Part 1: https://artofproblemsolving.com/community/c2532359h3109293_the_advice_post_to_end_all_of_my_advice_posts_part_1_of_2

(Definitely at least read the beginning of Part 1 first, thanks )

Section 4: College and Beyond

I did really well in math competitions!!! College math will be easy!!!

Haha.

Actually, I do agree that competition math experience can help in some aspects. If you've figured out how to write proofs well and have a good mathematical maturity, especially surrounding discrete subjects, then you'll have a pretty nice headstart. And, some competition tricks could be applicable in your homework sometimes (as was the case for me).

But if you didn't do much math competitions, you should not feel as if you are "behind" whatsoever. Several of the best math students I've known had zero math competition experience in high school!

Oh shoot college math is hard. What do???

Work with friends!!! I believe math homework should be done with math pals. It's great for learning. For best results, find a partner (or more) who you think is probably around your level of competence.

Is CMU good for math?

Yeah it's pretty lit, I honestly think I would not have thrived more at any other university. Our department strengths include analysis and uhhhh maybe discrete or logic stuff? I mostly know about the analysis.

Join our Math Club too!

yooo have you met Po

Heck yeah!!! He wrote one of my grad school rec letters!!! I also worked for him. He's great!

If you come to CMU there's a sunday session for the Putnam seminar where you solve hard problems for like three hours and then eat dinner with Po.

Should I do Putnam?

If you liked math competitions, sure, why not? It's fun and it's not like you need the score to get into a good college anymore, right?

Should I prepare seriously for the Putnam?

I think there is generally no point. Unfortunately these is one dumb reason to take it, and that is to make yourself look good to shallow companies. I'd say that the path of maximum utility is to prep enough to get top 500 (which usually entails just two problems and a bit more, which can be hard but isn't too crazy) and then voila you sound kinda cool. If you're reading this, you probably have a good shot at that.

After college, should I plan to go into grad school or should I get a real job?

I can't answer this for you. Instead I will ask you to consider the following questions:

• Do you need money?
• Do you care about money?
• Do you want to be rich?
• Do you really, really want a PhD?
• Do you think that what you're going to do in grad school will be vastly different from what you're doing in a "real" job?
• What do you want?
• What makes you happy?
• Where do you want to spend your 20s?
• Do you want your math to help the world? What, in your view, does that question mean anyway?
• Do you care if your mathematical work does not help the world?
• If so, are your field(s) of mathematical interest applicable for helping the world?

If you want to know how I decided: I realized that for me to live a fulfilling life, I need to use math to help the world in some way, and the best way to do that is to be the best at math I possibly can.

I can't decide. What do I do?

Ah, you're just like me! Worst case scenario, you can "play both sides" by looking at both grad schools and work opportunites. If that doesn't help, you can play both sides even further by applying to both grad schools and jobs. This can help you stall for time while you make a decision.

What you do during your summers can help you come to a decision. You can try e.g. doing a summer research project / REU or interning for a hedgefund to see how you like it. Doing research isn't looked down on by companies (I think), nor would doing an internship be looked down much by grad schools (so I've heard).

(Fortunately I eventually figured out what I wanted to do.)

Should I care about my grades?

Only if you plan to go to grad school.

I'm burnt out.

Very seriously consider dropping a course. That's what I did! It was fun.

I've decided that I want to go into industry. What should I do?

You asked the wrong person.

I've decided that I want to go to grad school. What should I do to maximize my odds?

• Get good grades.
• Get your professors to like you! Rec letters are pretty important (and some say that it's the most important aspect). One way to do this by going to office hours often (even if you aren't in dire need of help). Also consider taking several classes with the same professor, if that's feasible in your school. (CMU is relatively cozy, so here it's not hard to take like 5 courses with the same professor even if you did not intend to).
• Write a paper by doing research with a professor. I hear that most grad school applicants actually haven't done research, so this will make your application quite competitive, and if you're reading this, I bet that you can totally do research.
  
