## My First Ever Popular Mathematics Article

I am overjoyed to write a small article in the Turkish popular mathematics journal Matematik Dünyası (Eng. World of Mathematics). Of course the article is about integer partitions. It is actually called Parçalanış Teorisine Davet (Eng. An invitation to Theory of Partitions). I tried to give some overview of what common questions are asked in this field of study.

I purchased some volumes of this journal when I was an undergraduate student in Bilkent and while doing my first masters. This is no Sudoku magazine. You can compare it with the American Mathematical Monthly without the refereed papers. It always includes some heavy hitting articles and an extensive problem section (just like Amer. Math. Monthly). No shame in admitting that I always found some parts of this journal hard to read. They let active researchers write notes that they see fit for a general audience. Sometimes some of those articles stay a little too technical for me. Nevertheless, this journal is one of those things that kept the spark alive in me. I liked mathematics but I remember losing motivation here and there. Reading about peoples excitement about the mathematics they love was/is a great refresher even when I didn’t understand what was going on mathematically.

I hope that my article stays on the fun side. I taught a partition theory course just recently, I hope that helped in keeping things light. If it encourages anyone to read into partitions, that would be wonderful!

I want to thank Ümit Işlak for suggesting that I write something for Matemaik Dünyası. I also would like to thank the managing editor Olcay Coşkun and the Turkish language editors that helped me with the proofreading. Who could have known that writing something in my native language would be harder? Because I didn’t ever write math in it and don’t know the terminology, because I form many upside-down sentences (like Yoda, I wrote), etc.

It was a wonderful experience. Can’t wait to get the physical journal in my hands. Once the article becomes open access I’ll also put a copy of it here. For now it is exclusive to Matematik Dünyası subscribers.

## November 21 – 25, 2022

This is a research in teams event that will take place at Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences (OEAW) and Research Institute for Symbolic Computation (RISC) of Johannes Kepler University (JKU). Some talks are announced below to introduce the problems and the state of the art.

This event is supported by Doktoral Program Computational Mathematics at JKU.

