PRIMES: Testimonials
Testimonials from PRIMES students: 2022, 2021, 2020, 2019, 2018, 2016, 2015, 2014, 2013, 2012, 2011
Testimonials from PRIMES students at MathROCs2020:
 How to get started with math beyond the high school curriculum
 Research experience before PIRMES
 Math preparation and research knowledge before PRIMES
 Choosing your research topic
 The time demands of PRIMES
 Weekly meeings wih mentors
 The PRIMES research group experience
 The PRIMES individual research project experience
 What makes PRIMES unique
 Most surprising PRIMES experience
 Advice for prospective PRIMES students
2022 PRIMES students
Kevin Edward Zhao PRIMES was an amazing experience. Before PRIMES, I had general computer science knowledge, but the PRIMES program allowed me to go deep in specific research fields; my first project was in genomics, while my second one was in natural language processing (NLP). Both disciplines were new to me, but in both cases, my knowledgeable mentors started me off with background readings and simple experiments to introduce me to the field. Now, after at least a year of experience in both genomics and NLP, I feel very comfortable in these two fields. In addition to exposing me to different research areas, PRIMES has also taught me how to share findings with other researchers, both through writing papers and giving oral presentations at the annual PRIMES conference. PRIMES has been a wonderful opportunity that has magnified my enthusiasm for research, and it has convinced me that I want to continue conducting research in college. Kevin Edward Zhao worked on the projects, “The role of protein occupancies in DNA compartmentalization” (joint project with Vishnu Emani; mentors Sameer Abraham and Martin Falk), “Text is an image: Augmentation via embedding mixing,” “More than BERT: oLMpics on diverse language models,” and “Life after BERT: What Do Other Muppets Understand about Language?” (mentors Prof. Anna Rumshisky and Vladislav Lialin, UMass Lowell). 
2021 PRIMES students
Sophia Benjamin PRIMES was amazing because I got to experience the satisfaction and excitement of discovering new math and understanding and proving why it was true. I also got to experience the research process and all the pondering and playing around and making mistakes that you do to finally get results, which just makes getting the results that much more satisfying and exciting. Writing up results was also quite satisfying because we got to formalize our ideas, rigorously prove our results, and share them with the world. It was especially excellent to work with my mentor and my project partners. Our mentor was very knowledgeable and would point us in useful directions and help us understand things, and my project partners and I would build off each others' ideas and collaboratively make progress. We also just had a ton of fun working together and getting to know each other better (and collectively wondering why people kept asking us about guavas). Sophia Benjamin worked with Arushi Mantri and Quinn Perian on the joint project "On the Wasserstein distance between kstep probability measures on finite graphs" (mentor Pakawut Jiradilok). 

Neil Chowdhury When I started my computational biology project, I expected research to be like solving a math problem: tinker around with the given information and deconstruct it, step by step. However, my PRIMES work shook up my view. In the first weeks, I studied background information and repeated my mentor’s analyses to get accustomed to biological data. When I finally started exploring my research problem, I made little headway and felt discouraged by my fruitless effort. However, I occasionally made small breakthroughs. Perhaps I ran some computation that showed something unexpected, or I found patterns and correlations in the data rather than a Gaussian blob of randomness. I remained enthusiastic thanks to these discoveries and encouragement from my mentors. My mentors were not only patient but also kind and easy to approach. Though research could be hard, I found I enjoyed challenging problems, knowing that great rewards could lie at the end. My PRIMES project taught me many essential research skills. Communicating effectively with my mentors helped me go in the right direction, and organizing my ideas made it easier to track my progress and reach my goals. Also, the involvement of scientists in several different fields taught me about the broad reach of research. My work explored biology, physics, chemistry, programming, and linear algebra, making me realize the usefulness of knowing a wide range of topics. I have learned about my interests and found joy in contributing to the scientific community. Neil Chowdhury worked on the projects "A method to recognize universal patterns in genome structure using HiC" (mentor Sameer Abraham), "Modeling the effect of histone methylation on chromosomal organization in colon cancer cells" (mentor Dr. Maxim Imakaev), and “Interplay between loop extrusion and compartmentalization during mitosis” (mentor Sameer Abraham). 

Andrew Du Research in high school is typically something that I’ve thought of as impossible or simply faked, with the stereotype of high school “research” being a kid washing test tubes or doing manual work to contribute. But with PRIMES I’ve come to understand through firsthand experience that real research is indeed difficult, yet also not entirely inaccessible. Under the guidance of my mentor Dave, I found the support needed to navigate my way through the daunting task of reading other papers—written by people clearly more experienced than a high schooler, whose work also felt far above my level—before making our own contributions to the field. With the care put into each individual project, chosen to be accessible for high schoolers without sacrificing meaning, and the absence of tight time limits like summer programs, PRIMES offers the most realistic and enjoyable research experience a high schooler could get. Andrew Du worked on the project “Quaternionbased analytical inverse dynamics for the human body” (mentor David Darrow). 

Vishnu Emani I joined PRIMES in my Freshman year through the Computational Biology track, and right away, I found myself immersed in extremely relevant, exciting, and meaningful research. My mentors were extraordinary and the community and administration of PRIMES did an amazing job supporting me in the process. I especially enjoyed meeting the other students and learning about their research at the annual presentation conference. The skills and community I have gotten from PRIMES will be indispensable for years to come! Vishnu Emani worked on the joint project (with Kevin Zhao) "The role of protein occupancies in DNA compartmentalization" (mentors Sameer Abraham and Martin Falk), and on individual projects "Statistical ranking model for candidate genes in rare genetic disorders" (mentors Dr. Klaus SchmitzAbe and Dr. Pankaj Agrawal, Boston Children's Hospital) and “Computational fluid modelling for surgical planning of single ventricle congenital heart defects” (mentors Dr. David Hoganson and Dr. Vijay Govindarajan, Boston Children’s Hospital). 

