Our students go on to do great things: MathILy participants have attended MIT, Harvey Mudd, Princeton, Stanford, UC Berkeley, Swarthmore, Oxford, Carleton, Carnegie Mellon, Brown, U Waterloo, Cornell, Haverford, Caltech, Stony Brook, and other excellent colleges. They have participated in PRIMES-USA and in Research Experiences for Undergraduates, published papers, scored well on the Putnam exam, gone to graduate school at U Chicago, Rutgers, UC Davis, Boston U, U Washington, Columbia U, UC Berkeley, and Cornell, and presented work at national mathematics conferences.

*Annie Guo*: I was not even aware until Root of how much MathILy had to offer in terms of shaping me to think in an entirely different and more creative way, something that I will be able to carry with me not only as I continue with math, but in all aspects of my life.

*Michael Aram*: At MathILy, I'm constantly surprised by the depth with which any inquiry can be investigated, and my goal has shifted toward understanding the content deeply, rather than moving along rapidly.

*Felix Santiago*: Further, prior to this ca -- program, I had barely any exposure to proof-based mathematics, Because of this, it has felt like a great personal achievement to have worked on actually proving why and how things work rather than simply accepting conclusions made by others or taking guesses.

*Angie Huang*: I learned Linear Algebra this school year but I feel like going through the intuition for the building blocks of it helped me actually understand more intuition and what I didn't understand so well before (for example, I was not super familiar with reefs as opposed to plain spaces).

*Alex K.*: For 8+ hours every day, I've collaborated with others to solve some pretty daunting problems, and fail to solve significantly more, but nevertheless, it always ends up being a fun time. In fact, I would say that the day I enjoyed the least was Sunday, where I found myself disappointed that there was no class.

*An Y.*: ...the name "topology" used to sound scary to me as it is typically associated with advanced math research. But at MathILy, that type of distance is taken away by goofy names and discovery-based learning that made me feel that I can engage with and learn the material.

*Eva Strelchenko*: My goals coming to MathILy were basically to learn more mathematics, because my school forced me to take some easy and boring classes, I really wanted to actually learn something interesting and challenging. Also, I wanted to learn about more specific topics in math such as graph theory and group theory. My goals were definitely fulfilled, and I especially liked that we actually learned and discovered some theorems by ourselves. This is important to me, because almost all my other mathematical experiences were basically just preparation for math contests instead of learning the concepts in depth as in MathILy.

*James P.*: While previously I felt restrained in what I could learn next due to the limits of my knowledge, I was recently hit with a stunning feeling of freedom in how many doors are open for me to pursue through self-study and other classes, an ability I will no doubt take full advantage of. In a sense (as corny as the next sentence sounds), while MathILy has ended, it really feels like it was the beginning of a much deeper passion and pursuit of mathematics that will follow me throughout my life in this sense. ...I'm going to seriously miss the fantastic peer group MathILy provided me.

*Connor Lane*: Much of the properties we showed in class (like how the VC condition relates to a linear function being surjective) I would have no chance of proving before MathILy, but now I do know how to prove it. Also, the laser eye spider problem is an excellent example of how I feel I improved. At the start all I could use was hand-wavy arguments, but now I understand it much better. Even though I expected to learn a lot, this is far more than I imagined.

*Lillian Wang*: I feel like these 5 weeks have been the most hectic, productive, fun, and mind blowing (in the literal sense) 5 weeks I've ever experienced (I feel like two years have gone by in 5 weeks). ...I also gained much more than I came here for. I learned that math is a much bigger world than I thought before coming here (I think I enjoy learning math so much more than before), received valuable advice, experienced writing in a magazine (very fun), and I think I developed more as a person (character development! But in real life!). ...But I feel like I learned so much, and I never thought I'd laugh until my stomach hurt in a math class or talk with my dad about math proofs for all of an afternoon. So yay! Thank you for the wonderful experience! :D

*Nishka Kacheria*: In every case, there is a potential for curiosity, and math doesn't feel as stifling. It feels more like we can choose where we want to adventure, and I like it a lot. ... We're given the time and environment---as well as like-minded people---for coming up with new questions and new conjectures, and it allows me to increase my skill and ability, as well as gain more enjoyment for everything we cover.

*Owen Zhang*: MathILy, especially the philosophy of the root class, challenged me because it taught me things in a completely different way that I was used to.
I think over these past two weeks, whether intentional or not, MathILy has instilled into me a new mindset of approaching math... However, it also occurred to me in the middle of camp that perhaps more important than improving my math skills, this camp and the people in it have reignited my passion for math and is one of the few times I've actually had fun with math.

