1. How did you get started with getting acquainted with math beyond the high school curriculum? Any recommendations on specific resources?

Alvin Chen

I was exposed to math through the competition math system, and I was able to meet peers who were similarly interested in competition math, which helped me get exposed to a lot of ideas in combinatorics, algebra, etc. Eventually, I began to hear from others about learning abstract algebra and analysis, and I began to just explore some books on my own. I also attended Canada/USA mathcamp, which presents a lot of short introductory courses on various topics, which was really helpful in motivating the background needed for advanced math. Some specific resources that I used include Mathstackexchange for questions of any sort, other people in the math community that I was able to talk to, and a lot of textbooks, such as Axler's Linear Algebra Done Right or Rudin's Principles of Mathematical Analysis. There is a good list of textbooks on the PRIMES website!

Alvin Chen "K-semistability of smooth toric Fano varieties" (mentor Kai Huang)

Simon Sun

I started learning math beyond the high school curriculum through competition math; it exposed me to a variety of subjects and a community of peers who shared my interests. Through this, I learned of resources and opportunities for high schoolers interested in math. I had the opportunity to attend PROMYS, where I was exposed to axiomatic number theory and the research process. I also read some math textbooks, including some of the recommended ones on the PRIMES website.

Simon Sun "Product expansions of q-character polynomials" (mentor Dr. Nir Gadish)

2. Did you have experience with research before PRIMES? If so, which skills carried over to PRIMES the most?

Katherine Tung

No, I had not conducted original research before PRIMES, but I had written expository papers before. At SUMaC, I had written an expository paper on a coding theory topic. When I studied coding theory, I got better at reading graduate level mathematical texts. This skill helped a lot for background reading during my research process.

Katherine Tung "The Sperner property for 132-avoiding intervals in the weak order" (mentor Christian Gaetz)

Jakin Ng

I hadn’t really had serious prior math research experience. However, I was part of two group projects at Canada/USA Mathcamp, which were a lot shorter and less serious than my PRIMES project, and I also had worked in a local experimental science lab, which was definitely a good introduction to research, but quite different from math research. I think these experiences were helpful in learning about the research process and also working in a group, but PRIMES has been for me overall a pretty different experience, and I don't think it's necessary to have had prior research experience.

Jakin Ng "The Stembridge equality for skew dual stable Grothendieck polynomials" (mentor YiYu (Adela) Zhang)

Will Qin

Before PRIMES I did a decent amount of research on machine learning and autonomous driving. In CS research, I worked with math to do research, but that's very different from actually doing research in math. Later on, I did bits of math research in combinatorial game theory and differential equations which gave me a good intuition for how to approach my PRIMES project. Although the research content has varied between all my projects, my experiences were very useful in teaching me how to think like a researcher/talk to professors/read papers/write papers. That being said, you don't need research experience to do PRIMES!

Will Qin "Colored HOMFLY polynomials of genus-2 pretzel knots" (mentor Yakov Kononov, Columbia University)

Honglin Zhu

I had done math research before PRIMES. I competed in the Yau competition and went to the Tsinghua math camp. But it was either on more elementary mathematics or based on things I learned from camp. And I had only worked on group projects before. I was relatively familiar with the math research process and the longer timespan of research as compared to contest math when I started my PRIMES project.

Honglin Zhu Text "Irreducible characters for Verma modules for the orthosymplectic Lie superalgebra 𝔬𝔰𝔭(3∣∣4)" (mentor Arun Kannan)

3. What kinds of maths did you study before starting PRIMES? How much did you know about your research area when you first got your project?

Daniel Hong

Before PRIMES, my experience with math was mainly math competitions. I was unaware of my topic previous to receiving it from PRIMES, but it turns out it was mainly rooted in graph theory and linear algebra. At Canada-USA Mathcamp the year before, I had some entry-level classes in linear algebra, and I had some experience with graph theory from competition math. I didn't have much experience at all with advanced math prior to PRIMES; apart from entry-level classes at Mathcamp, I only had experience with high school calculus.

Daniel Hong Text "Optimal solutions and ranks in the max-cut SDP" (mentor Dr. Diego Cifuentes) Text

Jessica Zhang

I've done some math competitions and went to PROMYS for two summers, but what I learned from those was related to my project only in the vague sense of teaching me proof-writing and certain proof techniques. Most of the mathematical background I had came from reading books, actually. My most relevant background came from reading John Lee's books on topological manifolds and smooth manifolds (the latter of which I've only partially read). That being said, most of my project background I learned throughout the project and with the help of my mentor.

