# PRIMES: Testimonials

#### 2020 PRIMES students

 Fiona Abney-McPeek Before PRIMES I thought of math research as something slightly superhuman. But the heart of my group’s research was in the unglamorous hours spent combing through jargon-filled papers, turning ideas over and over in my head, chasing false proofs and red herrings, and every so often, taking a tiny step forward. For me the hardest part of research was the uncertainty — not knowing which ideas would work out and which were dead ends. But this uncertainty was also what made it so thrilling — each idea had the potential to prove our conjecture, or reveal something entirely unexpected. My mentor was amazing and she provided so much invaluable support and guidance, and it was the persistence of my collaborators that kept me going through the inevitable frustration. By the end we got to know each other both as mathematicians and as friends. PRIMES was an amazing experience and I highly recommend it. Fiona Abney-McPeek worked on the project "The Stembridge equality for skew dual stable Grothendieck polynomials" (mentor YiYu (Adela) Zhang). Mikhail Alperovich PRIMES was really amazing experience for me. Here I learned how to do research: reading scientific papers to learn about Genomics and Single Cell Sequencing, which I knew little about before, analyzing large amounts of data both computationally and by hand to summarise the best of it to present, and how to write a paper myself. What really makes me excited about this research is that its results can be potentially used by thousands of scientists across the world and help with research to improve people’s health. Mikhail Alperovich worked on the project "Data driven quality control for single-cell RNA sequencing analysis" (mentor Dr. Ayshwarya Subramanian (Broad Institute)). Serena An PRIMES was my first experience with research math, and I am very thankful to the PRIMES program for this one-year journey. We began with background reading, often involving hours on just a few pages and serious note-taking. Upon transitioning to the problem, I learned that research oftentimes involves spending a lot of time thinking on paper—I’ve filled up countless sheets with diagrams, numbers, statements, and questions. And whenever I felt stuck or had a new idea, I could always discuss with my groupmates. Additionally, my mentor Adela (YiYu) Zhang was very encouraging throughout the process by providing us with directions to consider during our meetings and thoroughly reviewing our paper and presentations. I sincerely enjoyed exploring the problem with my groupmates, and I am excited to continue pursuing mathematics. Serena An worked on the project "The Stembridge equality for skew dual stable Grothendieck polynomials" (mentor YiYu (Adela) Zhang). Adithya Balachandran I fueled my early interest and passion for mathematics by participating in competitions where I learned many new concepts, but upon entering high school, I yearned for something more. Over the last year, PRIMES-USA has provided me the opportunity to gain valuable experience in mathematical research. Conducting collaborative research through PRIMES provided a unique and exciting theoretical challenge, adding an entirely new dimension to my growth. After my mentor, Dr. Nir Gadish, introduced our group to the interesting problem of computing statistics on matrices, we dived in hoping to make a contribution to the field by learning more about prior work and deriving useful insights. Noticing elegant patterns emerge in smaller matrices, we formed general conjectures which required us to think creatively and work meticulously over many weeks to develop proofs. I realized that trying to prove a conjecture without knowing if it is true in general was quite different from competitions where solutions are typically known. While it was extremely fulfilling to finally have a rigorous proof, I still learned something new with every unsuccessful attempt. My mentor’s guidance and the highly collaborative nature of my PRIMES group research project made it an enriching experience. I also enjoyed learning from my mentor about many new topics in representation theory and character theory and how they related to our project. For high school students seeking to expand their mathematical knowledge and experience beyond any textbook or competition, PRIMES provides the perfect opportunity. It adds a new dimension to mathematical education by allowing participants to appreciate the beauty and elegance of mathematics while exploring previously unsolved problems. PRIMES teaches one that while progress in mathematical research is neither easy nor guaranteed, the growth in one’s learning and confidence is unparalleled. Adithya Balachandran worked on the project "Product expansions of q-character polynomials" (mentor Dr. Nir Gadish). Alvin Chen PRIMES was a very eye-opening experience to how difficult math research is conducted. I expected to have to do a lot of background reading for my project, but it turned out to be even more than I had anticipated. It was so important to really understand all of the background algebraic geometry in my project, and I now I see all of that reading helped me understand not just my particular problem, but also how it fit inside a broader picture of mathematics that I have found