  If you don't get to publish your research before grad school apps, that's fine. But at least put it on arXiv (which is not hard!).
• Alternatively to the above point, you can do an REU. I can't tell you much about that unfortunately, but it is a popular option.
• Take graduate courses if you can!
• Take an easier courseload during the semester in which you work on your applications (i.e. probably your Senior fall semester). The apps are really time consuming. It's ridiculous.
• Apply for the NSF GRFP grant. If you win it (which isn't that unrealistic), that will give a huge boost to your application.
• If you are a minority (e.g. a woman, a native Alaskan, etc.), mention it.
• For your essays, make good use your university's writing center if they have one. It was a huge help for me, since it can be hard to tell if you're on the right track with your writing.
• Participate in mathematical communities and organizations, and consider being a leader in said communities. For example, I was the VPE of CMU's Math Club, and I was also a TA three semesters. If you're a woman, consider participating in your university's AWM chapter (assuming it exists... don't know how that works tbh). These initiatives can be important - your outreach and community involvement can be a great talking point in your application. Bolstering the mathematical community is just as important as your mathematical contributions, in my opinion.
• Foster diversity, especially if you're a minority. Some applications (e.g. UCLA) may ask for a diversity statement. Even if you're an asian male, you should be able to find something of substance to write for such prompts if you actually care about the issue. (Source: I am an asian male.)
• Unfortunately, take the GRE (both general and math) and study for it. (See below below)
• (Very optional) Make a website. Your uni might provide a web hosting service, so use that if they do. Your website should include your CV, your research, any notes from courses you teach or TA (if you write notes), and then whatever you want. My site has a bunch of puzzles on it, as well as a list of math competition problems I've proposed.
• Start your application early.

How do I get into an REU?

As someone that didn't get into an REU, here is my advice:

• Check if the REU is likely to accept students of your year. Typically your odds aren't great for a lot of programs if you're a freshman.
• Finish your applications two weeks before their respective deadlines. This was some advice give to me that I did not follow, and I got rejected from all six programs I applied to. Oh well. A useful data point for you, though!

Which graduate schools should I consider applying to?

This will depend on your field of interest. For example, if you want to do analysis, the schools to consider tend to be (in no particular order):

• NYU
• UCLA
• Chicago
• Princeton
• Washington
• Berkley
• Minnesota
• Indiana
• (Quite a few more great options)

You should ask graduate students and professors that you trust for advice to suggest which schools would be a good fit for you.

How many graduate schools should I apply to?

At least 6. Apply to as many schools as you are comfortable with financially (applications are expensive!) and have the time for (applications are time-consuming!). My advisor told me to apply to like 20 schools, but most internet acquaintances I've talked to applied to like 6 or 7ish. But maybe they're all old and it's more competitive today. If you want to know: I applied to 13 programs.

Some advice I've heard is to apply to both competitive schools and "backups".

Also, while it is not recommended to go to your undergraduate institution for graduate school (in the interest of mathematical breadth and development), you should still apply. They know you well so you have good odds if you don't get in anywhere else. If that happens, you can stay for a year and then try applying again the next year. Not a terrible outcome!

What is the applying-to-grad-school process like?

It's definitely daunting, so let me outline what basically happens. The materials you need to prepare are:

• Your Statement of Purpose and/or Personal Statement (see below)
• Three Recommendation Letters (I would not go less or more. Only go more if you reaaaally think it'll help)
• General GRE and Math GRE scores (slightly optional but unfortunately recommended; various programs still require it)
• Your CV (Curriculum Vitae)
• (Optional) Apply for the NSF GRFP or other grants (these boost your application but are not necessary)
• (Optional) Your written research papers on arXiv

Here was my timeline, according to my journal:

1. (By Early August) Sign up for the GRE and Math GRE.
2. (August-November, or before) Do some research on grad schools and try to figure out where to apply.Write down all the deadlines. Consider making a spreadsheet for the grad schools you're considering (this was really helpful for me).
3. (September-ish) Ask your rec letter writers to write your rec letters for NSF (if you're doing it). Give them at least a full month's notice.
4. (September-Mid October) Write the NSF statement.
5. (September-Mid October) Don't forget the rest of the NSF application; look at it and fill it out as time goes. It is not unlikely that you'll run into at least one issue so don't wait until the day before.
6. (Near end of October) Submit NSF.
7. (Near end of October) Ask your rec letter writers to write your rec letters for the actual grad school apps.
8. (September and Octoberish) Study for and take the GREs. Aim to take them 2-3 months before your deadlines (most of the deadlines were December 15th when I applied in 2022, and the earliest deadlines were December 1st).
9. (Octoberish) You should start the applications for the schools you're interested in so that you can get a sense for what the applications require. There is no "Common App", unfortunately. Every program has a different application. This is partly why this is so time consuming.
10. (October-Early December) Write the template for your statement of purpose / personal statement (see below). Every grad school will ask for one.
11. (November) Finalize the list of schools you're interested in and send your GRE scores.
12. (Mid November) Definitely start the applications if you haven't!
13. (End of November) Spam your rec letter writers the list of schools to send their letters to...
14. (End of November) Apply! (12/1 deadlines)
15. (Early December) Apply!! (12/15ish deadlines)
16. (Mid December) Apply!!! (Weird deadlines)
17. (End of December) Apply!!!! (Early January deadlines)
18. (Early January) Apply!!!!! (Later deadlines)
19. (Mid January) IT'S OVERRRRRRRRRRR

Statement of Purpose / Personal Statement???

