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Info about applying to PhD programs in pure math

This is... basically what it sounds like. I’m sure a lot of this is applicable to other PhD applications but I’m only very familiar with pure math. This is aimed at current seniors about to apply, but there is a section about prepping for applications in advance.

The highlights:

Recommendation letters are the most important thing. Most schools ask for 3ish. Try to get people who know you well, not just a student in the class. Someone you've conducted research with and one from a different institution are ideal if possible

Ask for rec letters at least a month before the due date is a good rule of thumb.

Research experience is probably the second most important.

Get the opinion of multiple professors who know you in order to build a list of potential schools. Then widdle it down to your will-actually-apply list (probably 8-14 ish). 

My opinion but please apply to at least 3-4 safety/match schools. Even when you're fully qualified, acceptance rates are simply low enough that a bit of bad luck means getting rejected or waitlisted from a few of them.

Most pure math due dates are in early-mid Dec but a few schools are in Nov and some are as late as mid-Jan. 

Schools will generally have their own graduate application portals. Some are better organized than others. Some require you to submit all your material before you can send a request for submitting rec letters so plan accordingly.

Acceptance letters will very slowly start going out in mid-Feb but the vast majority of programs won't send out anything until like, March and not be done until later than that. Accordingly, wait until at least mid-March to begin freaking out if you haven't been accepted anywhere.

You should 100% be expecting a tuition waiver and stipend from a program if you're applying for a PhD.

The rest of the posts is.... ridiculously long so I’m putting it under a cut. I mention things to do in advance to help you decide if grad school is right for you and things that make your application look good, give a full time line of the process, a list of things applications commonly ask for, and some miscellaneous notes. (The points above are repeated in more detail).

In addition, some links to other resources math students may appreciate:

an old post of mine about grad school apps (overlaps a lot and features some ranting from during the application process)

about REUs including my addition specifically about math ones

summer programs for undergrads that aren’t REUs by @counter-example and @jungleuniversity

Tips for prospective grad student visits 

Also about prospective grad student visits by @thisurlhasbeenleftasanexercise

Also for context, I went to a large state school in the US for undergrad. I started as a CS major and added on math as a secondary major after my first year and dropped CS during third year. I’m primarily interested in discrete and algebra, though I have a significant topology background from undergrad too. I got most of my advice from people around the department, as I became pretty involved during my third year. Now, I’m a first year grad student at another large state school in the US, generally considered pretty decent though not a “top math program” at all. Not that much else has happened so far.

Things in advance (aka things to help you decide if grad school is for you and things that look good on an application)

Take the standard classes. For pure math, this is at least one semester of linear alg, abstract alg, and analysis each. Linear and analysis are also good for applied math but I'm not sure what else if anything is considered standard.

Take some grad classes if you have the option. Most people are not ready for this until senior year, but some do manage as juniors. Talk to people who know you well and the prof teaching the class before you do this though.

Try to get involved with research whether this is through independent studies at your home institution, REUs, internships, or other stuff.

Be involved in your department. This helps with getting you more personalized advice for applying.

The rough suggested timeline (assuming junior yr is your second to last year and senior is your last of undergrad)

Junior April: Take the math subject GRE so you can take it again in Sep or Oct if desired (perhaps not applicable atm). The general can be taken kinda whenever; I suggest fall of senior year.

Junior April/May: Start talking to professors/post docs/mentors/etc. about programs you may be interested in. Write/type it down. Don't worry if it gets long, you will shorten again later.

Summer: Do some research if possible; an REU or research at your institution (if an REU, also get your mentor's opinion on potential schools towards the end as well)

Senior Sep: Start whittling down your list. 8-14 seems to be the "normal" range of schools to apply to but some people panic and do more. Remember that asking for waivers is completely acceptable but applying is still just generally expensive (I spent around $800 for 10 schools)

Senior Sep: Apply for the NSF GRFP. You can apply as an undergrad senior and once during your first or second year of grad school if you didn't already get it. The due date is in mid-late OC but ideally you'll have a draft of your essays and ask for rec letters by the end of Sep, if not earlier.

