# 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)). Holder 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. Holder 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).

#### 2014 PRIMES students

 Yilun Du PRIMES was an amazing experience. At first, research seemed very daunting—there were so many possible directions to explore and everything seemed so technical. Not surprisingly, research to be difficult. I would often spend many hours staring at the same result trying to find some solution. Yet it was also incredibly rewarding, especially when I finally saw the solution to the question I had been working on for many hours. Furthermore, it was just so amazing to actually conjecture and prove results that you come up with. I also absolutely loved my mentor who would help guide my thinking and gave me resources to explore topics I was interested in. The PRIMES-USA program also allowed me to really experience the research process ranging from the PRIMES conference to writing a mathematics paper. I absolutely loved the PRIMES program and would highly recommend it to anybody interested in math. Yilun Du worked on the project On the Algorithmic and Theoretical Exploration of Tiling-Harmonic Functions under the mentorship of Prof. Sergiy Merenkov (University of Illinois at Urbana-Champaign). Niket Gowravaram My first experience with PRIMES was attending the PRIMES Annual Conference. There, I met people unlike those I had competed against. These high school students were not only willing to share their mathematical knowledge and original research, but they also treated the conference as a “festival of math”, where no one was competing with each other. The conference left an impression on me, because it was what math truly should be. Math should be a free and open collaboration among peers, not a collection of individuals actively working against each other like many of the competitions I participated in. Niket Gowravaram worked with Uma Roy on the joint project Diagrammatic Calculus of Coxeter and Braid Groups under the mentorship of Alisa Knizel. Read Niket's story Shashwat Kishore My background reading, which was a set of notes for a first year representation theory course, proved to be quite difficult. However, my mentor, Gus Lonergan, patiently met with me every week to go over the material and answer any questions I had. Slowly but surely I advanced through the readings, and by February I had learned enough background to start work on the problem. Although progress was slow at first, I incrementally moved forward, following Professor Pavel Etingof’s advice: “Have something doable on your agenda at all times.” I analyzed small cases with the help of computer programs, and eventually formulated conjectures for the general case. Proving my conjectures for the general case was one of the most amazing mathematical experiences I’ve enjoyed so far. The feeling that accompanies the discovery of a completely new theorem is one that cannot be replicated in any setting other than research. Shashwat Kishore worked on the project Signatures of Multiplicity Spaces in Tensor Products of sl2 and Uq(sl2) Representations, and Applications under the mentorship of Gus Lonergan and Pavel Etingof. Read Shashwat's story Jessica Li PRIMES introduced me to problem discovery in mathematics. Before PRIMES, my encounters with math had mostly been through solving problems other people had created in books, on websites, or in contests. Through PRIMES, I entered a new level of mathematical study by discovering the problems within mathematics that I would like to study and designing and implementing procedures to solve those problems. The research experience that PRIMES provides is invaluable and gave me the tools to approach new problems and thus use my knowledge to make discoveries. The project I completed through PRIMES on snowflake modeling is quite interdisciplinary, so I was able to learn and use information from physics and computer science as well as mathematics. The interdisciplinary nature of the project showed me the endless potential of mathematics. Jessica Li worked on the project On the Modeling of Snowflake Growth Using Hexagonal Automata under the mentorship of Prof. Laura Schaposnik, University of Illinois at Urbana-Champaign). Read Jessica's story Suzy Lou When I found out about PRIMES, I was really excited that a high school student like me had the opportunity to do math research. Math research has a far more organic feel than math competitions--you get to think about deep and meaningful questions. Unlike contest math which, though occasionally interesting, has little larger structure or coherence, when I did research I felt like I got to taste math. To explore its surprises and endlessly various nooks and corners; feel a little (or a lot) overwhelmed and inadequate at times; but enjoy it all the more for that. Math is really cool. The staff and mentors of PRIMES are also extremely warm, helpful, and knowledgeable. My mentor, somehow, was endlessly patient with my impatience and demoralization and false proofs. I can't believe that I got to work with such fabulous mathematicians, and I feel incredibly undeserving of this opportunity. I am still excited that this program exists and recommended