Alright this is basically the last thing you need to know about grad apps. It's pretty important. Basically every grad school will ask you for either a "statement of purpose" or a "personal statement". In either case, you seek to answer two questions, and everything you write needs to tie into these two questions:

1. Why do you want to go to graduate school?
2. Why should the graduate school you're applying to take you?

There need not necessarily be much of a difference between a statement of purpose and a personal statement. Here is my view:

• A statement of purpose is intended to be a more dry piece of writing. Briefly give your background and have a topic sentence that describes why you want to go to <insert university>. Then start elaborating on your qualifications - significant coursework and research. While your statements for different universities should look similar to minimize the amount of writing you need to do, you should always have a paragraph at the end personalized to the university you are sending it to. In that paragraph, select 2-3 potential advisors and explain why you think you'd be a great fit for working under them. In your conclusion or otherwise, explain what you envision doing with your PhD.
• A personal statement, on the other hand, is... basically the same thing, except you can embellish your writing a bit more. In my case, my personal statement template was the same as that of my statement of purpose, except for a more flaboyant introductory paragraph that described, in narrative detail, the exact moment that I realized that math problems were just like puzzles, which characterized my childhood, and hence this was the moment that I was certain that I wanted to be a mathematician. This added a bit of a "hook" to my essay. To forge a sense of a thematic narrative, I also included at the end a sentence (...or maybe just a few select words) that tie in what I've been writing back to the ideas I established at the introductory paragraph (i.e. puzzles).

Tips:

• I said this before but it deserves another mention: Don't be afraid to use a writing center. It's a really, really good idea to have professional writers and other eyes review your work. Writing center staff have seen hundreds of applications, so they know what things go in an application. Consider sending your statement to friends and family as well. You can also ask your professors and advisors for guidance.
• Your statement of purpose is NOT a contract. So don't be afraid to massage the truth as to your future goals.
• Don't give an impression of uncertainty. Try to be assertive and confident in describing what you want to do.

Who to ask for rec letters?

The people you ask to write your letters should like you and be able to speak to your ability to do research and advanced mathematics (but mostly the research). To give you an idea, here was my lineup:

• My advisor Leoni, because he knows me well, has worked with me quite a bit on research, and can speak to my ability to do research.
• One of my professors, Fonseca, because I did well in her class and can attest to my understanding of and ability to grasp highly advanced topics. Moreover she is famous and has clout.
• Po, because I worked for him so he knows me well and can attest to my problem solving ability and work ethic. Also, clout.
• (Only for NSF, before Fonseca knew I was not terrible at math) Another one of my professors, Tkocz, because I took several of his courses and he's a very cool guy that can attest to my ability to do hard math.

REUs are also an excellent source for a rec letter. The people that run them may have clout and getting a rec letter from a different institution can strengthen your application.

If you wish, when asking your rec letter writers, you can lightly suggest what you'd like them to write about, e.g. "Hello Professor X, I am applying for Y. Would you be willing to attest to my ability to Z in a recommendation letter? Thank you very much!"

GRE???

Basically this is the graduate version of the SAT. Meaning, it's a bad test designed to steal your money but you probably should take it.

There are multiple GRE tests. The ones relevant to you are:

• The General GRE, which tests critical thinking, your ability to write a cohesive argument, and your ability with English.
• The Math Subject GRE, which tests your ability to do basic college math. The most advanced topics that I've seen are basic topology, basic group/ring theory, the divergence theorem, and stoke's theorem. And everything I just listed is not that common and only at the back of the exam anyway.

If you're reading this, you probably will be fine on the math GRE, but you should take 1 or 2 practice exams just so you understand the rhythm. For general GRE, definitely do at least one practice. Essentially the purpose of math GRE is to make sure you're not a bumbling idiot that can't do a double integral, which makes sense but overall I still disagree with the test since getting a good score is more of a matter of test-taking skills and quick thinking, so it's testing for the wrong thing.