Ram Goel I learned a lot through PRIMESUSA. Everything from the rigorous application, to the reading phase, to the actual research, taught me so much mathematics and the general mathematical research process. Before PRIMES, I felt intimidated by how hard it seemed to do actual research that was relevant to work in a mathematical field. The application was very rigorous, and gave a taste of what this process would be like. When I got in, I spent a couple months reading specific papers and books relevant to my field. Then, for the rest of the year, I was able to make actual progress on a real research problem with guidance from my mentor, and eventually ended with a set of new results. The entire process was made accessible by the structure and rigor of PRIMES. I would highly recommend this program to those looking to have a challenging and rewarding research experience. Ram Goel worked on the project “Products of reflections in smooth Bruhat intervals” (mentor Christian Gaetz). 

Zifan (Carl) Guo PRIMES is the most important thing that happened to my computer science journey. Because PRIMES demands me to learn at an accelerated pace to match the graduate level of research we are conducting, I learned in the last year perhaps more than all my years of studying CS combined. While I began my PRIMES project knowing basic high school CS, I was unfamiliar with the two fields that my project explores, machine learning and systems. Still, as time progressed, I grew to be more comfortable in actively investigating novel questions in the field and, hopefully, contributing my work to the literature. I also really enjoyed the “learningbydoing” approach of research, from which I learned not only technical knowledge but also methodology for more rounds of selflearning. My mentor is amazingly flexible in his schedule and is always there whenever I need help. I’m grateful for everything that PRIMES offers. Zifan (Carl) Guo worked on the project “Studying the effectiveness of transformer models for analyzing lowlevel programs” (mentor William Moses). 

Alex Hu I am grateful to the PRIMES program for giving me an amazing research experience. When I started my project, I was awestruck by the number of concepts, theorems, and ideas in quantum computing, what we already know and what we do not know for certain. Compared to competition math, math research was an experience on a whole new level of mathematical elegance and creativity. I learned that an important part of research is to read many papers, and I was able to improve my paperreading skills. PRIMES has given me confidence that I can thrive in the rigorous research process and achieve original results. I am immensely thankful to my mentor, Andrey Khesin, for supporting me through all the ups and downs and sharing with me invaluable insights about not just working on research but also writing papers and giving presentations. I am excited to continue my math journey and work on more research projects in the future! Alex Hu worked on the project “Improved graph formalism for quantum circuit simulation” (mentor Andrey Khesin). 

Akhil Kammila PRIMES was one of my most rewarding and exciting experiences ever. Each step of the research journey was enjoyable and taught me valuable skills. At the start of the program, my mentor guided me through background reading and material. She was extremely helpful and was always available for me to ask questions to. I learned how to analyze past research papers and delved deep into the intriguing field of cryptography. The research throughout the remainder of the year was even more enthralling. I loved the process of brainstorming and testing ideas with my mentor, who constantly supported and encouraged me. I would undoubtedly recommend PRIMES to any peers – you won’t regret it! Akhil Kammila worked on the project “An analysis of the Tor Handshake” (mentor Kyle Hogan). 

Yanan (Nancy) Jiang Before I entered PRIMES, I thought math research was — like competitions — an independent endeavor. I often struggled through problems myself, communicating exclusively with my mentor until a possible solution dawned on me. However, having worked with Sophie and Ben under Dr. Felix Gotti this past year in a group research project, I realized just how precious and indispensable this experience was. My research partners helped me resolve standing questions and offered many new perspectives to our project. We bonded over revising each other’s works and sharing the frustrations of not yielding satisfactory results. Moreover, I am very grateful for my mentor, who has been incredibly supportive, encouraging, and inspiring all along. Although we weren’t able to meet or work together in person due to the ongoing pandemic, MIT PRIMES has provided me with an unparalleled and fruitful research experience that further consolidated my determination to pursue mathematics in the future. Yanan (Nancy) Jiang worked on the joint project with Bangzheng (Benjamin) Li and Sophie Zhu “Factorization invariants of algebraic valuations of positive cyclic semirings” (mentor Dr. Felix Gotti). 

Sushanth Sathish Kumar Prior to the PRIMES program, I was familiar with competitive math and the basics of higher math like abstract algebra. PRIMES exposed me to algebraic topology, an entirely new field of mathematics for me, and also to research. The math problems I worked on before were all solved (and could be solved in a relatively short time frame) so the prospect of working on an unsolved problem thrilled me. Aside from learning the background material, I learned how to break down a complicated problem and write my own paper on it. During my weekly meetings with my amazing mentor Adela Zhang, we talked not only about progress on the problem but also the bigger picture of my project, which captivated me. Discussions like these also piqued my interest in the other aspects of algebraic topology besides homology and spectral sequences. I’m extremely grateful to PRIMES for the opportunities it provided me in not only working on an unsolved problem but also realizing the beauty of a field of mathematics I had never seen before. Sushanth Sathish Kumar worked on the project “Extending the restricted Lie algebra structure on the homology of a double loop space” (mentor Adela Zhang). 