*Dries Rooryck*: I have learned more than I ever thought I could. My foreseen bundles of 200 pieces of paper were not enough for the ridiculous amount of notes I've made during the program.

*Sophia Zhu*: One of the best things about MathILy is that it changed my perspective of math. I realized that there's way more interesting math out there than what's in school math classes and competitions. MathILy also improved my ability to understand complicated ideas without feeling a need to bang my head against a wall. I am much more confident in my mathematical abilities than I was at the beginning.

*Mohit Hulse*: My main goal before coming to MathILy was to learn as much math as I could here, but as the weeks progressed (and especially in Branch) I discovered another: I wanted to have as much fun as possible while doing as much math as possible! Happily, this is what I ended up doing in MathILy. ... Perhaps the most important thing I have learned (non-mathematical) is how important it is to love math in order to do math.

*Ryan Ramos*: Overall, my experience at MathILy has been more than what I was hoping for... I can confidently say that MathILy has been the best learning experience I've ever had. There's just so many great and amazing things that go on around here that I can't compare it to anything else. All the techniques that I've learned about how to approach any problem that may seem impossible inspires me to boldly continue learning things outside of my comfort zone.

*Alex I.*:
I have a feeling that MathILy will (or already has) single-handedly make most of my future math classes boring. ...I would just like to say that I'm sincerely going to miss being surrounded by so many bright students and instructors. ...I feel like in the past 5 weeks at MathILy, I've learned more than I did during the entire last school year.

*Joyce Zhang*:
...Even though I was familiar with the base of what we were learning, root class definitely introduces a new perspective to things that I appreciate and helps me understand the topic more intuitively. ...these past two weeks have changed my mind about what I know and don't know. After all, there's always something to learn.

*Sebastian Oleksa*:
I have been to math camps before, but they were much lighter on math than this... I really enjoy how we cover a variety of topics while still spending enough time on each to thoroughly explore it.

*Emma Lee*:
The topics covered in MathILy are (for the most part) not even touched on by standard high school, but I have learned how to think(better, that is), how to approach a difficult problem, and how to convince people that I am right(unless I'm not).

*Quinn S.*:
When I first came to Mathily, I hoped to gain knowledge about the mathematical experience beyond high school, as I have never had the opportunity to have much interaction with professional mathematicians, and I have not had any formal math training beyond the standard high school curriculum. Over the last 5 weeks, I feel that I have gained a lot of knowledge not only about fascinating new math topics, but also about higher-level math in general, which has solidified my passion for mathematics. ... I think that the skills and knowledge I have attained here will help me better use and explain concepts in my classes, even though I am not yet at a level where they will be directly applicable, and they will also likely help me with classes in college.

*Winston Zhang*:
I like the feeling of waking up knowing that I have a full day of mathematics ahead of me, which is definitely an experience I had never felt before coming here.

*Oliver Hayman*: I think MathILy best taught me about the type of work mathematicians do. I think through some of the Daily Gathers and classes, I learned more about higher level math topics that are studied by mathematicians---topics which I hadn't encountered before at school or on competitions. MathILy gave me a sort of introduction of the type of experience I'd have if I chose to pursue a career in pure mathematics.

* Brian L.*:
I had a great time and got a lot out of MathILy!

*Lavinia Delfini*:
The math I do at school is really different from what we did here, but I feel like this experience will help me anyway. I've learned to think about math in a totally different way, and the fact that I learned how to write proofs will help me with math competitions. This experience also made me start thinking of math as something that I might like to keep studying after high school.

*Henry Hale*:
I came to MathILy hoping to find out more about the structure of learning as well as the content in higher math, and I now know a lot more than I did about what there is to know and how I can best learn it.

*Wen P.*:
One of my main goals in coming to MathILy was to branch out from my primarily contest math background... My studies in contest math would never have led me into Complex Analysis (which is the coolest math course I've taken), knot theory, or almost everything in my branch class. The branching out gave me the sense "math is awesome" again, since contest math tended to get somewhat repetitive (as in there were 4 main types of problems, using the same 4 major groups of technique) after my many years of doing it.

*Artur Amorim*:
I came here thinking that I knew more that I actually did. When I realized this, at first, it was a disappointment. But, after I saw that this was a good thing, because I could still learn a lot of things.

*Charlotte Kavaler*: I think that the program overall was fantastic. I really didn't know, when I first came here, if I would like it, but I definitely have, and it has taught me so much about things I had never even thought about... On top of learning new things, I have also learned how to think about subjects I already know about in greater depth and from a different perspective.