Anyway, more and more grad schools are not requiring these tests (but may "strongly recommend taking them"), but some grad schools may require one test, the other, or both. Do note that as of this writing, you can take the general GRE at home, but the math GRE is less accessible and you may have to travel far to take it, which can be a reason to not take it. When researching possible grad schools to go to, be sure to keep track of which schools require which GREs and factor this into your decision on which grad schools to apply to and whether or not you want to take each GRE.

Choosing to not take the GREs is not too unpopular, I believe. Several of my CMU peers did not (and "several" is rather large compared to the number of students choosing to go to grad school, percentage-wise).

Section 5: Miscellaneous

What is the difference between TeX and LaTeX?

Roughly speaking, TeX does the math stuff whereas LaTeX formats the rest of the paper.

How do I go about learning TeX/LaTeX

Practice typesetting various equations using AoPS's TeXeR. When you don't know how to do something, Google it.

How do I do X in LaTeX?

Google it. I swear it works 95% of the time.

What is Asymptote and how do I learn it?

Asymptote makes pretty geometry diagrams! Definitely a bit less accessible than LaTeX because it's more niche.

I find the official Asymptote tutorial to be a fantastic reference - look at it for some nice examples, then copy them down and play with them. As for where to play with Asymptote code for practice or otherwise, I recommend http://asymptote.ualberta.ca/ over TeXeR, since TeXeR doesn't really tell you what went wrong very well if you have a bug.

You could also try installing Asymptote on your machine, but both me and math senpai could not figure out how to get it to work so in my opinion this is not the best option unless you are a software genius. Feel free to try though.

Section 6: Deep Questions

What's the point of doing all these math competitions? Do they actually matter?

When I was in your shoes, I read a quote from some famous mathematician which compared math competitions to fantasy books (or something), in that fantasy books are not a very good representation of history whatsoever. And I remember being quite triggered by this quote. As I've aged, I've begun to agree more and more with the perspective that math competion math is... not really very much like math.

Sure, there are clever tricks that I like and still kinda use to this day. But math competitions have fundamental issues that prevent them from being mathematical. To name a few:

• The time limit is too short. Real math does not have such contrived time limits. Even the USAMO is arguably a speed competition.
• Being good at math competitions does not at all suggest that you would be a good mathematician. I know some bright students that could slay any math competition, but I was certain that their attitude would prevent them from being a good mathematician. On the other hand, I also know bright students that didn't find that much success in math competitions, but their patience and outlook on math would absolutely make them excellent mathematicians.

Here was another reading that triggered me back in the day: https://mathbabe.org/2011/07/17/math-contests-kind-of-suck/ But be rest assured that this isn't some crazy malarkey. I find myself agreeing with O'Neil's points more and more. Math competitions, especially the AMC and AIME, truly are flawed. They shouldn't have the prestige that they have today. Alas, colleges like CMU and MIT ask for your AMC and AIME score... and to what end? Honestly the only contest that doesn't have the bulk of the issues is the USAMTS.

Anyways, that's my philosophy on how much competitions should matter. At the present day, do they actually matter? In the short term, it is plausible that they may decide things like where you will be accepted for college. But 15 years from now, I hardly believe that you'd have wished that you didn't silly #5 on the AMC. I say this for two reasons.

First, if you have a goal in mind such as working as a hedgefund or shooting for a PhD, then I don't think your choice of college necessarily has a big impact on your ability to achieve this goal. If you are set to it, then you will find the resources you need to do it. In particular, I really don't think that there is much of a different in mathematical outcomes between going to like a top 3 college for math and going to like a college that's #80 on the list. I think decent opportunities exist everywhere, and if you're not at a college that provides the best opportunites, you can work around it. For example, a recent CMU alum wanted to do number theory, but CMU has no number theorists. So she found outside learning and research opportunities or learned it by herself or something. And now she's at a great grad school for number theory, despite going to a college that does not have great opportunities for number theory.

Second, even if "timeline where you silly #5" and "timeline where you don't silly #5 and qualify for USAMO and make MIT" are vastly different, that's not necessarily a bad thing. Introspection is difficult, and in particular knowing what your happiest future is is nigh impossible. Your dreams are unlikely to be as wonderful to live in as you think. A mild example of this is me not making MIT, even though it was my dream school. After spending 4 years at CMU, I'm now certain that being at MIT could have been extremely disastrous for my mental health and mathematical development and I'm really glad that I didn't put all my cards on MIT. My "MIT Timeline" and "CMU Timeline" would have ended up in two very different mathematical outcomes, but I do not think the MIT timeline would have been better, even though it was my dream.