to be very valuable and important to me. I really enjoyed having conversations with my mentor, Kai Huang, about not just different ways to approach the problem in our research, but also interesting stories in the fields we were working on. PRIMES wasn't about just working on one problem; through PRIMES, I've connected with a whole network of other high school researchers, gained important experience on the research process through presentations and papers, and really developed a better picture of how different areas in math can fit together. Alvin Chen worked on the project "K-semistability of smooth toric Fano varieties" (mentor Kai Huang). Quanlin (Andy) Chen PRIMES-USA provides me with the opportunity to meet like-minded peers and exposes me to some serious math researches. Sometimes asking a good question or formulating a vague observation could be even more important than solving problems in math research. My mentor Calder Oakes Morton-Ferguson helped us to distinguish the most interesting patterns and questions and introduces us to the representation theoretical background of the research problems. I started with a combinatorial group project, and after I finished, PRIMES-USA offered me the chance to initiate another project with the same mentor on a more algebraic topic. I greatly appreciate this flexibility and support of the PRIMES-USA program through my research process. Quanlin (Andy) Chen worked on the projects "On the Generational Behavior of Gaussian Binomial Coefficients at Root of Unity" and "The Center of the q-Weyl Algebra over Rings with Torsion" (mentor Calder Morton-Ferguson). Linda Chen PRIMES has been an amazing experience for me and I have learned so much about new computer science topics and the research process in general. Although it was difficult and the research process was filled with ups and downs, it was incredibly rewarding to achieve original results. I am grateful to my mentor and the PRIMES program for offering me this opportunity, and I hope to continue applying the knowledge I learned in future research! Linda Chen worked on the project "Reducing round complexity of Byzantine broadcast" (mentor Jun Wan). Richard Chen PRIMES allowed me to dig into topics I had never heard of before as I embarked on the research project. I was placed into a group with two other students; we all had different toolboxes in problem solving, so it was awesome how we were able to work off of each others' specific skills and interests. Our mentor, Feng Gui, was incredibly helpful; he guided us all along the initial stages as we read through the papers and resources he started us off with. Trying to do something new with the tools we had gleaned from the papers was intimidating at first, but I found that over time, there was a lot of new insight that creativity and the fusion of different ideas was able to uncover. We brainstormed many ideas, then found disappointment when nearly all of them fell through in one way or another; however, we then experienced satisfaction when a select few observations led to new results. PRIMES always had me thinking, as I went from one possible path to another, working through all the theorems that I thought might become useful for us. Curiously, we left with more questions than we had answered, as we started to create new ideas and concepts and see where they led. PRIMES has made me a much better problem solver and has provided continual support in my journey through this research project, and for that I will always be grateful. Richard Chen worked on the project "Few distance sets in $\ell_p$ spaces and $\ell_p$ product spaces" (mentor Feng Gui). Sarah Chen I come from a background in competitive programming, and through my two years in the PRIMES program, I’ve gone from knowing nothing about computational biology to analyzing hundreds of gigabytes of data in an effort to identify potential cancer neoantigens. I’ve also met mentors at the Broad Institute and the Dana-Farber Cancer Institute who are invested in my academic and personal growth and who have taught me the value of scientific communities. Moreover, the annual conferences have pushed me to not only face complex technical challenges but to also explain my research—and why it matters—to my peers. PRIMES has also laid an invaluable foundation for future opportunities, from internships to other research programs to any other academic pursuit. Sarah Chen worked on the project "In silico prediction of retained intron-derived neoantigens in leukemia" (mentors Tamara Ouspenskaia, Travis Law, and Aviv Regev (Broad Institute), and Nicoletta Cieri, Kari Stromhaug, and Cathy Wu (Broad Institute and Dana-Farber Cancer Institute)). Matthew Ding PRIMES has definitely been one of the most enjoyable experiences of my high school career. When I was first accepted to the program, I was incredibly nervous about not being “good enough.” Also, I was worried about not doing enough during my first few months. During this time, I did not have my own new ideas. Instead, I was just reading research paper after paper. But with the invaluable guidance of my mentor, Hanshen Xiao, I kept working at it. And sure enough, I ended up finding the inspiration that eventually led to the completion of my final project. If I were to give advice to an incoming PRIMES student, I