PRIMES to some of my younger friends. They applied. Will you? Suzy Lou worked with Max Murin on the joint project On the Strongly Regular Graph of Parameters (99, 14, 1, 2) under the mentorship of Dr. Peter Csikvari. Shyam Narayanan Research is challenging. It leaves you despairing after being unable to solve a conjecture for weeks. It leaves you confused as to where you can go next. It leaves you angry after you realized one tiny hole in your solution has caused your theorem to come crashing down. However, math has a nice property of branching out into many possibilities. While one method of solving a problem may not work, there are dozens of paths out there. With the help of my mentor, David Corwin, I was able to discover the numerous ways I could approach my problem, and while my methods were not the originally intended ones, they were successful nonetheless. For example, while I initially looked for empirical patterns among strong pseudoprimes for the Miller-Rabin Primality Test, I noticed one theorem I proved revealed an additional pattern I could not find from my empirical results. Shyam Narayanan worked on the project Improving the Speed and Accuracy of the Miller-Rabin Primality Test under the mentorship of David Corwin and the supervision by Dr. Stefan Wehmeier and Dr. Ben Hinkle. The project was sponsored by MathWorks. Read Shyam's story Uma Roy Research can definitely be intimidating—unlike contest math, there are no clear cut solutions and no easy tricks for solving problems. Instead, the problems you explore gnaw constantly at the back of your mind and at times not having a clear direction forward can be daunting. However, my weekly meetings and flurry of emails with my mentor, Alisa Knizel, helped guide my exploration of the project my partner and I were doing. Every step of the way I felt extremely challenged but also adequately equipped because of Alisa’s guidance. I was always impressed with her insight and ability to gently guide me and my partner—she gave us sufficient freedom to tackle the problem as creatively as we wanted but also gave us direction and motivation. It was inspiring to work together as a team on our project and have our ideas culminate in beautiful results that we had all contributed to. Uma Roy worked with Niket Gowravaram on the joint project Diagrammatic Calculus of Coxeter and Braid Groups under the mentorship of Alisa Knizel. Read Uma's story Alexandria Yu From a young age, mathematics has always fascinated me. The many different ways of solving a single problem were intriguing and appealing and I always happily dove into contests and classes to satisfy my wish to know more. However, math research is completely different from anything else done in school or contests. There are no solutions using another method you can compare to your own as a different viewpoint and no way of knowing if your idea will work until hours and hours of thought reach a conclusion and you either have a contradiction, or everything works out. It is both frustrating and rewarding in a vastly different way from working on a typical difficult contest problem. Alexandria Yu worked on the project Towards the classification of unital 7-dimensional commutative algebras under the mentorship of Sherry Gong. Read Alexandria's story

#### 2013 PRIMES students

 Leigh Marie Braswell Though my day-to-day work and correspondence with my mentor brought me hours of entertainment and joy, I saw the culmination of my efforts when I presented at the PRIMES conference and co-wrote a paper. PRIMES gave me the knowledge, confidence, and experience necessary to continue researching in mathematics. Everyone with an interest in problem-solving would benefit enormously from participation in such an excellent program. If you want to appreciate the beauty of mathematics by creating and solving new problems, I highly recommend applying! Leigh Marie Braswell worked on the project The Cookie Monster Problem under the mentorship of Dr. Tanya Khovanova. Read Leigh Marie's story Boryana Doyle My experience in PRIMES has been amazing; I did not really know what to expect since I had no previous experience with anything like this, but the hard-work and surprises were all pleasant. I enjoyed discussions with my mentors, Maxim Imakaev and Geoff Fudenberg, about my project and related work in the lab. Exchanging ideas and questions was an integral component of making progress on my project. Moreover, my mentors and other lab members were very useful resources for technical questions; a few minutes of help can save hours of confusion. Because of my participation in PRIMES, I am now considering a career in research. Boryana Doyle worked on the project Chromatin organization: from polymer loops to topological domains under the mentorship of Geoffrey Fudenberg and Maxim Imakaev. Read Boryana's story Kavish Gandhi PRIMES has been an invaluable experience for me, and most definitely would be an exceptional experience for any high school student interested in mathematics. There is no doubt that it is hard; research does not come easily or quickly, but rather requires hours upon hours of struggle. However, doing research is something that every mathematically-inclined student should be exposed to, and the PRIMES program does an