## Confirmed Participants

• Zafeirakis Zafeirakopoulos, 21 Nov 2022 – 2pm @RICAM Seminar Room (SP2 416-2)
Polyhedral Omega (+ Applications)
Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decomposition and Barvinok’s short rational function representations. This synthesis of ideas makes Polyhedral Omega by far the simplest algorithm for solving linear Diophantine systems available to date. After presenting the algorithm, we will see some applications and generalizations.
• Christian Krattenthaler, 22 Nov 2022 – 2pm @ RICAM Seminar Room (SP2 416-2) / Over Zoom
Identities for cylindric Schur functions
A well-known, but difficult-to-prove result is the determinantal formula for the sum of Schur functions over shapes with a restricted number of columns, due to Gordon, as realised by Bender and Knuth. I will represent affine refinements of these identities, which can be seen as identities for cylindric Schur functions.
This is joint work with JiSum Huh, Jang Soo Kim and Soichi Okada.
• Ole Warnaar, 23 Nov 2022 – 2pm @ RISC Seminar Room
Cylindric partitions
Cylindric partitions are an affine analogue of plane partitions. They were first introduced in 1997 by Gessel and Krattenthaler, and are closely related to the representation theory of the affine Lie algebra $\mathrm{A}_{r-1}^{(1)}$. In this talk I will try to explain why cylindric partitions have become such a powerful combinatorial tool for discovering new identities of the Rogers–Ramanujan type.
• Jehanne Dousse, 24 Nov 2022 – 1:30pm @ RICAM Seminar Room (SP2 416-2) / Over Zoom
Cylindric partitions and mod 8 Rogers-Ramanujan type identities
Cylindric partitions, which were introduced by Gessel and Krattenthaler in 1997, can be seen as a generalisations of integer partitions involving periodicity conditions. Since the 1980s and the founding work of Lepowsky and Wilson on Rogers-Ramanujan identities, several connections between representation theory and partition identities have emerged. In particular, Andrews, Schilling and Warnaar discovered in 1998 a family of partition identities related to characters of A_2. Recently, Corteel and Welsh established a q-difference equation satisfied by generating functions for cylindric partitions, and used it to reprove the A_2 Rogers-Ramanujan identities of Andrews, Schilling and Warnaar. In this talk, we we build on this technique to discover and prove a new family of A_2 Rogers-Ramanujan identities with modulo 8 congruence conditions.
This is joint work with Sylvie Corteel and Ali Uncu.
• Shunsuke Tsuchioka, 24 Nov 2022 – 2:45pm @ RICAM Seminar Room (SP2 416-2) / Over Zoom
A Fibonacci variant of Rogers-Ramanujan identities via crystal energy
We define a length function for a perfect crystal. As an application, we derive a variant of the Rogers-Ramanujan identities which involves (a q-analog of) the Fibonacci numbers.
• Shashank Kanade, 24 Nov 2022 – 4pm @ RICAM Seminar Room (SP2 416-2) / Over Zoom
On the Andrews-Schilling-Warnaar identities
Generalizing the usual $A_1$ Bailey machinery, Andrews-Schilling-Warnaar discovered an $A_2$ generalization of the notion of Bailey pairs and the Bailey lemma. Using this, they discovered identities involving principal characters of standard $\widehat{\mathfrak{sl_3}}$ modules. However, for every positive integral level l (barring a few low-lying cases), a majority of identities involving level l standard $\widehat{\mathfrak{sl_3}}$ modules were yet to be discovered. In a recent joint work with Russell, we have given conjectures that encompass all of the missing identities. Especially, we can prove our conjectures in levels 3 and 7 (and also levels 2, 4, 5, but these cases are not exactly new). Our proofs crucially use the Corteel-Welsh recursions governing cylindric partitions. Using a different circle of ideas, we were also able to produce various results when the levels are divisible by 3. In this talk, I will explain these developments. Time permitting, I will touch upon some exciting open problems that still remain.
• Walter Bridges, 25 Nov 2022 – 10am @ RICAM Seminar Room (SP2 416-2) / Over Zoom
Weighted cylindric partitions and sum-product identities
Corteel and Welsh recently showed how sum-product identities arise naturally from cylindric partitions.  Using work of Han and Xiong, we extend their ideas to more general structures like weighted cylindric partitions, symmetric cylindric partitions and skew double-shifted plane partitions.  In this greater generality, “all products” appear.  We discover new identities and reprove the Göllnitz–Gordon and Little Göllnitz identities, as well as some so-called Schmidt-type partition identities highlighted in recent work of Andrews and Paule.
This is joint work with Ali Uncu.
• Ali Uncu, 25 Nov 2022 – 2pm @ RICAM Seminar Room (SP2 416-2) / Over Zoom
On modulo 11 and 13 cylindric partition conjectures of Kanade-Russell
In this concluding (and to be informal) talk, I will show how we recently proved Modulo 11 and 13 cylindrical partition related conjectures of Kanade-Russell.

A legendary mathematician, amazing visionary, fun loving easy to chat overall nice person.

Bruno changes the game in everything he does. For example, we couldn’t stop at the 15 minute mark. It was an enjoyable conversation. I am glad we found the time.

## I received the ACA 2022 Early Career Researcher Award

It was a great honor to be given the Applications of Computer Algebra 2022 Conference’s Early Career Researcher Award. I want to thank the deciding committee.

This raises my motivation while also puts more responsibility on me to do the best I can in the advancement of mathematics and its interactions with computer algebra.

More on the award and the recipients are here.

Miklós is a known name for anyone interested in combinatorics. It is either due to his great introductory books that covers the fundamentals in such clear terms or it is due to his phenomenal research. Regardless of how you know the name, the person behind it is just as great.

I had the great pleasure of sit down at Istanbul and hear his opinions on some topics. I hope you’ll also enjoy it as much as I did.

## Ramanujan and Euler: Online Conference and Summer School

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.

In fact, I decided to build up these techniques in a way people fresh to the topic can understand and guide them to the last place these ideas were applied in my work and in others’.

#### Conference Portion

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.

#### Summer School

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.

Come on! I don’t need to introduce George E. Andrews, do I? I also frankly don’t have the space to write down half the things he accomplished.

Instead, I will say that I am so lucky to have met him. He is a regular visitor of University of Florida and the number theory seminars that my former advisor Alex Berkovich ran was always better with George’s presence. I was fortunate to have him in many of my academic milestones; I am honored that he was the one who announced I passed my PhD defense.

## Second Workshop on Integer Partitions

##### May 30 – June 5, 2022, @ Nesin Mathematics Village

It finally happened! Since 2018, I have been trying to find a time to organize this event. The first Workshop on Integer Partitions at the Nesin Mathematics Village (NMK, their Turkish abbreviation) was great fun and a major success. I wanted to keep the ball rolling and introduce these research topics (that I love) together with more people in a friendly environment.