would tell them not to worry too much. Most of us are completely new to research, so it’s completely understandable to have worries. The important thing is to not get discouraged and keep working. Also, I would tell them that the start of any research project is reading about existing work. Reading papers is not “doing nothing,” it’s an essential part of the research process. PRIMES has shown me how incredible research experiences can be, and I look forward to all the new opportunities I might find. Matthew Ding worked on the project "Relay protocol for approximate Byzantine consensus" (mentor Hanshen Xiao). Tianze (Peter) Jiang PRIMES has shown me what real math research was like and provided an opportunity that links from my previous experience in competitions to future math I will soon face heading into college. Starting from the stage of knowing nothing to expect, I got to learn advanced math research with my mentor quickly through those perhaps not very long months. Moreover, through this group project, on which I’ve collaborated with not only my mentor but also my classmates, I experienced the ecstatic sharing of ideas and collaborations of thoughts. PRIMES brought me to a new level of understanding of how math works. I wouldn’t think that there'd be any better opportunity other than PRIMES as my first glimpse into math research. Thanks again for the wonderful and precious opportunity that you and PRIMES have given me, and I look forward to continuing my math journey in MIT in the future! Tianze (Peter) Jiang worked on the project "On the Generational Behavior of Gaussian Binomial Coefficients at Root of Unity" (mentor Calder Morton-Ferguson). Matthew Ho PRIMES was an incredible experience: it allowed me to experience both the struggle to make progress on difficult problems and the feeling of exhilaration when we broke through. It was amazing to learn from and to collaborate with my group and my mentor. Beyond the really cool mathematics that I played with in my project, the PRIMES program also created a community of students and introduced me to many new friends. I'm extremely thankful to the PRIMES program for all the happiness it has brought to my life. Matthew Ho worked on the project "Group testing via zero-error channel capacity" (mentor Dr. Zilin Jiang). William Li PRIMES gave me the opportunity to play with mathematics in a way I’ve never been able to do before. Before PRIMES, my encounters with math had mostly been answering questions instead of asking them. PRIMES provided me with a sandbox where I explore math the way I wanted under the guidance of an experienced mentor. I was able to investigate mathematical topics in depth, and I learned how to draw connections between non obvious fields. Through PRIMES, I gained a new appreciation for the creativity involved in math research and the endless potential of mathematics. Beyond the academics, PRIMES introduced me to a wonderful community of passionate young mathematicians. It has been an absolute pleasure working with them on math or just chatting with them about life. Math Rocks! William Li worked on the project "Lebesgue measure preserving Thompson's monoid" (mentor Prof. Sergiy Merenkov, CCNY – CUNY). Srinath Mahankali When I first joined PRIMES, I was thrilled to have the chance to work on a single problem for a long period of time. Until then, most problems I've worked on could be solved within several hours at the most, whereas now, I've been able to study completely different areas of mathematics for a full year. PRIMES has been everything I wanted and more, giving me the chance to work with my amazing mentor, Dr. Yunan Yang. Through PRIMES, I've been exposed to exciting yet unfamiliar subjects in mathematics, and I'm extremely grateful to PRIMES for expanding my horizons. I've gained experience in math research, and I know the skills I've learned through PRIMES will tremendously help me in the future. Srinath Mahankali worked on the project "Velocity inversion using the quadratic Wasserstein metric" (mentor Dr. Yunan Yang (Courant Institute, New York University)). Holden Mui First of all, I want to say that I was quite lucky with the mentor assignment. My mentor was quite fun to chat with, and he was genuinely interested in helping me understand the material to the best of my ability. Especially since I want to go into research mathematics in the future, the experience was certainly an enlightening one which helped me better understand what a possible career in academia might look like. Secondly, I know that my research was nowhere near spectacular, but it was definitely a worthwhile experience due to the mathematics I learned, the practice I had with writing and editing a research paper, and my participation in a mathematics conference. I was not chosen as a Regeneron Science Talent Search semifinalist, but ultimately, there are bigger fish than STS. Holden Mui worked on the project "Generating symmetric functions of every arity" (mentor Dr. Zeb Brady). Read Holden's full story Jakin Ng I've had a great time and I'm very grateful to the people making the PRIMES program possible, and to my mentor, Adela. I was definitely intimidated at first by the prospect of conducting original research, and at the beginning of the year I felt like I was just staring at lots of tabs on