extraordinary job of introducing and enveloping you in the research experience. I cannot laud it highly enough; the mentors are outstanding, the projects are great, the support is exceptional, the experience is phenomenal. Oh, and remember to never trust a Sith lord, and may the Math be with you. Kavish Gandhi, together with Noah Golowich, worked on the project Inequalities and partition regularity of linear homogenous equations under the mentorship of Laszlo Lovasz. Read Kavish's story Ying Gao What I do with PRIMES is exciting in a completely different way from all the math I've done in competitions and classes. The things about research that were intimidating at first - working with a new and rather esoteric topic, having so many directions to go in, never knowing when I'd get a result, never being done - turned out to be some of things that made the math so beautiful and the results so rewarding. Through my project I've found an interest in an area of combinatorics that I hadn't even known about before; the support and resources that the program and my mentor provided were tremendously beneficial to that end. I've also learned how to write a paper and give a math talk, both of which are very important. To anyone who likes math and wants to be able to conduct research: PRIMES will definitely help make that happen.. Ying Gao worked on the project Depths of posets ordered by refinement under the mentorship of Sergei Bernstein. Noah Golowich Pursuing mathematical research can be quite challenging, especially when things appear so complicated that it seems impossible to make any progress. However, Professor Jacob Fox selected an appropriate and intriguing problem, and our mentor Laszlo Lovasz taught us the importance of being organized and trying to look at things in a clear way. With their guidance, my partner and I were able to make progress. When we finally proved the result that was our original goal, it was very rewarding. Furthermore, our method of solution inspired new directions of research that became the core of our project. I am very thankful to the PRIMES program and all the PRIMES staff for providing me with this great opportunity.. Noah Golowich, together with Kavish Gandhi, worked on the project Inequalities and partition regularity of linear homogenous equations under the mentorship of Laszlo Lovasz. Read Noah's story Ravi Jagadeesan My excellent experience at the Canada-USA Mathcamp motivated me to join the PRIMES program. The decision to do so was probably one of the best decisions of my life. PRIMES has allowed me to appreciate the monumental differences between math Olympiads, more advanced mathematical learning, and mathematical research. I enjoyed my first year in PRIMES greatly, and I was pleased to be able to participate again this year. During his second year at PRIMES, Ravi Jagadeesan worked on the project Belyi functions with prescribed monodromy under the mentorship of Akhil Mathew. Read Ravi's story John Long My mentor was very helpful in explaining to me the background and previous research. I knew little about purifying selection, genomic regions, the Unix command line, and scripting languages, but with the help of my mentor, I was able to learn all of these quickly and painlessly. Understanding where my project fit in with the rest of the research also gave me a greater appreciation for my project, and the material was interesting in its own right. John Long worked on the project Evidence of purifying selection in humans under the mentorship of Angela Yen. Read John's story Bryan Oh The great mathematician Georg Cantor said "The essence of mathematics lies in its freedom." PRIMES helped me realize and understand this. At PRIMES, I was fortunate to have met Rik, who is my mentor. He helped me with one of the more famous open problems in combinatorics, Conway's thrackle conjecture. It was a great time to explore the mysterious and unexplored world of thrackle, and I was able to experience the joy and freedom that underlies mathematics. I could think freely of anything on the thrackle; we were even able to make connections to ideas that were never related to thrackle before. Newly-learned and newly-developed terms effected me to think deeply because learning became interesting again. It was a great time. Bryan Oh worked on the project Towards generalizing thrackles to arbitrary graphs under the mentorship of Rik Sengupta. Read Bryan's story Ritesh Ragavender MIT PRIMES USA is an excellent program for those with mathematical curiosity. All the mentors and organizers do an exceptional job in organizing the conference, providing general assistance, and in helping young students realize their mathematical potential. The long hours and dedication were certainly worth it, and the last six months have been the best of my life. I strongly recommend MIT PRIMES USA to any student interested in mathematics. Ritesh Ragavender worked on the project q-analogues of symmetric polynomials and nilHecke algebras under the mentorship of Alex Ellis. Read Ritesh's story Raj Raina When I first heard that I was selected for PRIMES-USA, I was a bit intimidated; how could I, a student who had so little experience in math research, possibly contribute anything relevant to the world of mathematics? My