Bangzheng (Benjamin) Li Besides learning math knowledge through math books and courses, doing math research also excites me a lot. Since middle school, I have written down my findings. At that time, I arranged some of my findings into a paper. However, from my current point of view, that paper was totally trivial and did not contribute anything new. After realizing this fact, I developed a slightly pessimistic view toward doing research in high school. I even heard that only graduate students would be able to solve open problems. However, the PRIMESUSA program totally changed my mind. Under the brilliant instruction of my mentor, I successfully produced some new research findings, which made me extremely happy because for the first time I believed that even as a high school student I could still discover new theorems and propositions if I worked hard. Moreover, some of these results are so beautiful that whenever I thought about what I have done, I would smile. Through PRIMESUSA, I saw my growth and learned a lot about how to do research, and I think I am more prepared to be a mathematician in the future. As a result, I would highly recommend it to those who are interested in math research. Bangzheng (Benjamin) Li worked on the joint project with Yanan (Nancy) Jiang and Sophie Zhu “Factorization invariants of algebraic valuations of positive cyclic semirings,” and on the individual project “A new class of atomic monoid algebras without the ascending chain condition on principal ideals” (both mentored by Dr. Felix Gotti). 

Yavor Litchev MIT PRIMES was a phenomenal learning opportunity. I enjoyed reading research papers in the field of cryptography and discussing them with my mentor. I learned a lot about artificial intelligence and cryptographic concepts such as differential privacy, multiparty computation, digital signatures, and more. I was able to work on both theoretical and applied aspects of cryptography and learned a lot about doing research. Although challenging at times, I found it very thrilling whenever I understood or solved a problem. PRIMES was a great experience and I look forward to continue exploring and applying what I’ve learned. Yavor Litchev worked on the project “Distributed signature scheme with monotonic access pattern” (mentor Yu Xia). 

Tanisha Saxena PRIMES was a great opportunity for me to dive into computer science research. The unique experience of genuine discoveries and indepth collaboration transformed the way I thought about the field. Through background reading, I learned how crucial a solid understanding of past work is for creating new research. The research process itself was satisfyingly challenging, and while occasionally it felt like there was so much to do that it was hard to determine where to begin, the invaluable insights of my mentor Jun Wan and our collaboration helped our research flourish. I am extremely grateful for the freedom PRIMES has given me to explore and develop in my areas of interest, and I look forward to seeing how I can use this experience to grow more in the future. Tanisha Saxena worked on the project “A compromise between synchronous and asynchronous systems” (mentor Jun Wan). 

Abigail Thomas My experience with MIT PRIMES has been one of the most lifechanging and rewarding opportunities of my life. I had always had an interest in computer science, but through PRIMES, I had the opportunity to explore branches I was unfamiliar with, including cryptography, with the guidance of an approachable, knowledgeable, and dedicated mentor. Going on my third year of research with PRIMES, I also appreciate how each year has been a unique experience. In my first year, I was paired with a research partner, while in my second year, I conducted research individually. Both years, I studied entirely different subbranches of cryptography and also acquired different soft skills from working in a team setting and working individually. Having participated in other research programs, I can also say that the duration of PRIMES makes it stand out from all other high school research programs. We have an entire year to conduct research, much longer than other high school research programs, which makes a significant difference in the quality of research presented during the PRIMES conference. For any students interested in mathematics, computer science, or computational biology research, I encourage you to apply! It is a very enriching experience that you will thoroughly enjoy. Abigail Thomas worked on the project “The implementation of model pruning to optimize zkSNARKs” (mentor Yu Xia). 

Eliyahu Yablon PRIMES was a great experience in which I learned a lot. I was able to delve into applied cryptography by brainstorming difficult problems with others and coming up with new ideas. I came across all kinds of interesting concepts that I had never heard of before: zeroknowledge proofs, secret sharing, and more. Not only that, but I also gained a great deal of experience in research. I now know how to effectively understand other people's papers, and how to write my own papers such that they will be enticing and coherent. Overall, I'm very thankful to the PRIMES program and to my mentor for presenting me with this wonderful opportunity to learn and grow. Eliyahu Yablon worked on the joint project with Simon Beyzerov “Versatile anonymous authentication with Cloak” (mentor Sacha ServanSchreiber). 

William Yue My time in the MIT PRIMES program for the last three years has been the most incredible mathematics experience of my high school career. Before PRIMES, I had a lot of exposure to math competitions and olympiads, but hadn’t explored much college math or research. In my first two years, I studied many undergraduate topics in reading groups and even conducted a theoretical CS research project over the summer! In my research project last year, my mentor helped us understand the challenging graduate level background and provided helpful guidance throughout the research aspects of the project. The work was difficult, and it required many iterative ideas or improvements to our methods over several months to prove our results. While it was unclear which ideas would be helpful and which would lead to dead ends, the mystery was an exciting part of the adventure! In addition, it was a great pleasure working with my groupmates: we could see different approaches to tackle the problem and bounce ideas off of each other. In the end, our research project was quite successful, and I’m excited for my fourth year in the program! I highly recommend the MIT PRIMES program for anyone interested in higher mathematics! William Yue studied in reading groups on “Shor's algorithm and the Period Finding Problem” and “Markov chains and card shuffling” (both mentored by Chun Hong Lo) and worked on the joint project with Karthik Seetharaman and Isaac Zhu “Regularities in the lattice homology of Seifert homology spheres” (mentor Dr. Irving Dai). 
2020 PRIMES students
Fiona AbneyMcPeek 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 jargonfilled 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 AbneyMcPeek 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 singlecell 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 oneyear journey. We began with background reading, often involving hours on just a few pages and serious notetaking. 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, PRIMESUSA 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 qcharacter polynomials" (mentor Dr. Nir Gadish). 

Alvin Chen PRIMES was a very eyeopening 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 "Ksemistability of smooth toric Fano varieties" (mentor Kai Huang). 

Quanlin (Andy) Chen PRIMESUSA provides me with the opportunity to meet likeminded 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 MortonFerguson 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, PRIMESUSA 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 PRIMESUSA 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 qWeyl Algebra over Rings with Torsion" (mentor Calder MortonFerguson). 