Jessica Zhang "Tight contact structures on the solid torus" (mentor Prof. Zhenkun Li, Stanford University)

Ilaria Seidel

Before PRIMES, most of my mathematical experience came from competition math and classes at school, where I'd taken up through calculus and linear algebra. My PRIMES project is very accessible, so I've hardly used the math from school classes, but I've been able to apply some of the concepts and problem solving skills from competition combinatorics (induction, graph theory, and so on). I attended USA/Canada Mathcamp last summer, and beyond introducing me to all sorts of new subjects, I gained some intuition and practice working with abstract math. I also had the opportunity to do a mini research project while I was there, and because I enjoyed it so much, I decided to apply to PRIMES!

Ilaria Seidel "Upho posets" (mentor Yibo Gao)

Honglin Zhu

I did math competitions and went to the Ross math program for two summers. I've taken calculus and multivariable calculus at school. I have learned some abstract algebra, linear algebra and group theory on my own. At Ross I learned a lot of number theory, some Galois theory, and I took advanced lectures on dynamical systems and analysis. My research topic was on the representation theory of Lie (super)algebras, and I had not even heard of Lie algebras before PRIMES. But having learned about other advanced topics helped build my mathematical maturity for the project.

Honglin Zhu "Irreducible characters for Verma modules for the orthosymplectic Lie superalgebra 𝔬𝔰𝔭(3∣∣4)" (mentor Arun Kannan)

4. Do you get to choose your research topics/problems? How much freedom do you have in deciding the course of the project? Were the projects assigned similar to what you had requested?

Yunseo Choi

I was originally assigned to the project that my mentor proposed. When I was done with the project, I chose to go in a completely new path for my second project! I ended up coming up with the main idea of the project myself (although, without my mentor, my project definitely could not have come to fruition), and this really kept me motivated, because it was genuinely something that I wanted to study! This is probably not the typical case for most PRIMES projects, however. I am working on applied math, which doesn't take too long to get acquainted with the background.

Yunseo Choi "Racial impact on infections and deaths due to COVID-19 in New York City" (mentor Prof. James Unwin, University of Illinois at Chicago)

Justin Wu

I was assigned a multi-part problem statement with multiple directions to work on. After I finished reading some prerequisite material, I chose the direction as well as a subset of the papers to read. I think freedom in deciding the course isn’t necessary for my case, I’m happy with working on the interesting questions my mentor comes up with. My topic was the 3rd project topic I requested, but I’m happy I ended up with a project with a balance between difficulty and depth as well as being extremely interesting even though it wasn’t my top choice.

Justin Wu "Borel cohomology of Sn mapping spaces" (mentor Ishan Levy)

Simon Sun

While we get the option to select what topics we are interested in working on in our application, we are assigned a mentor and research problem. My topic was one of the top choices that I had requested, so I was extremely happy about that! I had some freedom in deciding what I wanted to pursue, but I also listened to my mentor to see what he thought should be done. I think that as high schoolers working on math research, mentor guidance is important in deciding the course of one’s project.

Simon Sun "Product expansions of q-character polynomials" (mentor Dr. Nir Gadish) Text

5. How much time do you spend on PRIMES each week? What do you do during those hours? How much of the year does it take to complete a research project with PRIMES?

Jessica Zhang

It's a little hard to say how much time I spend on average, since it varies a lot. I think I once spent almost the entire week working on PRIMES (making it something like 80 hours?) but at other times I will go a whole week without working on my project. I think the easiest way to put it is that I work how much I want to or am motivated to (and, luckily, I quite like my project and so generally quite enjoy working on it!). My project is very geometric, so a lot of this time is spent drawing pictures and trying to work out proofs of small cases. But most of the time, in truth, isn't spent doing research proper. Instead, I spent (and still spend!) a lot of time trying to understand the background; this involves reading books or papers, emailing my mentor with long lists of questions, browsing MSE, etc. My project was actually completed about halfway into the year (and it took maybe another month to write it up). However, there are still lots of related questions to ask, and so the project is not in any way "completed" yet!

Jessica Zhang "Tight contact structures on the solid torus" (mentor Prof. Zhenkun Li, Stanford University)

Will Qin

The research process is very different from person to person, from topic to topic, and from professor to professor. That's why it's hard to really track hours. During busy weeks, I don't do much PRIMES, but when I have more free time, I can just sit down for hours with PRIMES work. Sometimes you have to eyeball these things; understand your schedule and your limits. On average, I spent something that 10-15 hrs a week on PRIMES. In terms of what I do with that time, it varies. Generally with research, you put in what you get out, but it's not linear. Research doesn't always have a clear end goal or clear steps. For the first few months, much of the process was reading papers and asking my professor questions about the papers. Once we had that established, I started building on the questions I asked, but a lot of the time was spent just staring at a blank page, being confused, not knowing what to do. Progress often happens in bursts of inspiration. You think of a lead, and you follow it until you hit a dead end and start again. Only about 5 or so hours a week of research time was actually doing productive math which is fine.