my computer of papers I didn’t understand and scribbling down lots of examples I didn’t know what to do with. But by the end, it somehow all clicked together and we proved the original conjecture along with part ii, combining different techniques, after many months of reading new papers and trying out new directions with my mentor and my group. I’ve found it very exciting to explore mathematics from a much more open-ended view than in competitions or in classes, and I’ve had a lot of fun with PRIMES, especially being able to work with my group and bounce ideas off of each other during our video calls. (Also, I can no longer say the word “sure” without thinking of Schur functions.) Jakin Ng worked on the project "The Stembridge equality for skew dual stable Grothendieck polynomials" (mentor YiYu (Adela) Zhang). William Qin Research is a long and arduous process, but PRIMES provided a great structure and environment within which to learn how to do it well, and helped guide me through the ordeal. From finding a problem to writing up results, the staff and mentors are incredibly helpful and knowledgeable. While research is still hard work, the program is optimal for beginning to learn to do research, but also a great learning experience regardless of one’s level. I unreservedly recommend applying, and participating if admitted. The program is a must for any mathematically-inclined high school student who wants to explore mathematics further. William Qin worked on the project "Colored HOMFLY polynomials of genus-2 pretzel knots" (mentor Yakov Kononov (Columbia University)). William Shi I found participating in PRIMES to be an invaluable experience. Research was intimidating for me at the start of my project, and it took me a while to adjust to the process. It's daunting because there often aren't standard tricks or techniques that can be applied to solve problems, and it's difficult to see if hours of time and effort will eventually yield fruitful results. However, for these same reasons, I found the PRIMES program immensely rewarding. Trial, error, and revision are inherent parts of math research, and my mentor Professor Alejandro Morales has been incredibly helpful in guiding me throughout the entire process. It's so satisfying when my observations consolidate into conjectures and eventually results, but perhaps more fulfilling is thinking about how much I've grown as a student and as a thinker throughout the course of the past year. William Shi worked on the project "Refinements of product formulas for volumes of flow polytopes" (mentor Prof. Alejandro Morales (UMass Amherst)). Siwen (Simon) Sun PRIMES was a supportive, interactive, and stimulating research and learning environment! At first, our problem seemed overwhelming. However, exchanging directions to pursue and ideas with my mentor (Dr. Nir Gadish) and my research partners (who have become very close friends!) during our weekly meetings, slowly but surely, we made significant progress on our project. When finalizing rigorous proofs, I was always overwhelmed with feelings of joy and excitement; PRIMES allowed us to make our first mark on mathematical knowledge. In addition to the research, my mentor made sure that we were learning mathematics by incorporating relevant mini-lectures into our meetings; it was amazing to learn new concepts and see them in action as we used them in our own research. I highly recommend and am incredibly grateful for the PRIMES program! Siwen (Simon) Sun worked on the project "Product expansions of q-character polynomials" (mentor Dr. Nir Gadish). Kenta Suzuki I was always interested in research, and my dream has always been to discover some extraordinary theorem. I had attempted to do some research prior to PRIMES, but those were mostly mathematically trivial, looking back (it was really interesting to me at the time.) PRIMES was my first opportunity to experience "real" research, and I've learned much about the research process. My mentor would talk about his experience doing research with other co-authors, as well as talk about some of the dynamics of academia. Perhaps most importantly, PRIMES has reinforced my interest in mathematical research. Kenta Suzuki worked on the project "Value sharing of meromorphic functions" (mentor Prof. Michael Zieve (University of Michigan)). Jason Tang PRIMES has been an incredibly meaningful and enjoyable experience for me the past couple years. Under this program, I have not only greatly enhanced my advanced math knowledge, but also developed many invaluable skills and had lots of fun. As I worked through the reading and research process, I have become more adept at quickly learning new material and working through challenging problems. Along the way, I have also had the privilege to meet and work with other high schoolers who share my love for mathematics. I am extremely grateful for the opportunities and experiences that PRIMES has provided me and I will carry all that I have learned into college and beyond. Jason Tang worked on the project "Few distance sets in $\ell_p$ spaces and $\ell_p$ product spaces" (mentor Feng Gui). Abigail Thomas I am very grateful to the MIT PRIMES program for giving me such a wonderful research opportunity. The program guides students through the research process with the help of approachable