As always, I felt the welcome of NMK from the start of the organizations. This time we got a new ally. Turkish Math Society (TMD, their Turkish abbreviation) also decided to support our event.

I would like to thank the Village and its wonderful team –special thanks to Asli, Aycan, and Tugce– and the TMD MAD fund for their support. I appreciate the recognition and this encourages me to do more.

### How did it start?

It was early 2020, and we didn’t know what was there to come, I first contacted NMK. Their program was full. We aimed this workshop to take place right after the Antalya Algebra Days (the major algebra conference in Turkey, which used to be in Antalya but now takes place at NMK). We couldn’t get the time we wanted. The village was busy for the whole summer. Even then they were able to carve us a small space to have this event. We started the organizations slowly only to cancel later due to the lock-downs.

In the beginning of 2022, I started by contacting my previous co-organizers Zafeirakis Zafeirakopoulos (Zaf) and Kağan Kurşungöz to see if they were up for another round of our workshop. The responses were encouraging. I knew I wanted Jehanne Dousse to be a part of this event too. I approached her. Voila! A positive response came and it got things officially moving.

It was time to contact, the Queen-Bee of the Village, Asli. I contacted her and requested some time and accommodations for this event. She was friendly as ever. It has been 12 years since we first met each other and to this day our conversion has been friendly, polite and professional. I find myself impressed by her professional conduct. We started with the basics, decided on the number of people, the level of participants we expect, the dates, etc. After some emails back-and-forth we decided to have a week-long workshop followed by couple more days for the organizers to stay and collaborate on their research projects. We aimed for the week after Antalya Algebra Days once again (although that event later got cancelled). Once our plans were solidified, Aycan took over and handled our organizational needs. She was just as great as Asli and helped me along the way leading up to the event itself.

After the organizations were done, Walter Bridges said he wanted to join. Back then, we were expecting a mixed group of students from upper level undergrads to early PhD candidates. So, I suggested that he joins us as a lecturer. He accepted and that finalized the list of people to give lectures in this event

### The Workshop

Let’s start with the timetable and the videos. You can watch all the lectures on YouTube on NMK’s channel, or directly from the links below:

Day 3

Day 4

Day 5

Honestly, we didn’t put much thought on how the program should be after the teaching group was formed. I knew what I wanted to teach, I knew it wouldn’t interfere with the others’ lectures and that was good enough for me. One thing I know for sure was that, in the last workshop we had four 90-minute lectures everyday and that was way too much for the students to endure.

Maybe a month before the workshop, I had separate conversations with Zaf and Jehanne. Hour long lectures seemed to make sense the most. That would also leave some time for us to chat and possibly collaborate. With this vague understanding we all made our way to the Village on 29th of May. We finalized the program to have 5 lectures every day (except for the research talks and the social event on Thursday) and a later 1 hour obligatory meeting time with the participants to exchange ideas and answer any and all questions.

The lecture videos above would speak for themselves. The quality of the lectures (excluding a judgment on my own lectures) were quite high and they were all at an accessible level to a wide audience. We required that accessibility because we ended up with a group of participants all the way from freshmen to postdocs. It needed to be both interesting to everyone and to be understandable. I was impressed with how the instructors (again excluding a judgment on my lectures) understood this and kept the level so right. I am so glad that the Village asked us to record the event and upload it all on their webpage. I hope these lectures will be watched and more importantly bring the theory of partitions to the attention of math-loving audience.

### The Participants

We had a mixed bag. It was quite diverse. Including the lecturers, we were 18 people from 8 nationalities coming from institutes that lie in 5 different countries. We had 3 participants that were in the in the first workshop as well. One of them, Halime, finished her PhD studies in the meantime under Kağan’s supervision. It was a really proud moment to have a former participant to join us again, this time as a colleague. The other two were at the end of their doctoral studies too. Clearly, this bunch stayed busy in between these two workshops and it was pleasant to see that mathematics were gaining some new blood, especially in this topic of study.

As mentioned earlier, we had undergrads, Masters students, PhD candidates, and postdoctoral researchers. This gave me a good opportunity to see what was being lost on different levels of participants. For example, due to the fast start, some undergrads got slightly overwhelmed on the first day. I and these couple of undergrads sat down together after dinner, while other people were having drinks and getting to know each other better. We had a chat about what are some common goals of the theory of partitions research, how things connected, what the new and weird notations meant, etc. It was good to see that they became more comfortable. It would have been an utter shame for them to come all the way to the village and not get anything out of it. I didn’t want to let that happen. These two undergrads participated more and stayed interested in the lectures, so I think it worked.