Will Qin "Colored HOMFLY polynomials of genus-2 pretzel knots" (mentor Yakov Kononov, Columbia University)

Jakin Ng

The amount of time spent on my PRIMES project varies from week to week, particularly since it's concurrent throughout the whole year with the rest of your life, so I learned to work it in with everything else. I actually think an important part of PRIMES is that since it's a year-long program, there's a bit more flexibility in how much time per week is spent on your project. It's definitely a time commitment, and on average I spent around 10 hours a week on PRIMES. For the first part of the project, my group and I mainly worked on reading background material such as textbooks and various papers, and understanding the conjecture and finding different potential approaches. As the project progressed, we started branching out, and I would work on a different approach than my group members, reading various papers and trying out different examples. I found that the weekly meeting with my group members and our mentor was super helpful. We haven't completed our project yet, so it definitely varies from project to project how long it takes to complete.

Jakin Ng "The Stembridge equality for skew dual stable Grothendieck polynomials" (mentor YiYu (Adela) Zhang)

6. What happens during weekly meetings with mentors?

Alvin Chen

Everybody has a different experience with their mentor, because every mentor is different, and every project is different. Especially in the beginning, a lot of the meetings will be longer and just about understanding the material and prerequisites needed for the project. Depending on the area, this can vary in time very heavily. When research begins to happen, more work will be independent, so you will meet with mentor to talk about your progress and talk about new ideas.

Alvin Chen "K-semistability of smooth toric Fano varieties" (mentor Kai Huang)

Katherine Tung

For my project, January and February meetings were more focused on background. My mentor selected exercises, textbook excerpts, and papers for me to read, which I read/worked on between meetings and discussed at meetings. However, in March and April, we spent more time alternating between reading research papers and working on our research questions. Compared to earlier in the year, we spent more meeting time discussing research questions.

Katherine Tung "The Sperner property for 132-avoiding intervals in the weak order" (mentor Christian Gaetz)

Daniel Hong

Starting in January, my mentor would give my group readings during meetings, and we would convene to discuss the background. Around March, we began talking more about our research direction. From then on, the meetings would be a place for us to relay ideas to the mentor and discuss what to do next. Since our project was highly based in experimentation and was not focused on a particular research question in general, the meetings were a good place to regroup.

Daniel Hong "Optimal solutions and ranks in the max-cut SDP" (mentor Dr. Diego Cifuentes)

7. What is it like to conduct research in a group? (Sub-questions: How does the work get divided? Advice on communication? Can you go in your own directions? If you have worked individually before, how does it compare with doing research individually? What is your relationship like with your mentor?)

Ilaria Seidel

I really enjoy doing research in a group: we can check each other’s work and exchange ideas, but most importantly, we all take a slightly different approach to our problem. More often than not, when one of us is stuck, another has a completely different approach that might work. Since we meet weekly (and often text outside of meetings), we’re always caught up on what everyone else is working on. Especially because this is my first research project, working with peers makes the times when we’re stuck less frustrating, and I’ve been able to learn a lot (both about math and about collaboration) from my group mates.

Ilaria Seidel "Upho posets" (mentor Yibo Gao)

Jakin Ng

I’ve really enjoyed working in a group; it’s a lot of fun and it’s definitely nice to have other people working with you. We usually work independently throughout the week, and then meet every week with our mentor for an hour or two where we are able to share what we’ve been working on, and from there get thoughts from our mentor (who is very helpful!) and each other on how to proceed or what to keep working on. Personally, I felt intimidated at first by the project, but it was definitely nice to have other people in the same boat. In the beginning, during the reading period, we mostly learned the same background material, but as the project progressed we’ve been able to each explore different directions. We also have a messenger group chat where we can update each other during the week about what we’ve been thinking about and a LaTeX document where we type up our work. For me it’s definitely been super helpful to have other people to bounce ideas off of or to ask for clarification when I was stuck or confused about anything. I would definitely advise anyone in a group project to take advantage of that!