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 DanaFarber 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 intronderived neoantigens in leukemia" (mentors Tamara Ouspenskaia, Travis Law, and Aviv Regev (Broad Institute), and Nicoletta Cieri, Kari Stromhaug, and Cathy Wu (Broad Institute and DanaFarber 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 MortonFerguson). 

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 zeroerror 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 openended 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 mathematicallyinclined high school student who wants to explore mathematics further. William Qin worked on the project "Colored HOMFLY polynomials of genus2 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 minilectures 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 qcharacter 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 coauthors, 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 conferencestyle 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 zeroerror channel capacity" (mentor Dr. Zilin Jiang). 

Katherine Tung The MIT PRIMESUSA 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 132avoiding intervals in the weak order" (mentor Christian Gaetz). 

Yuxiao (Tom) Wang For me, the PRIMESUSA 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 PRIMESUSA 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 LusztigVogan Bijection for $GL_n$ on Dominant Weights" (mentor Calder MortonFerguson). 

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 highenergy 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 tightknit 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 COVID19 pandemic" (mentor Dr. Jesse Geneson, Iowa State University) and "Graph alignmentbased 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 openended, 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 predesigned 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 slowplaced, 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 Highorder cumulantbased sparse ruler for improved lag generation" (mentor Hanshen Xiao). 
2019 PRIMES students
Sooyoung Choi I would like to say that Kaavya always mentored us with the kindness. If I have a question, she would always show her care and tries to answer any question. Personally, she taught me so many new math concepts such as analysis, taking log of a complex number. For some entertaining detail, which happened last time, I needed a ride from Alewife station to my home but bus was not available. Luckily, Kaavya was also there and got me an Uber and paid for it. Sooyoung Choi studied in a reading group on “Two proofs of Dirichlet's theorem” (mentor Kaavya Valiveti). 

Nithin Kavi Unlike contest problems, requiring a clever solution within a relatively short time frame, PRIMES is much more realistic: students are given a long period of time and have access to a mentor to help them tackle a difficult, previously unsolved, research problem. Since joining PRIMES in 2017 as a freshman, I have had the opportunity to complete two research projects, present them at the MAA Undergraduate Student Poster Session, and submit them to professional math journals. None of this would have been possible without PRIMES or my mentor Zhenkun Li, whose patient, unwavering guidance helped me navigate the world of math research and become a better problemsolver. Nithin Kavi worked on the projects "Width and trunk of satellite knots" (joint with Wendy Wu) and "Cutting and gluing surfaces," and studied in a reading group on “Topics in knot theory” (all mentored by Zhenkun Li). 

Aneesha Manne Thank you so much for the opportunity of being a part of the PRIMES program. With the help of my mentors and partners, I have been able to grow not only my knowledge in mathematics but also my love for the subject. I am excited to continue that journey as a math major this fall! Aneesha Manne studied in a reading group on “Pell's Equation and Diophantine Approximation” (mentor Zhulin Li). 

Minghan Sun In the past year, I had the distinct honor and good fortune to be mentored by one of the kindest and most intelligent people I have ever met: Kaavya Valiveti. I must admit that initially, the transition from school math to proofbased mathematics was a bit challenging, but Kaavya’s careful instruction and guidance made it a lot easier than it would have been for me. Thanks to Kaavya, not only did learn a tremendous amount of mathematics over the course of the year, I also became convinced that I really enjoy the beauty and rigor of mathematics and should major in it in college. I want to express my immense gratitude to Kaavya for her superb mentorship, patience, and humor. Minghan Sun studied in a reading group on “Two proofs of Dirichlet's theorem” (mentor Kaavya Valiveti). 
2018 PRIMES students
Rinni Bhansali Before beginning my research, it seemed exciting, but also incredibly daunting to work on an open problem no one had yet solved. However, my mentor, Prof. Laura Schaposnik, made this seemingly impossible task doable. By first offering background reading and a way to approach this literature, she allowed me to not only better understand my topic, but also to see how research is made feasible. Through our weekly meetings, she guided me through each step of the research process, as I went from background reading to computational exploration to creating and proving conjectures. Rinni Bhansali worked on the project "Modelling epidemics on polluted networks and mstate networks" (mentor Prof. Laura Schaposnik, University of Illinois at Chicago. 

Sanjit Bhat To best explain what I learned from my research experience, I’ll invoke a commonly used data structure that’s probably familiar to many of you reading  a tree. You start research at the root node, learning about a topic and often proposing directions early on for how to solve some loosely defined goal. However, instead of following an immediate path from root to leaf like in competitions, research deals with unsolved problems and thus fails to show you such a path. The key to research then is exploration. Every day, I learned to ponder what information I’d found and, if I was lucky, answer some previously asked question. The more I’d think about and solve questions, the more I’d realize that the path I had traversed was more nuanced than previously imagined, bringing me even deeper into the tree. Finally, after reaching a leaf node, writing a conference paper, and triumphantly traversing my way back to the root, I realized that there were still dozens of child nodes I hadn't explored (what we researchers call future work). Sanjit Bhat worked on the projects "Towards efficient methods for training robust deep neural networks" (mentor Dimitris Tsipras) and "VarCNN and DynaFlow: Improved attacks and defenses for website fingerprinting" (joint with David Lu; mentor Albert Kwon). 

Leo Dong I have to say that doing PRIMES is one of the best experiences I had in high school, because it is not intimidating for a high school student, yet rigorous enough to give me an introduction to research that allows me to produce quality work (after sticking to it long enough) and inspires me to keep doing research in university. In fact, I am doing research at the Stanford AI Lab this summer as well on computer vision! Leo Dong worked on the project "Novel feature learning method of gene expression data based on an optimized denoising autoencoder" (mentor Dr. Gil Alterovitz). 