fear was quickly dismissed as my mentor Andrey guided me through the beautifully complex world of Ramsey Theory. Together, we made progress on branches of Ramsey Theory that had largely been undiscovered. Although the work was certainly difficult, it was even more rewarding; the moment when you solve something that no one else had considered, when you find some order in the apparent chaos of math, when you put your brick into the wall of mathematical contributions, is undoubtedly an unparalleled feeling. Raj Raina worked on the project Minimum degrees of minimal Ramsey graphs under the mentorship of Andrey Grinshpun. Read Raj's story Ajay Saini When I originally started working on my project, I thought I knew what direction the project would take throughout the process and had an end goal in sight. However, my research took an unexpected turn almost every step of the way. During our weekly meetings, my mentor and I always discussed alternative approaches to the project based off of current findings. New observations suggesting unexpected ideas would often come up, causing me to rethink my approach and make the necessary adjustments to what I was investigating. Consequently, I soon found myself immersed in an idea that had been scarcely touched upon in my research field. Knowing that I was researching a topic of interest that others had not significantly worked on made the prospect of discovering something new all the more exciting. Ajay Saini worked on the project Modeling the opinion dynamics of a social network under the mentorship of Dr. Natasha Markuzon. Read Ajay's story Gabby Studt The relevant literature to my topic is somewhat technical, but [my mentor] was able to pose thought-provoking questions so that our meetings were always productive and enlightening. Although it can feel daunting to confront a research problem, I've found quite a bit of excitement in the small steps that I've made so far. Participating in PRIMES has made me even more eager to study other areas in math, and to collaborate and discuss my work with peers as well as those with more experience. That's why I think that anyone with a similar passion for math would also enjoy this program greatly. If you love math, you should apply, even if you don't think you will get in! Gabby Studt worked on the project Higher Bruhat order on Weyl groups of Type B under the mentorship of Daniel Thompson. Read Gabby's story Nathan Wolfe Coming into the computer science program last year I knew some Java, and not much else. My project last year was based on working in an experimental language that was different from anything I had ever seen before. At first it seemed to me like learning it would be a big challenge. It turned out, though, that my mentor had everything planned out. My partner and I ended up being able to learn the new language easily, since our mentor was willing to put in a lot of time at weekly meetings to help us understand new concepts. Nathan Wolfe, together with Istvan Chung, worked on the project A collaborative editor in Ur/Web under the mentorship of Benjamin Barenblat. Read Nathan's story Junho Won I really loved the fact that I met with my graduate mentor, Chiheon, almost every week. We discussed so many problems and cases leading to some original results, but it was also simply awesome to build a strong personal relationship with someone who guided me so thoroughly and patiently through abstract concepts, and sometimes, just chatted (in Korean!) about random things. Also, given that combinatorial mathematics was rather new to me, he showed me why graph theory - the field for my project - is quite beautiful and initiated me to study topological graph theory. The problems we worked on were very simple-looking and yet sinisterly difficult due to the virtual non-existence of organized tools to tackle them (which were relatively young), but coming up with my own insights and ways to resolve problems was the day's delight. Junho Won worked on the project Highly non-convex graph crossing sequences under the mentorship of Chiheon Kim. Read Junho's story William Wu From late-night edits to weekly meetings, PRIMES plunged me head-first into real-world research. I had a taste of the entire research process, from the conception of the idea to writing and publishing a research paper. I was fortunate to be assigned to a group with a wonderful partner and supportive mentors, not to mention an enjoyable and intriguing project. My favorite part about research is the creative aspect of the problem solving process, as computer science questions can often be answered in a multitude of ways. I was excited to see how the mechanisms I came up with compared with our existing ones. At times we would experience a "researcher's block," where the project would remain untouched for weeks. Other times, ideas would come one after another. All the while, everything had to be meticulously documented, from the thought process to the finished mechanism. I'm looking forward to another rewarding year with PRIMES, performing research once-reserved for college and post-college students. William Wu, together with Nicolaas Kaashoek, worked on the project How to teach a class to grade itself under the mentorship of Christos Tzamos and Matt Weinberg.