Jakin Ng "The Stembridge equality for skew dual stable Grothendieck polynomials" (mentor YiYu (Adela) Zhang)

Simon Sun

Working on research with a group is a lot of fun! We can pursue our own directions for the project as well as work collectively on problems. Each week, the three of us would meet with our mentor to check-in and go over the progress we’ve made. These meetings are nice as they allow us to communicate with each other and see all the progress for that week. At this time, we also set up goals for next week. These goals can be smaller individual goals or larger group goals. In terms of communication, we kept in constant contact through a group chat and had a shared LaTeX document where we typed up any meaningful work. For me, the best thing about working in a group is having peers to consult. These consultations were vital in getting “unstuck" and moving our research forward!

Simon Sun "Product expansions of q-character polynomials" (mentor Dr. Nir Gadish)

8. What is it like to conduct research individually?

Yunseo Choi

I can definitely control my pace with PRIMES, because I work individually. I work on my project a lot during the weeks that I am free and don't as much when I am not. I think that having such freedom is a huge advantage that comes with a year-round project. I have previously worked in a group with two undergraduate students, which was a huge learning opportunity for me. The camaraderie that I built with my group members is priceless. But as much as I enjoyed working in a group, I am grateful that PRIMES gave me the opportunity to apply the lessons that I learned from group work through spearheading an individual project.

Yunseo Choi "Racial impact on infections and deaths due to COVID-19 in New York City" (mentor Prof. James Unwin, University of Illinois at Chicago)

Katherine Tung

My mentor and I meet once a week for about an hour. Each meeting, we check in, discuss the progress we’ve made over the past week, and sketch out a plan for the next week or so. My experience with research is that it rarely proceeds linearly, and working individually allows for a lot of flexibility with pacing.

Katherine Tung "The Sperner property for 132-avoiding intervals in the weak order" (mentor Christian Gaetz)

9. How does PRIMES stand out compared to other research programs or math camps? (Sub-questions: Feel free to connect to your previous experiences.)

Yunseo Choi

PRIMES is for sure a unique experience compared to any other program that I have been a part of in that it is a year-round research opportunity. Specifically, I am grateful that I can make as much out of it as I can over a span of an entire year! The longer time span allows us to be flexible with scheduling compared to summer programs that are compact. And because mentorships last an entire year, it is a great opportunity to get close with your mentor, both academically and personally!

Yunseo Choi "Racial impact on infections and deaths due to COVID-19 in New York City" (mentor Prof. James Unwin, University of Illinois at Chicago)

10. What was the most unexpected about PRIMES? What were some of your most fulfilling moments?

Alvin Chen

One of the most surprising things about my project is that students and mentors have a lot of freedom to work around their schedules. There are only really two deadlines for PRIMES, a shorter report in May and a full report due at the end of the following year. This gives projects a lot of freedom, which makes sense, because every project is different. My project was definitely slower in the beginning, because I had to learn a lot of background. However, this was okay because I was still able to come up with some small results by the first deadline.

Alvin Chen "K-semistability of smooth toric Fano varieties" (mentor Kai Huang)

Jessica Zhang

Honestly, I hadn’t realized how helpful it would be to have a mentor who understands the project and background material, who can give high-level or pictorial explanations of complicated concepts, and who takes the time to answer all my questions (even when they’re very elementary!). My regular communication with him has not only helped me learn math much more quickly than I ever could alone, but has also given this project more direction than previous projects that I’ve done. Probably the most fulfilling moment has been, predictably, the moment when I saw how to generalize our proofs to the general case—that is, the moment when we effectively solved our problem. A surprisingly close second is when we were writing up the preliminaries to the paper. It was nice to revisit these preliminaries and to realize that I had finally understood (at least on some level) how the concepts related to one another and to my project.

Jessica Zhang "Tight contact structures on the solid torus" (mentor Prof. Zhenkun Li, Stanford University)

11. Any advice to the prospective PRIMES students?

Daniel Hong

One thing I learned while doing my project is that especially for PRIMES research, communication will drive the success of your project. Make sure that you are on the same page as the people you are working with, whether that be your mentor or your team members, at all times. Important advice I would give is to be proactive and resourceful about asking for help.

Daniel Hong "Optimal solutions and ranks in the max-cut SDP" (mentor Dr. Diego Cifuentes)

Ilaria Seidel

Think about your project whenever you can! That might be on a commute, during a class, or just at your desk for a bit before you go to sleep. Playing with my problem for a few minutes here and there has led to some interesting ideas, even on days when I don’t have time to sit down for an hour and work.

Ilaria Seidel "Upho posets" (mentor Yibo Gao)

Simon Sun

Don’t be afraid to ask your mentor (or groupmates if applicable) for help! Research can be tough and the concepts may be hard to grasp, but know that you will not be judged for asking questions to your mentor/peers!

Simon Sun "Product expansions of q-character polynomials" (mentor Dr. Nir Gadish)