Perhaps you silly #5, go to school X, and then meet a professor who inspires you to do mathematical physics, and so you ultimately become a reknowned physicist who could not be happier? Or perhaps you silly #5 and go to school Y instead, where you meet a wonderful friend that changes your outlook on life completely? (Almost) Every timeline has its ups, and I hope you can spot and appreciate the boons of the choices you have made, especially those that you may have viewed as less than ideal.

Should I really only be doing math competitions to have fun? Is it wrong to do math competitions to get into college?

Ideally, you should be doing math competitions for the fun of it. And most competitiors do say that this is their main motivation. But deep down, I do believe it is true that we hope that math competition can give us a boost in life, particularly for college admissions.

As I suggested above, your choice of college may not matter so much in the long run of it. But the choice at least matters a little, certainly, and it is natural to be worried about getting into one where you can thrive and be happy. So, while I don't like the idea of college being a major reason for doing math competitions, I can certainly respect it. That being said, I think it is unhealthy to let math competitions rule your life in this way. If you're working hard for some achievment in math competitions in order to give a major boost to your application, it can cause a lot of preventable stress that may end up worsening your performance. For that reason, you should try to focus more on the fun part rather than prestige, even if prestige is a goal. Having a healthy relationship with math is key for thriving in math.

Also consider finding other ways to convince colleges that you are mathematically mature and/or have a particularly strong interest in mathematics. Doing math research, participating in summer programs, participating in the USAMTS, taking leadership rolls in math competitions, taking college courses, etc. can all contribute to this, especially if it is clear that these efforts are due to your own initiative.

Women in math...?

Sigh.

One time, during a certain PUMaC GUTS round, I sent my friend, Y, to be the team's runner. When Y went to the front of the room to place her hand on the problems, I noticed that she was the only girl there. I then looked around the room full of teams, and realized that we were the only team with more than two girls (and we had five!). Seriously?

At CMU, it arguably got worse. In my advanced multivar class, there were maybe 3-5 girls out of 30. When we proceeded to the next course in the sequence, that dropped to 2 out of around 15. In the last course that I was a TA for, there was a single girl.

There are few things that I despise more than the gender gap in mathematics. I hate it for many reasons. The first is a personal one that I will keep to myself. Second, I was inspired to do math by women. The fact that there is any sort of gender gap is nonsense to me. Third, diversification in mathematical views is essential for mathematical progress. Fourth, this is a problem that causes itself, particularly through inherent sexism rooted in society that has no reason to exist. I would rant more about this but that would not be appropriate for this blog.

In any case, here is some deeper insight into why this is such a major issue that we should be angry about. I am not a girl, so if you want a girl's perspective, read Wanlin's experience in the middle of this infamous thread. It is very eye-opening, and anyone that thinks that this gap should be allowed to continue should read this. If you want my perspective: I sense that the main reason as to why these gender arguments keep coming up time and time again is because men simply do not understand the implicit and explicit oppressions that women experience from society (and while I do not claim to do so, it is important to at least acknowledge that you may not see the full picture by virtue of being a certain gender). To wit, here is a study done in 1999 in which researchers gave Asian women math tests under various conditions: https://journals.sagepub.com/doi/10.1111/1467-9280.00111

To sum up the study: The term "salience" refers to when a person is made aware of one of their social identities. For example, the gender distribution of my PUMaC team back then was very salient, since basically everyone else was male except for the disproportionate number of women on my team. Essentially, the study found (among other things) that by making a participant's female gender identity salient (by way of asking certain personal questions beforehand), their performance decreased significantly, due to existing negative stereotypes pertaining to women in math.

Hm, don't various tests ask for your gender beforehand?

Sidenotes

Tanya Khovanova also has written some interesting pieces on this, e.g. https://blog.tanyakhovanova.com/2008/11/a-room-full-of-women/

So... what can we do?

I have no idea. Many universities have AWM chapters, and great competitions like MPfG exist. You should support efforts like these if you can. But I don't have any good solution to this. It really sucks and I don't think I will live to see the day when sexism in mathematics is eradicated.

Why is there no boys-only contest?

We don't need more men in STEM. It's not that complicated and trying to argue against this is stupid.

Why does school math suck so much?

Read Lockhart's Lament. Although it is too radical in my view, it will do a better job of explaining the issues than I ever will.

Am I too dumb to do competition math?

I'll quote myself here.

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Alright, that's it from me! If you feel like anything I wrote is blatantly wrong, feel free to let me know and I'll consider fixing it.