and knowledgeable mentors. What differentiates PRIMES from other high school research programs is its duration. We have an entire year to conduct research and write a formal conference-style paper. This is a lot longer than other high school research programs, but the extra time makes a significant difference in the quality of research presented during the PRIMES conference. Despite the difficulties 2020 had to offer, my mentor still found ways to continue to meet with my partner and me, and we were able to conduct some very interesting research, which shows how dedicated the mentors are to helping you succeed. For any students interested in doing serious research projects in mathematics, computer science, or computational biology, I encourage you to apply! It is a very enriching experience that you will thoroughly enjoy. Abigail Thomas worked on the project "Hybrid privacy scheme" (mentor Yu Xia). Rahul Thomas Participating in PRIMES has been the one of the most phenomenal and rewarding experiences in my life. Before the program, I had little experience in math research and was daunted by the prospect of delivering substantive results; PRIMES has given me an excellent foundation for tackling research problems. Though the initial readings and first attempts at our optimization problem were quite difficult, I was soon deeply captivated by the mathematics and the beautiful integration of various methods and approaches. Working in a group, I have come to realize that research is a deeply collaborative experience, not an isolated setting. One of the most enjoyable parts of the program was discussing ideas and findings with my mentor and groupmates, who were wonderful and supportive throughout the process. Because of my experiences in PRIMES, I am strongly considering pursuing a career in research. I would strongly recommend PRIMES for anyone passionate about mathematics! Rahul Thomas worked on the project "Group testing via zero-error channel capacity" (mentor Dr. Zilin Jiang). Katherine Tung The MIT PRIMES-USA program gave me the opportunity to feel the joy of working on my own math research project and the distinct satisfaction of crafting an original argument that resolves an open question. I am indebted to my mentor, Christian Gaetz, for his steady guidance and wisdom as I explored the rich and fascinating subject of algebraic combinatorics. He introduced me to current questions of interest to algebraic combinatorialists, allowing me to better understand where our work fits in. Katherine Tung worked on the project "The Sperner property for 132-avoiding intervals in the weak order" (mentor Christian Gaetz). Yuxiao (Tom) Wang For me, the PRIMES-USA is a challenging, rewarding, and amazing experience. Having never done math research in the past, the reading materials seemed overwhelming at first glance. I started to realize that different from math competitions, research is filled with uncertainties, lots of possible directions, and –– unavoidably –– failures and frustrations. Yet, the sense of accomplishment after making a major progress to the research problem after months of investigation always outweighs all difficulties encountered in the process. The PRIMES-USA also provides me valuable experience as to how to work in a team, write papers, and prepare presentations. Yuxiao (Tom) Wang worked on the projects "On the Generational Behavior of Gaussian Binomial Coefficients at Root of Unity" and "Asymptotics for iterating the Lusztig-Vogan Bijection for $GL_n$ on Dominant Weights" (mentor Calder Morton-Ferguson). Zheheng (Tony) Xiao PRIMES was an incredible experience. It provided an invaluable opportunity for me to delve into an advanced field of mathematics that I would have normally encountered in college. I was able to learn about the representation theory of algebras and finite groups, and it was just so amazing to see convoluted theorems developed from simple ideas. At PRIMES, math evolved from awards and competitions into something beautiful. It really means a lot to me to work with a group of people who are passionate about math for the sake of it. I also learned how to present my ideas and how to write a paper, both of which are important skills in mathematics. I am very thankful to PRIMES for making all of this happen, especially during this difficult time. Zheheng (Tony) Xiao studied in a reading group on the topic of "Induced representations of finite groups" (mentor Dr. Alexander Vitanov). Yuxin Xie At first, I thought PRIMES would be hundreds of hours of intensive math, piles of scratch paper, and tons of “nerdy” math buddies. However, PRIMES turns out to be much more chill and fun! Our small reading group includes me and two other students at my school that I am already friends with by taking math and physics class together at Andover, and we meet weekly with our mentor, Chun Hong Lo. Initially, I simply enjoyed being able to go to MIT campus once a week (and have some Poke Bowl eight minutes away), while also doing some interesting math, but then the pandemic made us visit Zoom instead. Nonetheless, I still enjoyed the program a lot and learned as much as I would offline. We studied differential geometry and Markov chain with our mentor this year. It is really helpful to learn these sophisticated concepts and theorems that I would