### The Mini-Symposium and The Social Event

We wanted to have a break day after the 3rd day of lectures and see the ancient city of Ephesus. It might have been a one time opportunity for international participants to be this close to that site (although I hope they will all come to NMK many more times). We decided to have our social event after lunch and this allowed us to organize a quick symposium in the workshop. After all, we had 8 participants that do research in discrete mathematics; some doing research directly in the theory of partitions.

The Thursday morning session was filled with really nice talks. I am thankful to all our participants that volunteered to give a talk. Their talks sparked many new ideas in the group and, for example, I already started to collaborate with one of them. This is what we wanted for the workshop. It created new connections.

### Post-Workshop

Jehanne, Zaf and I stayed for another three days to collaborate after the workshop. I need to admit that I was drained with all the organizational responsibilities and especially the other work that I needed to address during the event. Zaf had a lot on his plate as well. Either way, I didn’t want to slack. This was a great opportunity to work with Jehanne on our common goals. I had a wonderful time working with her and I wish we could stay there longer.

After our workshop, CIMPA summer-school came to the NMK and filled it with a new batch of young mathematicians that were hungry for research. We met the organizers of this event back in 2018 when the first workshop on partitions was taking place. It was a great coincidence that they came back to the Village 4 years later at the same time as us.

Their event was a much larger than ours and the idea was to have small groups of 4, 5 students with a researcher to tackle some new research problems that the group leader chose. I should say that it gave me some ideas for the future of the workshop on integer partitions. We already had a strong group with many researchers in it. We could have also approached the partitions workshop in the CIMPA summer-school way.

On our last night, CIMPA event had their warming up and welcome party on the Village deck. We crashed their party an hour after it started and had a blast. We let loose and danced with the Village employees, the other event’s organizers  and participants into the night (dramatization). It was a superb ending to our event and our time at NMK.

Once more I left the Village longing for more; wanting more for the Workshop; wanting more for the future; wanting more from the times to come… I believe, I wasn’t the only one that felt this way either, and –in my book– that means there will be more to come.

##### Some trivia:

• Elaine and Vishnu were the pioneers of the WhatsApp group. They fostered such a friendly environment from the get go. I couldn’t have thought of doing such a thing. Thanks to them we had great transfer of information and a fun group chat too.
• The group photo was taken by a middle-schooler named Mert that was at the Village for a couple of days on a school trip.
• Caption of the group photo: Front-row left to right: Nicolas Smoot, Vishnupriya Anupindi, Ali Kemal Uncu, Kağan Kurşungöz, Walter Bridges, Shamus Albion. Second-row left-to-right: Kevin Allen, Yalçın Can Kılıç, Alp Eren Yılmaz, Jehanne Dousse, Halime Ömrüuzun Seyrek, Elaine Wong. Third Row left to right: Ömer Selçuk, Zafeirakiz Zafeirakopoulos, Mohammad Zadeh Dabbagh, Zohreh Aliabadi, Murat Ertan. Top row: Hasan Bilgili.
• NMK approached us and requested to record our lectures right after Jehanne’s first lecture. We needed to record her lecture again at the end of the second day. It was incredible to watch Jehanne recreate the same board and teach the class almost verbatim. She should be awarded an academy award for the best reproduction of a lecture.
• I decided to organize the next event not to be on the week after Antalya Algebra Days but to organize it to be on the week before a CIMPA school. They sure know how to have fun. 🙂

Matthew England is a great collaborator. I am glad to be in the same project as him. Coventry University is also a great host. I enjoy my visits to this group a lot. After working for 3 days non-stop, we had a short time to have this chat. I am so happy that we get to talk about important questions like what motivates him to do research. I learned a lot, I think you can too.

Matthew also has great humor and such a quick wit. It is quite fun to chat with him. That is why, this time, I am not editing the early minutes of our conversation out of this video. I think you’d like it. I am so dissapointed I don’t seem like I am enjoying the jokes but I love how he “threatened me with a pen”. 😛

I was just too preoccupied, thinking how I am supposed to start the video, what to ask, and in which order, etc. These early jokes deserve much more laughter that I didn’t bring to the table… I hope you will enjoy them more than I couldn’t back then.