#### 2012 PRIMES students

 Nihal Gowravaram At first, I was skeptical; how could I, a high school sophomore, conduct mathematical research? Soon, however, PRIMES made me realize that this goal was not out of sight. Working with my brilliant mentor Dr. Joel Brewster Lewis, I began an extensive study of pattern avoidance, a branch of combinatorics, with my research partner Ravi. Though at first I was intimidated by some of the background reading, with reassurance from my mentor, I continued to persist onwards, reading a single paper dozens of times for hours or even days. Math research problems are not engineered like competition problems, and so, I initially found myself frustrated. However, I soon learned to grow patient, as the process of mathematical inquiry itself grew to satisfy me, and that eventual "Aha" moment was all the more satisfying. Nihal Gowravaram, together with Ravi Jagadeesan, worked on the project Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux under the mentorship of Dr. Joel Lewis. Read Nihal's story Steven Homberg I have always found both mathematics and computer science beautiful in similar but subtly different ways. The structure of some abstract math concept or the logical flow of a computer procedure each carry the same flavor of appeal to me, and one often echoes an instance of the other. Despite this correspondence, the true extent of the fundamental similarities of the two never struck home until given the opportunity to conduct research through the PRIMES program. Before, computer science was mostly writing programs and math was largely abstractions lacking context in the real world. After my first year in PRIMES, the distinction is not quite as clear as it was before; the project, in merging two of my greatest interests, served to enhance each. Steven Homberg, together with Eli Sadovnik, worked on the project Improving the efficiency of fault-tolerant distributed shared-memory algorithms under the mentorship of Dr. Peter Musial. Read Steven's story Ravi Jagadeesan At the beginning of the PRIMES program in January last year, I was mildly nervous that I would not be able to discover anything new. However, such fears were certainly unmerited. During the first few meetings, my mentor Joel Lewis provided my partner Nihal Gowravaram and me with background readings to become familiar with the common techniques. Within two months, we were formulating some of our own conjectures based on computer simulations, and before long, we were even able to find proofs of some of these conjectures. Throughout the process, Joel offered incredibly helpful insight, guidance, and support. Ravi Jagadeesan, together with Nihal Gowravaram, worked on the project Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux under the mentorship of Dr. Joel Lewis. Read Ravi's story Saarik Kalia My project this past year has given me a whole different experience because this time I was working with a partner. I think when we first saw our project this year, we were both completely baffled, for one thing, just by trying to understand the project, but even more importantly, trying to conceive of some way that we could possibly make any progress on it. I think if I showed myself in January all the progress we've made on the project by now, he would be shocked that that work could have been our own. Working with a partner has been great. Having a second head to run your thoughts by is immensely helpful. It forces you to check every counterexample. It forces you to figure out how to properly explain what you're trying to say. And it gives you a new perspective and a new way of looking at everything you do. Saarik Kalia, together with Michael Zanger-Tishler, worked on the project Schmidt games and a family of anormal numbers under the mentorship of Tue Ly. Read Saarik's story Skanda Koppula PRIMES gave me an excellent opportunity to conduct real world research. Research in Computational medicine requires study of various disciplines such as Biology, Mathematics, and Computer Science. Working with my mentors not only gave me the unique opportunity to study the subjects, but also taught me the importance of dedication and perseverance that is required to conduct good research. Working on PRIMES project is a unique opportunity for any high school student. The experience is unmatched in every aspect. I strongly recommend PRIMES to any high school student who is interested in Computer Science research. Skanda Koppula worked on the project Prediction-based Bayesian network analysis of gene sets for genome-wide association and expression studies under the mentorship of Dr. Gil Alterovitz and Dr. Amin