Ezra Erives In a group with two others, my research project was centered around maximizing the number of visible objects in three dimensions. The premise of our research project was simple, and after a week of intense work, and a flurry of perceived progress towards “answering our question”, we began to realize that our task was not so simple. Fortunately, our mentor, Dr. Zarathustra Brady, was able to help us with this issue. At first, he emphasized that we should ignore our question completely. The basis of our research described a world which could be explored, and before any questions could be reasonably asked, or an answer expected, it was necessary to better understand the arena in which we were working in. After we had poked around for a few weeks, he was able to suggest a method by which we could generate a true lower bound for our problem, and through this process, we were able to begin to cross the void that had looked so intimidating at first. Ezra Erives worked on the project "Asymptotics of visibility in threedimensional grid worlds" joint with Srinivasan Sathiamurthy and Theodore Baas; mentor Dr. Zarathustra Brady). 

William Fisher PRIMES has been a great, albeit challenging, experience for me. There are traits about research which are invaluable to be exposed to that you can't really anticipate until you just try it. Getting complete freedom on a project is something you don't get from textbook exercises and traditional math courses. It is also in this sense that research is intimidating—you have so many routes to take, none of which are guaranteed to produce any sort of result. There were a lot of routes I decided to pursue that ended up nowhere. Even worse, there were a lot of ideas I believed worked that only after hours of parsing through the scratch work I found were flawed. This is at times demotivated me and caused me to put off the problem. However, there were also times where I gained immense excitement over finding something new that did work. It is a skill set all of its own to be able to look at math and see what you can add to it, cement that goal, then actually work towards achieving it, and it's precisely this skill that PRIMES offered me. William Fisher worked on the project "Polynomial Wolff axioms and Kakeyatype estimates for bent tubes" (mentor Robert Burklund). 

Aayush Karan I was initially daunted by the unpredictability of research, but I decided to apply to PRIMES nonetheless. What resulted was one of my most exciting and instructive endeavors into math. With the kind help of Dr. Jianfeng Lin, my mentor, I was able to access and appreciate the advanced theory of homological algebra and incorporate it into my set of problemsolving tools. I was quite astounded by how large of a role homological algebra played in the seemingly unrelated knot theory. PRIMES allowed me to examine how wildly different flavored ideas overlapped at a fundamental level, a phenomenon that is truly remarkable. Aayush Karan worked on the project "Mutation invariance of the Szabó spectral sequence" (mentor Dr. Jianfeng Lin). 

Ryan Kim The PRIMES program taught me that research is the process of blazing one’s own trail in math. PRIMES showed me that math is beautiful not only in a competitive setting, but also in researching to create a unique and original product. Ryan Kim worked on the project "Cohomology groups of the dual Steenrod algebra" (mentor Sanath Devalapurkar). 

John Kuszmaul PRIMES is absolutely brilliant. Through PRIMES, I have had the opportunity to work on three research projects while in high school. One of the greatest parts of research is understanding an area in depth. For my freshmanyear PRIMES project, the first research paper my teammates and I read took us more than a month to digest. We went through the paper paragraph by paragraph. When we finally understood it, and everything clicked, it felt great. John Kuszmaul worked on the projects "Verkle trees: Ver(y short Mer)kle trees" and "Keychat: Secure messaging via Bitcoin" (joint with Robert Chen and Yiming Zheng (both mentored by Alin Tomescu), and "Exploring proof of space with hardtopebble graphs" (jojnt with Vivek Bhupatiraju and Vinjai Vale; mentors Ling Ren and Albert Kwon). 

Ethan Mendes For my partner and I, the problem involved improving the Direct Anonymous Attestation
(DAA) protocol by lessening the reliance on the host operating system which could potentially
be a corrupted party. Being thrust into the world of advanced cryptography was at first rough. I
remember scanning through the reading material provided to me by my mentor that seemed to be
written in a foreign language due to the heavy use of cryptographic notation and the universally
composable (UC) framework. However, as time went on, with many hours sifting through
various scientific papers and conference videos, I began to understand the crux of what aspect of
DAA needed to be improved. Ethan Mendes worked on the project "Maintaining the anonymity of direct anonymous attestations with subverted platforms" (joint with Patrick Zhang; mentor Kyle Hogan). 

Andrew Shen At first, the everything about the PRIMES program felt very overwhelming from tackling a significant problem to the large time commitment. However, I soon realized that working with my mentor and my partner, we could break down the problem into smaller, more manageable pieces. It was an incredible experience to be able to assemble the larger picture by combining and applying each of the smaller pieces to each other. As I became more invested and passionate about our work, I found that spending time on our project wasn’t bothersome, but rather enjoyable and incredibly rewarding at times. Although I spent many hours debugging and fixing code, the fulfillment that resulted from my perseverance and efforts made everything worth it in the end. I really appreciated how our work built on top of our research; what I learned from readings was actually very applicable to our project and seeing that connection felt truly magical. It was extremely satisfying to be able to build something with actual realworld applications that someone else might use. Andrew Shen worked on the project "A Practical analysis of Rust's concurrency story" (joint with Aditya Saligrama; mentor Jon Gjengset). 

Shashvat Srivastava I just wanted to thank you for offering me the chance to work on MIT PRIMES project. I really enjoyed working on both the Monero last year and later on the Algorand project this year. They were both really interesting projects. I fondly recall our long discussions when I started working for the first time at Boston University under our mentors' guidance. It was tough but their help made it both possible and easy enough for us to crack the puzzle. During this year's Algorand project under Ms. Kyle Hogan, I further enriched my knowledge about cryptography and her insights helped me immensely in solving the problem. Shashvat Srivastava worked on the projects "AnonStake: An Anonymous proofofstake cryptocurrency via zeroknowledge proofs and Algorand" (mentor Kyle Hogan) and "Monero, Moproblems: Defending Monero against temporal analysis" (joint with Henry Heffan; mentors Ethan Hellman, Dr. Jason Hennessey, Kyle Hogan, and Dr. Mayank Varia, Boston University). 