*Eugene Shao*: I've experienced graduate-level math to math that I didn't even know existed... MathILy doesn't have the normal school environment one would expect; the classes demand students to discover many of the things by themselves. I love this technique of teaching because, instead of a teacher lecturing, we can discover theorems on our own, learn the topic better, and know when to use it.

*Laurel N.*: I got exposed to areas of math that I'd only ever heard about, discovered some areas that I was really interested in, and got to work with people who were interested in math and learning and delving deeper into the problems... the Week of Chaos classes managed to be surprisingly detailed given the amount of time available, which was awesome.

*Hugh G.*: My goals were to... learn how to be the coolest cool kid and with what I know now, well, no one can say that I'm not the swaggiest person at school.

*Yiting Y.*: I gained a deeper understanding of math and realized what real math learning should be---generating conjectures and creating new things. I am extremely horrified at the thought of returning to the traditional math class at school. ...I learnt a great amount of math, realized that math is way cooler than I thought, and have become a bolder and more energetic person in class... The experience here at MathILy has been inspiring and transformative.

*Jamie Gu*: I had a very good experience... The program is different and a lot better than what I was expecting before coming. I like this format of learning because we have more interactions with others and we all have fun.

*Esther Plotnick*: When I was originally applying, I was a little hesitant to spend five weeks away from home to do math... I am so glad that I made the decision to apply... I definitely learned a lot over the past five weeks---way more than I ever anticipated. I now have a great curiosity about mathematics that I didn't have before, and I am very happy about that.

*Kye Shi*: One of my main aims before coming to MathILy was to learn about all sorts of different subjects in math---throughout the past several weeks, that has definitely been covered... I had a ton of fun developing math skills and learning how cool math topics relate to each other and explain things. I'm sure that, later in life, whatever skills I develop(ed) here I will use again whenever I do math.

*Nadav Kohen*: Over the past five weeks, I've gotten better and more comfortable writing detailed and sometimes long proofs more rigorously. ...I feel much more prepared than I was five weeks ago for this coming year and its proof classes... I can't believe I've been here for a full 5 weeks and I'm really sad that it's almost time to go but I'm certainly even more excited about math now and look forward to doing more!

*Andy Zhang*: I'm so happy that our classes are being taught the way they are being taught. Here, the things that we come up with ourselves matter... It feels great and even motivational to see our own conjectures appearing on the problem sets.

The past five weeks have been so amazing. I feel like I have never had a more productive summer than this one, and I certainly never learned this much stuff and had this much fun at the same time.

*...from a parent*: I can't tell you how much MathILy has meant to [my daughter] - it's like she discovered a whole new world. It was the first time in her life that she was truly academically challenged and she loved being part of a community where people were so excited about math.

*A. Wickard*: MathILy will help me in the future in 1000 ways.

*Brandon H.*: One of my main goals when coming into MathILy was to learn new types of math. I have known for a long time that I want to do math when I get a job, but I never knew about the many fields of mathematics. I'm happy to say that this goal was fulfilled... I was especially excited for applications of mathematics in game theory and economics, but I was surprised to find myself extremely intrigued by Markov chains and combinatorial optimization as well. I believe now that I have a wider view of the different fields of mathematics and will be better able to make choices regarding what I want to do later in my life.

*Josh Mundinger*: There's no BS here, no showing off, none of the surrounding clouds and fog that shroud any ordinary attempt to understand math.
... I have been having an amazing time outside of the classroom, whether talking about math in the dining hall, playing Pandemic (we won today!) or throwing a frisbee.

*Rashmika Goswami*: I've met a lot of great people who love math, and been able to see the many interesting variations and applications of math in other fields through the Daily Gathers.

*Jonathan Hess*: During Saturday's life seminar the following phrase was said: "There is nothing like math to make you feel dumb." I want to start by saying that I wholeheartedly agree with this statement. One of the main reasons I really wanted to go to MathILy, is to feel dumb for once in my life. Going from a school where I am top of my class to a summer program where I spend two hours on a problem to get it wrong in the end has been nothing short of an eye-opening experience. This has humbled me incredibly, but has also expanded my knowledge of mathematics more than I could have ever imagined.

*Heidi Z.*: The community that MathILy fosters of cooperation and friendliness is incredibly motivating without being destructive. ... In coming to MathILy, I wanted to rediscover what I enjoyed about math: the revelations and realizations, the exploration of new material that had been missing from my math experience for the past couple of years... Coming here made me realize how incredibly deep the world of mathematics is. MathILy has made me excited to do math again and I'm endlessly grateful for that.

