I had the utmost pleasure of being invited to give some lectures at the Euler Institute’s summer school on Partitions, Mock Theta Functions and q-series. I want to thank the main organizer Eric Mortenson once again here for inviting me to this event. The conference portion was scientifically fulfilling and the summer school went really smoothly. Moreover, although it was an online event, I felt that the audience was engaging, which made the teaching experience ten-fold more enjoyable.
Short History Leading Up to the Event
This event wasn’t organized this way. Merely a year back, I was hoping to go and see the International Congress of Mathematics (ICM) at St. Petersburg. I was so happy when I got the invitation to the Euler Institute to give a scientific talk. This was supposed to take place in a satellite event of ICM 2022. Well… then things happened, foreseen and unforeseen… sad things… things that are not in the hands of mathematicians. This led to the eventual cancellation of the in-person ICM event.
Thankfully, this satellite event decided to become a stand-alone affair. They not only kept the conference (not in-person anymore) but to expand it with a summer school. I was just done with the Second Workshop on Integer Partitions at the Nesin Mathematics Village, when Eric asked me if I would be interested in giving three lectures in the summer school.
Knowing the dividing nature of the country name where Euler Institute resides. They said that they will keep a neutral website that mentions minimum of such ties… look at all the weird things conference organizers needs to deal with, I thought to myself.
How could I say no?
Teaching and sharing new research is a part of the job that I cherish. So when then invitation came from Eric, it was a no-brainer. I just wanted to teach something new. My lectures in the Workshop on Partitions were fresh. They were all recorded anyway. (I talked about Schmidt type partitions there.) I didn’t want to repeat myself. I picked one of my papers that I didn’t/couldn’t promote in the pandemic times. The ideas were fundamental and applicable and versatile.
The scientific talks from mathematics legends like George Andrews and Bruce Berndt are never to be missed. There were world leading scientists like Jeremy Lovejoy and Atul Dixit as well. I had the opportunity to talk about my recent work on cylindric partitions, and my collaborator and friend Walter Bridges followed my talk with our spin on these problems and the results we recently found.
We had some great questions from the audience. It is clear some people got interested in the topics. Let me try to add some encouragement to that interest: I am here to collaborate if you want! Just send me an email.
I learned that Walter was going to be one of the other lecturers in the summer school. We also understood that we were the ones that will stick with partition theory topics. We discussed what we wanted to teach and made sure that we don’t overlap. I was going to teach the basics of partitions and move on with combinatorial ideas to write generating functions. Walter was going to stay within bijective combinatorics to prove partition inequalities, etc.
The first lecture was mine, so I better gave a good impression and clear definitions of how we treat the objects.
Three lectures is a lot of time on paper but they go by really quickly. I was able to cover maybe 80% of what I intended on the first lecture. I couldn’t define what a partition identity is. No problem though, this made me time myself better. The second lecture was building up on how we write generating functions using combinatorial constructions and writing double sum generating functions for partitions with uniform gap conditions.
The questions after each lecture also helped me understand where the audience was, what they wanted, etc. You gotta give the audience what they want, right? At the end of my second lecture I was asked how or if the way I was writing generating functions had anything to do with my papers (joint with Alexander Berkovich) in Capparelli’s identities. The answer is, of course, or course! I added a mention of Capparelli’s finitizations to the last lecture as well.
The last lecture was there to tie things together. We wrote generating functions in different ways, wrote their polynomial refinements and mentioned how we can reflect them. In my recent work with Wadim Zudilin that’s exactly what we did anyway. This also led us to new conjectures. I wanted these new conjectures to be the highlight and a good ending of the lecture series. I wanted to pay homage to the St. Petersburg university and the Euler Institute. That was easy in this case, because a recent student of this university, Stepan Konenkov, worked on my paper with Wadim and wrote down 4 more conjectures (and 2 more theorems) using the very technique I was teaching. Turns out there were 2 more modulo 9 conjectures discovered by D. Hickerson. You can look into those even newer conjectures here.
It was a great conference and a great summer school to follow that. I was a bit disappointing at first to have this event online, but the engagement from the students, both in lecture and after the lectures via e-mail gave me a feeling of fulfillment. I was happy to be a part of this wonderful event and it was a great honor to be a summer school lecturer at the Euler Institute. Hope to visit the location, the Sketlov Institute, Chebyshev lab, and so on in a peaceful future.