Senior early Nov: Ask for rec letters if you haven't already. The rule of thumb is a month before the due date. Provide them a list of schools you want to apply to including due date and where/how to submit as soon as possible (as well as anything else they request of course; many ask for a resume and a draft of your personal statement).

Senior Dec-Jan: Submit stuff! Pure math programs typically have deadlines in Dec or early Jan. I think the big days are Dec 10th, Dec 15th, and Jan 15th but some are earlier or later. (applied math masters tend to be earlier I think; in Nov). I suggest putting them all into a list or calendar. In addition, some schools won't let letter writers submit until all of your stuff is submitted so start applications early, even if you don't finish them immediately.

Senior Feb: Programs will slowly start sending out offers in early Feb and pick up in mid Feb, but don't fret until AT LEAST the beginning of March! Grad programs are just way too slow at getting out offers for it to be worth worrying until then (and even then, it's definitely not time to panic but mathematicians are frequently anxious people so I get it). Waitlists are slower to come out; usually starting in early March. Also note, there are many programs that don't actually send out replies to everyone unfortunately.

Senior late Feb-early April: prospective student days! They might be online in 2021 unfortunately but try to attend whatever form they're in if you can (only one of my visits during spring 2020 was online since the others happen to be very early and safely beat covid in the US). Be warned, it's very possible to get offers of admissions and to visit very last minute. I do not have advice for how to make that less stressful.

Senior April 15th: Common reply deadline. If you got your offer in the first round or two, this is probably your deadline to accept. In addition, this means more offers will likely come out shortly after once more people have declined. 

Senior summer: graduate. Send a completed, official transcript to your new institution. Check your new email account for stuff you're suppose to do. Some programs have some sort of program during the summer for in-coming students. Most places have graduate student training of some sort for a week or two before semester starts. 

Some common things to be asked for in applications

Not actually a thing asked for but many graduate schools have their own portal for which you will have to make an account to submit an application. A few use a common system that kinda sort shares a database of accounts? Some are fine and some massively suck.

Personal Statement/Statement of Purpose: Occasionally called something else and once in a while actually separate things; will usually have a prompt of wildly differing specificity. Sometimes, the prompts come from the department itself and sometimes from the university's graduate school. I suggest having one or two "base" essays then tweaking them for each school. Sometimes a word/page limit is specified but if it's not, around 2 pages/1000 words is pretty reasonable.

Transcript. Some accept unofficial but some require official but generally not an unsealed one. I ordered myself one official transcript and sent it to multiple schools instead of paying for them to be sent to each school during the application process.

Resume or CV: Most ask for either a CV or is fine with either, in which case I give them my CV. I sent more or less the same one everywhere.

Some other notes

Yes, ask for application waivers. Just be polite about it.

Your goals for your essays are primarily to show that you're interested in math and math research and are capable of like…. writing things that make sense

Do not start out an essay with either "I loved math since I was little" or "I actually didn't like math when I was young" or any variations of those. (I had one essay that started with a mildly humourous anecdote from undergrad combinatorics and another that talked about how my undergrad department has greatly affected me).

You should 100% expect to get a tuition waiver and living stipend as part of a TA fellowship (or more rarely, an research fellowship) as part of your offer of acceptance for a math PhD program (pure or applied). Health insurance is also frequently part of the package. This is not true of masters programs unfortunately.

How schools do waitlists depend wildly though most don't have super long ones like prestigious undergrads do. If you're still interested in a place you're waitlisted at, follow their instructions to confirm your placement on the waitlist then wait until April before following up again, expressing your continued interest and asking for an update. You might even want to wait until around the common deadline, April 15th. The number of people who declined before April is just really really low so nothing really happens until then.

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when u go to ur professors office hours but the door is closed and ur too shy to knock so u just stand there like

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how to survive college:

1) impulse buy things you can’t afford

2) play stardew valley when you have important things due

3) cry

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uhhh... no comment

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when the professor says you need to buy an access code:

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me during the school year

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my last two brain cells when I’m trying to study:

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everyone: college will open so many new doors for you

me, after receiving my tuition bill:

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friend: hey, how have you been?

me:

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me after successfully completing any basic lab procedure:

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always an et al., never a first author 

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me waking up at 2pm after saying i would fix my problematic sleep schedule

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a summary of my undergraduate career