*Vincent Chen*: These two weeks have caused a major shift in how I think about mathematics. The rigorous, proof-based classes contrast starkly with the corner-cutting of competitive math and assertive, memorization-based nature of most math classrooms.

*Kira Lewis*: I became much more comfortable with presenting a proof to the class, which I really enjoyed. ... I also found the teaching style of MathILy very valuable... I now see that actually doing something myself cements it in my brain much more permanently than just being told how to do it. It also makes the math somehow more exciting. I think that this type of learning also allowed me to focus with a lot more energy and think harder about the math.

*Matthew Q.*: I loved being here. It was great to hang out and play games with people, and my brain grew a ton. I'm going to be pretty sad and bored at home...

*Shlok Bansal*: At MathILy, I spent some 7-8ish hours a day doing math and the time flew by. I barely even noticed when we started and ended class and I learned about areas of math that I couldn't even fathom were math... MathILy gave me a completely new approach to math that I really enjoyed and somewhat reignited my passion for math.

*Celene S.*: The math I learnt here feels very applicable and just deeper than what I had experienced before. ...MathILy has made math feel like a much more fun and social activity, since everyone here is so excited about it... Beyond doing math together, we spent time working on RoM articles, singing off-key 2010s pop, torturing ourselves with a 2000 piece puzzle, and so much more. MathILy gave me a math community and it has increased my interest in math and completely changed my perspective on math.

*Dieter Y.*: The number of different topics covered during the past five weeks blew my mind away. From counting probability to writing programs in Mathematica, the amount of information flowing in my brain is at an all-time high.

*Charlie B.*: MathILy has been so different from anything that I expected from my first math camp, yet it has been so much more meaningful than any preconceived program I had dreamed up before attending. I sure expected to be taught some cool math with some cool people, but I had never anticipated that me and my classmates would be the ones recreating this mathematics from almost nothing. ...I think that this kind of education helps me create a far deeper connection with the math.

*Hellen Z.*: I had come to the camp hoping to stuff my brain with new ideas, which definitely happened. I also picked up better notation and structure in my proofs, not to mention better thinking/reasoning in my proofs... Overall, MathILy has been very fulfilling for me... I may have stayed indoors (and basically in just one room) for five weeks, but I feel like I've discovered a whole new world!

*Dylan X.*: The main thing at MathILy that has differed from my expectations so far is how much math we do collaboratively. I knew we were not going to have lectures every day, but the amount of interactivity in every class (especially the Daily Gather) and how, in Root Class, our class needed to build everything we learned from the ground up was surprising...

*Aria Tang*: The instructors and AIs clearly have put a lot of effort and attention into reaching out to every student, including me, and the comments on the HW problems and the relatively small class size has helped a lot in my learning and overall experience at MathILy so far.

*Jiakang C.*: Although I was suspecting this before coming to MathILy, now I definitely know that there are way more fun and wacky areas of math out there than what the education system tends to lead you to believe!

*Sophie Xuan*: Working with others was a lot funner and less frustrating and awkward than I thought it would be. ... I know I like doing math, but I didn't know if I was learning math because I liked learning math or because I needed to learn more math to do better in competitions (and that's been bothering me for while). Attending MathILy helped me see more clearly that there is so much more to math than competitions.

*Pablo C.*: Discussing un-intuitive concepts and debating how certain definitions should be phrased has brought me great joy, it created a feeling of comradery between myself and the other MathILy members. Real human-to-human connections like that are difficult to come by when everything's over the internet.

*Ellina Zhang*: Overall MathILy has been a great experience. I constantly feel motivated to do math everyday and am generally more energetic with the constant enthusiasm from others. ... I realized that listening to and learning the final conclusion is not nearly as satisfying as hearing it after having spent a lot of time trying to do it yourself.

*Matt Neissen*: Coming into MathILy,... I was eager to discover a lot more of the infinitely many things I don't know. As my root class delved deeper into our curriculum, I was hit hard with the realization of just how little I knew... After just a week and a half of MathILy, I was able to understand what most of the questions on the IMO were asking! I am positive this wouldn't have been the case prior to MathILy.

*Ian Cheung*: The themes I learned from MathILy--asking questions, speaking up, presenting well--apply to any form of education. I am not just a more confident mathematician: I am a more confident student, period. Put shortly: In my 5 weeks here, I learned that I can learn how to learn.