Vinjai Vale Thank you so much for the PRIMES program; it has been a formative experience for me over the last three years and has strongly influenced me towards pursuing research (in some shape or form) throughout my life. Vinjai Vale worked on the projects "A new paradigm for computer vision based on compositional representation" and "Vision as inverse graphics: Machine learning techniques towards a programbased model for scene understanding" (both mentored by Kevin Ellis), and "Exploring proof of space with hardtopebble graphs" (joint with Vivek Bhupatiraju and John Kuszmaul; mentors Ling Ren and Albert Kwon). 

Archer Wang Upon learning enough material from my background readings, I began the journey into my PRIMES project. Although I had a vague idea on where to begin, my weekly meetings with my mentor Xiaomeng Xu helped solidify my understanding of the project and gave me guidance to solve the research problems. Dr. Xu gave me the appropriate research papers to learn from, and whenever I had questions, he would discuss the possible approaches with me. For example, he suggested me to look into specific cases concerning the Hilbert series of a special class of quasiinvariant polynomials, and my experimentation ultimately led me to proving a theorem that was a generalization of the cases. From my PRIMES experience, I learned to make use of small cases, often from computer programs, to formulate conjectures based on the patterns I observed. PRIMES gave me an invaluable experience in the process of doing research math and developing my research paper. I would highly recommend PRIMES to anyone with an undeniable passion for mathematics! Archer Wang worked on the project "Hilbert series of quasiinvariant polynomials" (mentor Dr. Xiaomeng Xu). 

Wendy Wu I first got into math by solving math competition problems. I love the challenge of picking apart and carefully investigating a question. Tackling each problem brings a spike of delight, feeling accomplishment by remembering the process, the determination and the creative thinking. For me, MIT PRIMES translated such problem solving to a larger scale. Instead of short, simple questions meant to be solved within hours if not minutes, when doing research, I put in effort over an extended period of time. To me, it was wonderous that each step and development in mathematical research was real, an indication of actual progress on an asyetunanswered question. Solving a problem, instead of merely being my personal success of investment, dedication, and ingenuity, becomes a concrete success in the real world. Wendy Wu worked on the projects "Width and trunk of satellite knots" (joint with Nithin Kavi; mentor Zhenkun Li) and “Second gonality of ErdösRényi random graphs” (joint with Andy Xu; mentor Guangyi Yue). 

Guanpeng (Andy) Xu Being able to perform legitimate, mentored research in mathematics while still in high school has been a dream come true, and a wonderful experience in general. At first, I found the entire process incredibly intimidating: I was to wade into unmapped mathematical seas, where, metaphorically speaking, nobody knew how deep the waters were! This intimidation, however, quickly gave way to months and months of enjoyment tackling engaging, challenging, openended, unsolved problems and tasks; it’s this element of uncertainty and adventurousness that separates typical problemsolving from genuine research. I soon found mathematical research to be extremely exciting, even if the pace of discovery did slow down at times; after all, not many professionals, let alone highschool students have the opportunity to work on the horizons of our mathematical knowledge and push the frontiers of enlightenment! Guanpeng (Andy) Xu worked on the projects “Approximating the Hurwitz zeta function” (mentor Hyun Jong Kim) and “Second gonality of ErdösRényi random graphs” (joint with Wendy Wu; mentor Guangyi Yue). 
2016 PRIMES students
Yatharth Agarwal PRIMES helped me understand my place in the world. I mean that quite literally. Research is about persistence and technical pursuits and all that, yes, but there’s so much else that goes into the process, unknown unknowns. My panic at finding quite related preexisting literature, for example. That was a kind of riteofpassage, and through it my mentors held my hand and patiently explained what a literature review was. Then only did I understand how to carve out my (work’s) place in the world. Yatharth Agarwal worked on the project "Moving in next door: Network flooding as a side channel in cloud environments" (joint with Vishnu Murale; mentors Dr. Jason Hennessey, Kyle Hogan, and Dr. Mayank Varia, Boston University). 

KaiSiang Ang Competition math is fun, but I sometimes feel like there is a standard list of techniques that are applied, and creativity is limited. There is a lot more to math then math competitions, and PRIMES provides a way to experience this larger world, which I feel is more challenging, exciting, and rewarding. In this program, one can learn how difficult yet satisfying it is to find the right question, to learn the necessary background, to find the right connections to solve a problem, and to write a paper that effectively communicates your findings. My favorite parts of PRIMES are the regular meetings with my mentor, who has been very enthusiastic, supportive, helpful, and knowledgeable, and the PRIMES conference, in which I was exposed to a wide array of projects and was able to meet fellow students who share a love of math and research. KaiSiang Ang worked on the project "On the geometry of icosahedral viruses" (mentor Prof. Laura Schaposnik, University of Illinois at Chicago). 

Zachary Chroman Coming from a background of primarily competition math, PRIMESUSA was a great way for me to learn about and try mathematics research. Having never done any sort of research before, PRIMES served was a way for me to experience it for the first time, and I got a lot out of this experience, because my mentor and the PRIMES staff were very helpful. For someone interested in potentially pursuing mathematics at higher levels, PRIMES is a great way to get a taste of what it’s like. Zachary Chroman worked on the project "Rational embeddings of convex polyhedra" (mentor Sheela Devadas, Stanford University). 

Louis Golowich Our mentor introduced us to the research process and guided us through my partner's and my first math research experience. We quickly learned that both the methods we were able to use and the question we were trying to answer were flexible. Early on we decided to investigate a generalization of our original question that was more interesting and approachable. This exploration allowed us to make progress on the more general problem, and also approach the original problem from a new angle. Louis Golowich worked on the project "Maximum number of pairwise Gdifferent permutations" (joint with Richard Zhou; mentor Chiheon Kim). 