not otherwise dare to touch with an undergrad student, as he understands where we might encounter difficulty, and I do not fear asking some “stupid” questions. Also, a lot of the math I learned at PRIMES even helped me in my astronomy research during the summer, as I used holomorphic functions in KAM theory to validate my discovery of high-energy quasiperiodic orbits. Additionally, gathering together weekly and working on the expository paper and the presentation collaboratively have allowed the three of us to form a tight-knit group and learn hugely from each other. We also formed a close connection with our mentor (through bombing him with endless questions on Discord every week). If I was not yet sure about if math could be my thing before PRIMES, I could now shout out a solid yes. I wish I could also become a PRIMES mentor someday and similarly dissipate others’ doubts or concerns about math! Yuxin Xie studied in a reading group on the topic of "Markov chains and card shuffling" (mentor Chun Hong Lo). Daniel Xu When I first joined PRIMES, I had no experience in research and what it would be like. PRIMES taught me about math beyond competitions. With the guidance of my mentor, I learned how math could be applied to real world problems. I learned about the entire research process from setting up the approach to writing the final paper to presenting my work at the PRIMES conference. I found that research was surprisingly fun and fulfilling. The problems I worked on left an impression on me as they were challenging and highly applicable. Finally, the PRIMES community was incredibly supportive and I’ve met many friends along the way. I would highly recommend PRIMES to any students interested in research. Daniel Xu worked on the projects "Network based digital contact tracing and testing strategies for the COVID-19 pandemic" (mentor Dr. Jesse Geneson, Iowa State University) and "Graph alignment-based protein comparison" (mentor Younhun Kim), and also studied in a reading group on the topic of "Walks on Young's lattice" (mentor Yan Sheng Ang). Jason Yang Before PRIMES, I had only done programming on Scratch and math in school competitions as casual hobbies. In my first year at PRIMES, my mentors worked out a project in computational biology, in which I got to try out data science for my first time and learn special libraries in Python. In my second year, which was in the computer science group and was more open-ended, I initially struggled to select between multiple interesting topics. However, I was able to work out this issue with my mentor and eventually choose a topic we were both happy with. In all my time at PRIMES, I was able to meet with the mentors almost every week, and learn things from them that I never knew before. But most importantly, PRIMES has turned my casual hobbies of math and computer science into my primary professional and intellectual interests, more so than any other program I have ever attended. Jason Yang worked on the projects "The relationship between gene expression correlation and 3D genome organization" (mentors Sameer Abraham and Martin Falk) and "On updating and querying submatrices" (mentor Jun Wan). Jessica Zhang When I began my PRIMES project, I was excited to conduct research, to answer open questions, to learn new math. In the end, the entire experience exceeded my expectations. But what I couldn’t even have expected—couldn’t even have imagined—was how much math I wouldn’t learn, how many questions I wouldn’t answer, or research I wouldn’t conduct. PRIMES showed me how much more math there still is for me to research and learn and love. The research I performed in PRIMES isn’t, I hope, the end of my math journey; it has merely pointed out a hundred different paths forward, a hundred different reasons to keep learning math. Many thanks for giving me this opportunity—I've really loved being a PRIMES student this past year! Jessica Zhang worked on the project "Tight contact structures on the solid torus" (mentor Prof. Zhenkun Li (Stanford University)). Beining (Cathy) Zhou I am truly grateful for PRIMES for the amazing research mentorship experience that I could not have gained elsewhere. At first, research seemed beyond my reach. Unlike competitions, research problems lack pre-designed solutions or structure, which is left to the explorer to construct, little by little. For my research, in particular, I generated various ideas in different directions, but I was rendered furious as many failed. Unlike competitions, my research was slow-placed, including hours of contemplation on the same idea and even contemplating more on further improvements after I have already reached a result. Yet, I found it incredibly enjoyable when I taste both the endless challenges and surprises. In this process, I am most grateful for my mentor, who has guided me with much patience as I plowed through the technical background reading and as I talked through my underdeveloped ideas (many of which are incorrect). My mentor has also guided me to use a key idea of generalizing results, from simple to complex, when investigating every aspect of my research, which I will continue to use in my future studies. Beining (Cathy) Zhou worked on the project "A High-order cumulant-based sparse ruler for improved lag generation" (mentor Hanshen Xiao).