Zollanvari. Read Skanda's story Ashwin Murali My mentors introduced us to the basics of simulating organic polymers demonstrating how their behavior and structure is similar to the behavior and path of random walks. It is very fascinating how complex organic molecules' behavior can be approximated with such a simple mathematical construct. Working at the intersection of programming, biology, and mathematics, we learned methods of simulating organic polymers, a technique applied toward our individual research problems, one of which evolved into my work recently presented at the Cold Spring Harbor Conference on the Dynamic Organization of Nuclear Function. The experience of presenting a poster was surreal and I owe this experience to my mentors, my teachers, and the PRIMES program. To say the PRIMES is life changing is an understatement. Ashwin Murali worked on the project Global positioning of interphase chromosomes mediated by local chromatin interactions under the mentorship of Geoffrey Fudenberg and Maxim Imakaev. Read Ashwin's story Alex Sekula Some of the most exciting times that I had while working on the project were my meetings with my mentor: I would come to these meetings with several new ideas for the direction of our project, and we would then use them to form new ideas to better tackle the problem. Sometimes, some of these ideas wouldn't work but at some other times, they would lead to a key progress in our project. Regardless of the outcomes though, I was always leaving our meetings with so much energy and enthusiasm to code and see an impact on our project. Alex Sekula worked on the project Natural language processing for spoken dialog under the mentorship of William Li. Read Alex's story Jonathan Tidor Research can be intimidating at first; there are so many directions to pursue and some might not lead anywhere while others might be too difficult to make any progress on. One of the most important things that PRIMES taught me about research was to always have a short-term goal. Knowing what you want to prove in the long run is great, but you can get lost easily. If you always know the next little thing you want to do there's always something to work on and you keep making progress, bit by bit. PRIMES is a wonderful opportunity for anyone who loves math. Even though the thought of doing actual research can be a bit scary at first, I found it to be a wonderful experience. Jonathan Tidor, together with Rohil Prasad, worked on the project Staged self-assembly under the mentorship of Jesse Geneson. Read Jonathan's story Dai Yang The leap from competition math to research is one that causes many to stumble. This was both one of my fears about and one of my reasons for applying for PRIMES. I feared that I would be one to stumble, that my inexperience beyond math Olympiad would make a fool of me. But I also hoped that this undertaking might teach me how to make the leap gracefully and successfully. Luckily I chose to believe in the latter, and I was correct. My mentor Tanya Khovanova showed me that the gulf was wide but not impassable. Although it took far longer to make progress on halving lines and underlying graphs than on an USAMO problem, the spark of joy upon discovering something new - something that possibly no one else had noticed - was more than enough wind under my newly-grown wings. Who knew that two words, two words briefly written en passant in an earlier paper, could erupt into a plethora of new questions and answers about halving lines? Who knew that graph theory, an old companion from the competition days, would come to visit me time and time again during this journey? Truly PRIMES was an adventure full of worthwhile surprises and lessons. I have but few words of advice for younger students. I looked. I ran. I jumped. Will you? Dai Yang worked on the project Halving lines and underlying graphs under the mentorship of Dr. Tanya Khovanova. Michael Zanger-Tishler Most adults look back at high school and think about their prom or the homecoming football game as the highlights of their youth. Not me. The best part of my high school experience has undoubtedly been conducting research with PRIMES. Nothing else I have done in my life so far has changed the way I think quite the way PRIMES has. My mentor, Tue, met with me and my research partner once a week and did so for as long as we wanted. He met with us on weekends, over the summer, and whenever we needed the time. He conveyed a love of math that was contagious. PRIMES is a unique experience. It is not easy, but it is rewarding beyond belief and definitely is something I will appreciate for years. Michael Zanger-Tishler, together