# Postdoctoral Fellow Spotlight: Sarah Dijols

**Can you tell me about your research background, how you became interested in this field and why you chose UBC?**

I work in representation theory, and more precisely on the representation of reductive groups over p-adic fields. This is also closely related to automorphic (or modular) forms which can be thought as functions on reductive groups over global fields with many symmetries. A quote attributed to the mathematician Martin Eichler is that "There are five fundamental operations in mathematics: Addition, subtraction, multiplication, division and modular forms.”

I think I accidently ended up working in this area! As a master student in Jussieu, Paris, I asked one of my favorite instructor from undergrad, who specialized in hyperbolic geometry, if he could be my master thesis advisor. I told him I was looking for a topic at the intersection of number theory and geometry, which were my favorite topics. I may have heard about the Langlands program at the time, but had no idea what it was, and he told me: "If you are looking for something at this intersection, then you should work on the Langlands program". He then sent me to meet Jens Funke who became my main advisor for the master thesis and taught me a lot about theta series and L-functions. I loved that topic, and felt I had been sent in the right direction. I chose UBC mostly because of my postdoc mentor here, Julia Gordon. It happens that UBC was also familiar to me as I was an exchange undergraduate student here in 2010!

**What is the focus of your current research, and what are the key questions or problems you are addressing?**

I am currently working on three different projects. I will describe them in chronological order:

The first one addresses the question of classifying the representations of the exceptional (p-adic) group G2 which are distinguished by its subgroup SO4 (meaning they have a SO4 invariant linear form). I have already written a paper on this case, but the current work, joint with Nadir Matringe, aims at completing it, and bringing in some new perspective. The quotient G2/SO4 being a symmetric space, we can mostly apply some known recipe to discover those representations. The difficulty lies in the fact that the group G2 is very bizarre, for instance this quotient corresponds to the set of quaternionic subalgebras of the p-adic split octonions!

On another project, joint with Taiwang Deng, we are exploring how a certain root system known as \Sigma_sigma could be related to certain endoscopic subgroups in the context of the local Langlands correspondance.

In a third project with Mishty Ray, we are working on the description of the ABV-packet (a packet of representations) associated to a specific Arthur parameter of G2 (again!) that Mishty studied in a different paper.

**What methodologies or techniques are you using in your research, and why did you choose them?**

I think my area is good at combining techniques developed in different other areas. For instance, in the G2/SO4 project, there is a lot of geometric intuition coming in, and some linear algebra. Recently, I have been learning techniques and concepts coming from geometric representation theory that we can use in the project with Mishty (in particular) and there I feel that again algebra won't be as useful as developing one's geometric intuition and finding mental pictures for a number of objects. I think each project's theorical constraints pushes the need to learn and use a limited set of techniques, and in my experience, I have rarely found different techniques to tackle the same question (albeit that has happened!) so I would say that my choice mostly lies in what question/topic to study.

**What have been some of the biggest challenges or obstacles you've faced in your research, and how have you addressed them?**

Surprisingly maybe, the biggest obstacles I have faced in research were more related to the context of my research than to the research itself (which can be challenging too but in a richer way). I was very isolated during my PhD years, and in an environment that, back then and retrospectively (based on observing other PhD students and advisors around me in the past many years) I see as uncaring, and even hostile toward the end of my PhD as my independance was becoming more obvious.

After my PhD, I therefore had to win back the confidence in myself I had progressively lost, relearn the capacity to discuss mathematics in a relax way, and to enjoy myself doing it. This may sound very surprising from a North-American perspective, where the well-being and respect due to students or learners, or even professionals in general, seem quite central, and their opinions is being asked and listened to. This is not as much the case in French universities, and my story is not so unusual there.

**Can you share any significant findings or outcomes from your research so far?**

I find it difficult to calibrate what is "significant" in my area, and it might be significant to someone who knows the history and the context of a problem, and very technical and narrow to someone who doesn't. I guess results which have been significant to me were the ones that either took me quite some time to reach (for instance the proof of the generalized injectivity conjecture) or contradicted some initial intuition I had. For instance, I have finally ended some computations on the root systems \Sigma_sigma using SageMath in the context of exceptional groups, and the results are surprising, and reveal some interesting patterns (and even some possible connection to a seemingly unrelated concept, but this is still a secret, as it is part of the work in progress with Taiwang Deng).

**What are your future research plans and career goals, and how do you see your work evolving over the next few years?**

Outside the three projects I already mentioned, I have started some discussion with my mentor Julia Gordon here, and we are exploring the possibility to characterize some properties of Langlands parameters using nilpotent orbits. I also started discussing with Felix Baril Boudreau, a former PIMS postdoc at the University of Lethbridge, some ideas toward a joint work. I would love to continue in academia, and to get a job in Europe or in France, to be closer to my relatives. The last couple of years in Canada have been enriching and productive for me and I am trying to take as much as possible advantage of the excellent work conditions and opportunities given here!