*Jonathan Shoung*: I really enjoyed my time here at MathILy, the environment was exactly what I hoped it would be like. It was great to be around people who shared my interest in mathematics, as well as in other topics and areas. I feel like being here has not only expanded my view on mathematics, but also my world view.

*Annika Mauro*: I learned that all the math I know is not as deep or intense as the real stuff here. I ended up learning real deep hard fantastic math... My experiences here will completely change how I experience math in the future. I feel like I understand proof and rigor in math which I did not before.

*Allen Wang, referring to Week of Chaos*: But in all seriousness we covered a freaking semester in like that week!

*Ian Shors*: I've come to understand the necessity of being able to prove everything you say you know, because if you can't prove it, how do you know you know it? And, furthermore, if you know it, but can't convey your knowledge to others, then what is it worth?

*Joyce Yuan*: I wanted to be exposed to mathematics that wasn't the bland and repetitive tactics taught at school. At school, I felt the lack of opportunity to get exposed to more interesting mathematical topics, or to develop my own ideas. Coming to MathILy, I have certainly learned a lot... I met many great minds whose stimulating ideas will never let me view the world the same way, and I met many amazing people who I hope to keep in touch with.

*Jonathan H.*: Root classes have taught me a whole ton of new stuff, (and I really like how you do correctly expect that our perceived understanding of topics is far less than what it actually is) so my expectation for serious maths was met. The activities that I've participated in after classes have made me appreciate how nice this community this is and how we share a similar sense of humor.

*Hayden Lieb*: These five weeks have probably been the best weeks of my life, so I have decided that I am almost certainly going to major in mathematics...

*Minkyu S.*: First, before coming to this program, I struggled to self-learn areas of math... However, in this program, I think I found answers to a lot of my struggles. In this perspective, I think I was very successful and I feel more confident about self-studying.

* Athina Avrantini*: ...over the five weeks, I have learned more stuff than my last few years of school combined! After spending five weeks here, I realized how learning math should be and it became clear to me that this is what I want to do. ... As someone who is used to being the best in my class for all my life, coming here has been a very enlightening, eye-opening experience. I realized how little I know and how much there is to learn!

* Michael Tang*: When I first arrived in Root class, I was taken aback by the very free and student-driven style of teaching. Since then, I have been continually surprised at the talent of my fellow students for finding patterns and meaning in problems that they and I have never seen before. ...

* Hussain Waris*: Not only did I learn so much more than I thought I would, I also learned that math is definitely what I want to do in the future.

* Luke M.*: ... I think that loads of doors have been opened, or at least pushed wider than they were before... I'm often amazed at how easy many obscure topics, such as graph theory, can be, if only they don't tell you what you're getting into until you're having so much fun drawing spaghetti monsters that there's no turning back.

* Éva Siladji*: I didn't know it was possible to learn so many things in such short period of time! I think I understood majority of it, and I won't forget them easily, mostly because of the way we learned them, discovering all the things by ourselves, and not having everything handed to us on a plate.

* Kaili Liu*: This summer is definitely a valuable experience, and it is the first time I have learned so much in one summer (and also had fun!)... I now find attempting extra-hard problems a fun activity...

* David Gonzalez*: ...MathILy has been a life altering, transcendental, suburb (I didn't mean 'suburb,' but 'superb'---No. suburb. I meant 'suburb.' I made no mistake.), experience that I will inevitably associate introducing me to proof-based math for the rest of my life. Just experiencing this program has grown my love of math to an unprecedented level.

* Tracy Wan*:
The MathILy community consists of people from various culture and geographical backgrounds. During one of our talks, we noticed that together, we speak more than 10 different languages. I really enjoyed the diversity of the group and learning about different cultures.

*Dennis Yatunin*:
I think the most valuable thing I've gotten from MathILy is the ability to think properly. Rather than believe anything that appears correct, I now look towards proving it using the techniques I've learned here... I've gotten more out of this than I ever dreamed I could, and I loved every single minute of it.

*Jo B.*: I also wanted to make sure that I did indeed want to do a Math major at college because I felt like I had not had enough experience to really know, but now I am sure! I definitely do.

*Cat O'Donnell*:
The teaching style at MathILy gives me the support I need to be comfortable asking questions... This creates an environment where I can be sure I understand what is going on enough to be able to contribute... The really enjoyable and funny group of students and instructors at MathILy are also great.

*MathILy, MathILy-Er, and MathILy-EST are projects of the nonprofit organization Mathematical Staircase, Inc..*