Nicholas Guo I really love the fact that in PRIMES, I have the luxury to ponder on different ideas for a long period of time to observe minute details that make mathematics so beautiful. I can research extensively and think independently to come up with novel ideas. Sometime this experience reminds me of entering into a giant labyrinth of math. Even though not every path leads to the goal, I know there is a way somewhere through thoughtful thinking and experimenting that will lead me to a successful finish, and I just need to be patient enough to get to the enlightening moment to discover. Nicholas Guo worked on the project "Rational hyperplane arrangements and counting independent sets of symmetric graphs" (mentor Guangyi Yue). 

Matthew HaseLiu Our mentor introduced us to the research process and guided us through my partner's and my first math research experience. We quickly learned that both the methods we were able to use and the question we were trying to answer were flexible. Early on we decided to investigate a generalization of our original question that was more interesting and approachable. This exploration allowed us to make progress on the more general problem, and also approach the original problem from a new angle. Matthew HaseLiu worked on the project "Counting points on curves of the form $y^{m_{1}}=c_{1}x^{n_{1}}+c_{2}x^{n_{2}}y^{m_{2}}$" (mentor Nicholas Triantafillou). 

Karthik Karnik I am very grateful to have been part of the PRIMES Program for these past two years, an experience that has provided me an extensive introduction to the world of research mathematics. Thanks to the dedicated mentors, Darij Grinberg and Akhil Mathew, who I have worked with, I was able to gain a deeper understanding of challenging concepts that are otherwise inaccessible to high school students. PRIMES has fostered my passion for mathematics, along with teaching me valuable skills such as writing a paper and giving a presentation at the conference. I hope to use these skills in my future mathematical journey, as I head to college. Karthik Karnik studied in reading groups on the topics "Combinatorial Fundamentals of Algebra" (mentor Darij Grinberg) and "The Outer Automorphism of $S_6$" (mentor Akhil Mathew). 

Rafael Saavedra I myself have never done competitive
mathematics, but in the program I have also met students who have
excelled in them. What matters is the ability to autonomously
investigate a problem that one has never seen before related to topics
one has never heard of before. The program provides every student with
a mentor, who helps us in learning the background and the techniques
of mathematics. My mentor Tanya Khovanova has been very supportive,
patient, and willing to help and collaborate. Thanks to my mentor, I
have become acquainted with how to write formal mathematics, search
the literature, and deal with roadblocks and deadends. Rafael Saavedra worked on the project "Discreet coin weighings and the Frobenius problem" (mentor Dr. Tanya Khovanova). 

Felix Wang Over the past year, I have had the opportunity to collaborate with two partners, a mentor, and a professor from an acclaimed university. It has been quite a challenge  certainly the most difficult I have ever faced. As I reflect upon this experience, I realize how much I have progressed in this short period of time. Two years ago, I held a completely inaccurate understanding of the foundations of research. I did not fully appreciate the patience and dedication necessary. But PRIMES has allowed me to conduct groundbreaking research, and has given me a wonderful glimpse into the world of math. I would advise all young mathematicians to fully invest themselves in such an enriching program, in order to broaden their minds and hone their mathematical skills. Felix Wang worked on the projects "Equal compositions of rational functions" (joint with Kenz Kallal and Matt Lipman; mentors Thao Thi Thu Do and Prof. Michael Zieve (University of Michigan)) and "Ramification of solutions of functional equations" (mentor Prof. Michael Zieve, University of Michigan). 

Richard Zhou Progress on our problem was slow at first, as we struggled to understand vital concepts in our field. Fortunately our mentor Chiheon Kim had a remarkable ability to explain things clearly and patiently, which made the process much easier. Within a few months, we were finally confident in our field and were able to start obtaining results pertaining to our problem. Richard Zhou worked on the project "Maximum number of pairwise Gdifferent permutations" (joint with Louis Golowich; mentor Chiheon Kim). 

"Annie" Siye Zhuu From the annual conference to arXiv.org, the daunting vocabulary of academic research, often associated with college if not beyond, has become accessible and familiar to me thanks to PRIMES. What I enjoy the most about the program is the professional setting that it provides for its participants. When I first applied to the computer science track, I never imagined that I would one day be presenting in front of a full audience and answering questions thrown at my group by MIT professors. In many ways, PRIMES offers high school students a unique prevision on research in graduate school and academia. It helps math and computer science enthusiasts gain an insight of the magic they could conjure with the knowledge at hand. Unlike traditional classrooms and high school contests, my PRIMES project has helped me grow through working on the infinite number of tasks to be solved and questions posed. Admittedly, research is challenging if not frustrating, but I am grateful to my mentor for encouraging and inspiring me as I embark on one intellectual journey after another. "Annie" Siye Zhu worked on the project "Scalable logging algorithm for inmemory database systems" (joint with Henry Liu and Justin Kaashoek; mentor Xiangyao Yu). 
2015 PRIMES students
Nick Diaco Even though it only resulted in a couple papers, the long mathematical journey leading up to and including my involvement in this yearlong program completely changed my perception of both myself and the world around me. I don’t mind admitting that I hardly understand many of my fellow PRIMES students’ projects; that is only a testament to how sophisticated and farreaching this program really is. PRIMESUSA offers a oneofakind experience that primes its students for an entire life of exploration and discovery, and there is nothing else in the world quite like it. The endless devotion of the mentors and staff put forth on a daily basis leaves nothing to be desired. My mentor, Tanya Khovanova, is truly one of the most amazing women that I have ever met, and I will be forever grateful for having the opportunity to work with her. My only complaint is that PRIMES can’t last longer than a year. I am extremely fortunate to have been a part of this amazing community of student researchers, and I will take the lessons I learned wherever I go in life. I will never look at any problem the same way again. Nick Diaco worked on the project "A new coin weighing problem and concealing information" (mentor Dr. Tanya Khovanova). 