with Saarik Kalia, worked on the project Schmidt games and a family of anormal numbers under the mentorship of Tue Ly. Read Michael's story Peijin Zhang My first few weeks starting in the PRIMES program were definitely some of the most difficult and overwhelming experiences I've had, but the challenge had piqued my interest, and I was inspired to succeed. Over the next nine months, with much help from my mentors and other members of the lab, I was able to shakily dive into the forefront of bioinformatics research, or the application of computer algorithms on large amounts of biological data such as gene expression studies or protein sequencing. In the span of nine months, I've performed analysis on a cohort of critical-care ICU patients for the prediction of hospital acquired illnesses, developed a program that will soon be on the forefront of clinical genomics, and identified several promising target regions for new drugs to counter MDR-TB. Peijin Zhang worked on the project Identifying Clostridium difficile in the ICU using Bayesian networks under the mentorship of Dr. Gil Alterovitz and Dr. Jeremy Warner. Read Peijin's story

#### 2011 PRIMES students

 Surya Bhupatiraju It was always really satisfying to be able to work on problems that no one else had attempted. Every time I walked into the computer lab at MIT and started writing the programs, I never wanted to leave. I loved the feeling of being able to sit and think about problems without having anything else in my mind. It was a stress-free environment, and I thrived here. PRIMES is an excellent program - it's a remarkable way to start research at a young age with the help of incredible professionals and mentors who love the math and science that you do, and will help you learn more and more. I'm very glad I chose to come to PRIMES, and it has truly changed my life as a student and a mathematician. Surya Bhupatiraju, together with William Kuszmaul and Jason Li, worked on the project  Lower central series of associative algebras in characteristic p under the supervision of Pavel Etingof and David Jordan. Read Surya's story Christina Chen Mathematical research is such a worthwhile and valuable experience. It is very different from problem solving because it does not have the security of a known solution. Sometimes, you are worried about not obtaining anything particularly useful or interesting. However, in the end, when you do, that feeling is empowering because you have determined something nobody ever has before. During the process, you both acquire new knowledge and the experience of confronting the inevitable setbacks of such a project, which can then be applied to other subjects. Christina Chen worked on the project Hiding behind and hiding inside under the mentorship of Tanya Khovanova. Read Christina's story Fengning Ding PRIMES is by far the best math or science program I have ever participated in. Unlike transient summer camps or programs devoted solely to competitions, PRIMES concentrates on real mathematics - an extended inquiry of the unknown. By having students spend a few months instead of a few weeks exploring unsolved as opposed to solved problems, PRIMES instills in students a far greater comprehension of mathematics than any other program. Before PRIMES, I have done numerous math competitions and read many advanced textbooks, but PRIMES gave me a special intimacy with abstract math that competitions and textbooks could not. Working at the frontier of mathematics, I gained not only a better understanding of abstract algebra but also an invaluable insight into mathematical research and the mathematical community. Fengning Ding worked on the project Infinitesimal Cherednik algebras under the mentorship of Sasha Tsymbaliuk. Read Fengning's story Dash Elhauge Computational Biology is the sort of thing one only gets to experience through a program like PRIMES. For programmers (like me), computational biology research is inherently satisfying because it lets us flex our programming skills to bridge that gap between abstraction and real world dynamics that only we can. Not only that, but it allows you to think about biology not as a memorized mass of terms and processes, but rather as a sophisticated system predicated on subtle balance. It's the sort of style of thinking you simply can't find anywhere else. Dash Elhauge worked on the project Modeling the role of cell fusion in cancer development under the mentorship of Christopher McFarland. Read Dash's story Caroline Ellison At first, I was definitely nervous; I didn't think I would find anything at all, and I didn't think I could really write a paper or give a math