langlands correspondence. In both cases, I can build on my previous knowledge and also broaden my research scope.
My wife is Mexican, and so, on a family trip there one summer, I had the opportunity to visit CIMAT (Centro de Investigación en Matemáticas) in Guanajuato, where I met with my future collaborators, Cristhian Garay and Pedro Luis del Ángel. Actually, Cristhian is currently visiting me at ULethbridge to finish working on a joint paper. It has flavours of combinatorics, algebraic geometry, and tropical geometry.
Give us more insight into your field of research and what you are currently working on.
My current main playing field is in function fields. From this, I like to grow and think about questions in Arithmetic Geometry and Analytic Number Theory. As I mentioned earlier, I like to study elliptic curves (and more generally Abelian varieties) and their L-functions (but also L-functions defined by characters such as Dirichlet L-functions and more generally Artin L- functions). Under the right set of assumptions, these L-functions are polynomials with integer coefficients.
Finding ways to determine these coefficients explicitly as well as computing their order of vanishing at some special values are questions that I started studying during my PhD thesis, and am now considering new ways to study these objects with Prof. Amir Akbary at ULethbridge. When we consider an Abelian variety defined over a given function field, part of the famous Birch and Swinnerton-Dyer conjecture says that the order of vanishing of its L-function at some specific value equals the finite rank of the group of points of that Abelian variety in that field. The former is called the analytic rank of the Abelian variety, while the latter is called its algebraic rank. In a joint work in progress with Aaron Levin (Michigan State University, USA) and Jean Gillibert (Université de Toulouse 2, France), we study upper bounds of algebraic ranks of Abelian varieties over function fields.
In fact, my research interests are broader than questions over function fields. With Prof. Andrew Fiori, we have begun to explore various questions of classification and computability of perverse sheaves that appear in the Langlands correspondence. Additionally, I’m working with Cristhian Garay and Pedro Luis del Ángel (CIMAT) on problems involving some form of Tropical Geometry, together with Combinatorics and Algebraic Geometry.
I also really enjoy the social aspect of collaborative research in mathematics. I see doing research with others as a wonderful opportunity to enrich one’s perspective from diverse points of view, to travel and discover the world while working toward a common goal and, in some cases, to build durable human relationships. This academic year, I am the organiser of the Lethbridge Number Theory and Combinatorics Seminar. It gives me the opportunity to be exposed to fascinating works from others and possibly find opportunities to start new research projects.
I also taught Math 1560 — Calculus I in Fall 2022. I enjoyed the overall learning experience and its challenges working on ways to transmit meaningful information and knowledge to students. I appreciated the opportunity to receive feedback from students during the semester as it helped me to fine-tune my teaching. There is certainly room for improvement with this first teaching experience, and I am now looking forward to Summer 2023 where I will be teaching Calculus I again.
What has life been like since arriving at Lethbridge, Alberta?
My best discovery since arriving at ULethbridge has certainly been the people. We have met many kind and generous people, both at the University and outside. It has helped a lot to feel welcome in this new city. Mathematics is wonderful and I feel passionate about it, but it does not define me. So I find it important to balance my family life, external social life, and research and teaching. I really enjoy spending time with my family, reading, playing board games like chess, as well as going fishing and other outdoor activities. On Sunday mornings, my family attends fellowship at a local church. In the afternoons, we either have family activities around or near Lethbridge, or we simply rest and catch up with friends and family. Currently, I am looking forward to learning new and exciting mathematics from my ongoing research and through collaboration with others in Alberta and all across the PIMS network and beyond!
Félix will be speaking at the PIMS Emergent Research Seminar Series, on March 1, 2023, at 9:30 AM Pacific. Details on his talk and poster, L-Functions of Elliptic Curves Modulo Integers, can be found here.
