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I just received my first offer to a Mathematics PhD program at a reasonably good US institution, where one of my recommenders is also working. He sent me an email saying he "hopes I can accept the offer quickly". I love working with him, and have told him his school is my top choice, which is true, but I have some very close second choices whose results are not yet out (expected within next two weeks). The official deadline to accept the offer is in April. Should I politely tell him that I wish to wait for a month before accepting this offer, stating "I'm a bit nervous to accept the first offer without taking a look at one other first" (which is also true)?

In the US, "hope you can accept the offer quickly" has become a set phrase in graduate admissions letters. You have the right and their explicit permission to wait to accept an offer until the deadline they give (such as the April 15th common US graduate school deadline), and you almost universally will not be penalized in any way for taking that time if you need it (i.e., waiting to hear from other schools, going to visit days to see the place in person, etc).

When I say set phrase, I mean that every offer I received from every graduate school had this term in it, so it is actually part of form letters now for many (most?) places in the US.

What does it mean? Simple - everyone would ideally like to know as soon as possible what your decision is, so they can know who is joining. That's it. They would naturally want to know, so if you have already made your decision they would like to know and have you lock it in rather than exhibit the common behavior or waiting until the day before the deadline to submit your decision.

That's it! So don't worry about, at all. All you need to do is is reply and thank them for their offer, perhaps state your excitement at having received the offer (if you genuinely have that reaction!), and let them know you will let them know of your final decision as soon as you have made it. No need to give excuses or be sorry, that's just how this works. No need to feel pressured into deciding sooner, its a big decision and you have the set deadline for lots of good reasons. Make the decision when you have the responses you need, and if there are visit days feel free to wait to accept the offer until after you have gone and met them (especially true if you have multiple offers you are trying to decide between).

Congratulations on your offer, and good luck in your future career!

Many graduate schools have subscribed to the April 15 Resolution which says that accepted applicants are not required to make a decision until April 15. If the school you have applied to is one of these schools, then you should feel absolutely no compunction to report a decision until then.

I would basically just tell him that you hope to be able to make a decision soon and will keep them informed. You don't need to mention that you're waiting to hear from other schools if you don't want to say that; it's implicitly understood that most applicants may have to decide between multiple offers.

Before you do anything, ask yourself what is the best possible outcome here. You have an offer from your top choice school and someone you like to work with. If the other parts, such as grants etc. are acceptable, this may be it. But I can't make that judgement. But if it is the best choice no matter what, then why risk it? What more do you hope to achieve?

I would slow pedal it. You don't need to explain anything. It is very unlikely the offer gets pulled. It's OK to communicate some gentle positivity and keep #1 positive. But I would not let yourself get influenced to early decision given "close seconds" exist. You need to string #1 along.

Just wait and see what comes back from the other schools, especially on the money front. Check on deductions also, some places pull for healthcare, gym access, etc.) This kind of dynamic with offers, competing offers is normal, especially for top candidates. Without making a big deal of it, you need to look out for you. It is possible some places offer you small incentives. You may also have the option of paid or at least subsidized visits to some schools to assist your decision.

Same dynamic will hopefully occur when you job interview in 4 years.

My first guess as to the reason that he "hopes I can accept the offer quickly" is that he genuinely wants you to come to his school. If so, any delay on your part would only increase the likelihood that you end up going somewhere else. If he is at all reasonable then there should be no problem in politely explaining to him that since you applied to multiple schools, you want to see how it plays out before making your final decision. I thus interpret his statement as an indirect testament to you potential, rather than an unreasonable pressure that should cause you anxiety. It is nice to be a sought-after student. No need to overthink it.