Akiva Gordon I loved my year with MIT PRIMES. It was my first experience with academic research, so I came into the program barely knowing what I was doing. However, my mentor and my research partner helped me to understand our project, and I greatly enjoyed the opportunity to contribute to current computer science research. At first we had difficulties trying to implement and understand our project, but once we passed the initial hurdles, I had a ton of fun working on improving ORAM systems and helping our mentor to review papers, some of which introduced interesting new approaches which we further investigated. I loved my time at PRIMES and am strongly encouraging my underclassmen friends and little brothers to apply for the program. Akiva Gordon worked on the project "Improving oblivious RAM protocol through novel eviction and access strategies" (joint with Krishna Suraj; mentor Ling Ren). 

Daniel Guo The experience I had at PRIMES was unlike any other. Not only was I able to perform mathematical research that was rigorous and intense, I also attended a conference in May at MIT, meeting a number of students who were extremely gifted at and passionate for mathematics. By learning about their findings, I gained a greater understanding of the unsolved problems in each subfield and how to approach them. For my research, I was able to apply different areas of mathematics, including statistics. I also utilized what was essentially a combination of many different subfields to prove my theorems. Daniel Guo worked on the project "An infection spreading model on trees" (mentor Prof. Partha Dey, University of Illinois at UrbanaChampaign). 

Jacob Klegar The MIT PRIMES Program was my first real foray into math research, and it was an absolutely amazing first experience. Dr. Merenkov, my mentor, gave extraordinarily valuable guidance, helping me understand the background and tools for working with tilingharmonic functions, our object of study. I grew comfortable testing and experimenting with them, and there was no shortage of aspects of the problem to explore. PRIMES gives you the full research experience, from working with a mentor to presenting at conferences to writing a research paper  all of which were not just new and unique but very fun as well. I would highly recommend the program to anyone interested in pursuing research mathematics. Jacob Klegar worked on the project "Tilingharmonic functions" (mentor Prof. Sergiy Merenkov, CCNYCUNY). 

Ahaan Rungta I love mathematics because it provides new challenges to us every day. The more you play with it, the more it responds to you in an inviting manner. However, research mathematics is a completely different ballgame. I didn't even realize it was possible for me to approach a major problem in mathematics without a menagerie of prior knowledge. Nevertheless, my passion for seizing opportunities to delve further into mathematics made me want to apply for PRIMES. I was privileged and surprised to be accepted. After several months working with my mentor Andrew Rzeznik, I discovered and justified some significant results on a major problem in biomathematics and fluid dynamics. Furthermore, I gained invaluable skills while turning my research progress into a presentation at the PRIMES conference. Our head mentor, Dr. Tanya Khovanova, guided us through the process of making a mathematical presentation and how to write a research report and transform it into a paper. PRIMES was an experience I'd love everybody with a zeal for mathematics to be able to be a part of, one that can turn you from a student into a researcher and eventually into a true mathematician. Ahaan Rungta worked on the project "Mathematically modeling the motion of cells in porous media" (mentor Andrew Rzeznik). 

Richard Yi I would like to thank you very much for allowing me to be part of the PRIMES program  I really appreciate the support you all have given me this past year to make it an enjoyable experience. It was a rare opportunity to be able to work on a research project and learn about others' work at the MIT conference. Thanks to the program, I have developed a stronger interest in math, and plan to pursue a math major in college. Richard Yi worked on the project "Continuous model for twolane traffic flow" (mentor Prof. Gabriele LaNave, University of Illinois at UrbanaChampaign). 
2014 PRIMES students
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 PRIMESUSA 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 TilingHarmonic Functions under the mentorship of Prof. Sergiy Merenkov (University of Illinois at UrbanaChampaign). 

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 sl_{2} and U_{q}(sl_{2}) 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 UrbanaChampaign). Read Jessica's story 

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 competitionsyou 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 MillerRabin 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 MillerRabin 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 7dimensional commutative algebras under the mentorship of Sherry Gong. Read Alexandria's story 
2013 PRIMES students
Leigh Marie Braswell Though my daytoday 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 cowrote a paper. PRIMES gave me the knowledge, confidence, and experience necessary to continue researching in mathematics. Everyone with an interest in problemsolving 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 hardwork 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 mathematicallyinclined 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 

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 CanadaUSA 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. Newlylearned and newlydeveloped 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 qanalogues 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 PRIMESUSA, 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 thoughtprovoking 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 simplelooking and yet sinisterly difficult due to the virtual nonexistence 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 nonconvex graph crossing sequences under the mentorship of Chiheon Kim. Read Junho's story 

From latenight edits to weekly meetings, PRIMES plunged me headfirst into realworld 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 oncereserved for college and postcollege 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 faulttolerant distributed sharedmemory 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 ZangerTishler, 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 Predictionbased Bayesian network analysis of gene sets for genomewide 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 shortterm 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 selfassembly 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 newlygrown 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 ZangerTishler 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 ZangerTishler, 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 criticalcare 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 MDRTB. 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 stressfree 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 mindboggling 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 subtheory 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 realworld systems with the more elusive abstractions of other fields of mathematics. Xiaoyu He worked on the project RotorRouters 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 chipfiring problem under the mentorship of Yan Zhang. Read Ziv's story 

DongGil 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. DongGil 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 
With questions, please contact PRIMES Program Director Dr. Slava Gerovitch at primes@math.mit.edu