talk. It turns out I was wrong on all counts. Coming from a math contest background, research has been very different. I still sort of expect everything to have an elegant solution that can be found in a half hour or less, and I have to remind myself that isn't always the case. Things are often messy, but when I do find something that might be interesting or useful, the fact that it exists at all among the chaos makes it doubly beautiful. Caroline Ellison worked on the project Polynomial coefficients over finite fields under the mentorship of Giorgia Fortuna. Read Caroline's story Xiaoyu He Math research is deep and slow: you spend all your time thinking. We tested dozens, if not hundreds, of conjectures, and worked out a mind-boggling number of small cases. Each time I thought I had reached a complete solution, new problems would arise and a new layer of complexity developed. Without solving the original problem, I was able to develop a rich, complex understanding of the system that was more than worthwhile. A whole sub-theory emerged from the study of one specific case. I am happy a combinatorics project was chosen for me because combinatorics combines the physical intuition of real-world systems with the more elusive abstractions of other fields of mathematics. Xiaoyu He worked on the project Rotor-Routers under the mentorship of Tanya Khovanova. Read Xiaoyu's story William Kuszmaul The most important thing that my mentor David Jordan ever told me was that math research is confusing, but that the key is to grab onto the moments when you feel like you understand a little something, to the little victories, and hold them. When the paper was finally finished, we'd proven much of what we wanted to prove, and David said that our theorems were surprisingly original. It gave me an incredible feeling to have the paper come together in the final days of it being written, and I came to cherish the feeling of just putting everything in life aside and "primesing" for the rest of a day. William Kuszmaul, together with Surya Bhupatiraju and Jason Li, worked on the project  Lower central series of associative algebras in characteristic p under the supervision of Pavel Etingof and David Jordan. Read William's story Ziv Scully I recommend PRIMES to virtually anybody who loves math. Research is perhaps the most exciting thing about mathematics, but it's difficult to go at it alone with no prior experience. Having a mentor to suggest a possible problem, discuss ideas with and provide direction if those ideas run dry is what PRIMES offers, and it turned me from someone who was curious about research into someone who was capable of doing it. There are very few places where high school students have this type of opportunity. Try it! Ziv Scully worked on the project Progress on the parallel chip-firing problem under the mentorship of Yan Zhang. Read Ziv's story Dong-Gil Shin I came to be interested in computational biology as it intrigued me to be able to apply mathematical methods in a field like biology and combine the rigor of mathematics with the more tangible applicability of biology. In my project, I studied a model of DNA chromosome condensation using molecular dynamics simulations. Simulating a system with mathematical models and analyzing its consequences in biological contexts were challenging and highly fascinating. At PRIMES, it was great to be exposed to, and to learn the intricacies of, research using powerful computers. My mentors were tremendously helpful in providing me the necessary background knowledge and tools in biological/polymer physics and programming in order to pursue research. PRIMES has been a stimulating learning experience for me, and I would highly recommend it to those interested in mathematics or applied mathematics. Dong-Gil Shin worked on the project Scaffold Assisted Chromosome Condensation: Molecular Dynamics Simulations under the mentorship of Geoffrey Fudenberg and Maxim Imakaev. Michael Zhang A few things surprised me over the course of doing mathematics research. The first is how collaborative mathematics as a practice is. The image that I had conjured up of math research prior to PRIMES was the image of Wiles in his attic, working alone for 7 years. This was shown to be an anomaly to me over the course of the project, as I worked with an excellent partner, Yongyi, and had long discussions with my mentor and partner in which all sides offered contributions. Michael Zhang, together with Yongyi Chen, worked on the project Poisson homology in characteristic p under the mentorship of David Jordan. Read Michael's story

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