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During my high school career, I had the unique opportunity to take multiple mathematics courses at my local university, and after graduating, I took two graduate level math courses: Measure theory, and a class on topological manifolds. This was a great experience, but it has left me in a shifty situation.

I asked the department head if I could be admitted into the program since I took a full time course load receiving a B+ and A respectively, but it was denied due to the fact that I did not have a bachelors. The only option given to me at this university would be to enroll in the undergraduate program and take graduate courses.

I would be open to that idea, but it has multiple pitfalls. First of all, I am not guaranteed any research opportunities, which view as highly restrictive on my growth as a mathematician. In addition, I am not guaranteed funding opportunities like I would be in the graduate school. This in combination with the fact that I would have to take 50% more credit hours of course work would mean I would end up having to take out more loans than I would like. This is all very disappointing for me.

Instead of enrolling in courses again, this semester I began independent study in algebra and differential topology using Aluffi's and Lee's book respectively on the subjects. In addition, I have been exposing myself to more theoretical physics, such as Yang-Mills theories and Supersymmetry. Because these subjects are capturing my interest as well, it seems natural that I take my studies towards Algebraic Geometry and String Theories. I really want to get involved with a program to help enable me with my studies, but I do not think an undergraduate program would be adequate. For those of you who are still reading, how can I work towards getting into a program which would enable my studies an empower me with supervised research?


EDIT

I did not think my undergraduate work would to be too relevant, since most of it was not rigorous; that is, calculation based. I took undergraduate courses in ODE's, PDE's, probability theory, analysis, mulitvariable calculus, and linear algebra. Unfortunately, I did not do well in the ODE, calculus, and linear algebra courses because they were during the summer, and I had not yet acclimated to the required amount of work. In this specific program, my course work is nearly equivalent to a bachelors degree, sans single variable calculus and a class in discrete math.

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It's going to be very difficult to convince a graduate school to let you in on the basis of having a grad class or two, if you don't have the expected background equivalent to a bachelor's degree in mathematics. –  aeismail Apr 10 at 21:32
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I would question a graduate mathematics program that took on a student that did not do well in undergrad calculus, no matter the reason. –  corsiKa Apr 11 at 1:26
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If you didn't do well in undergrad calculus or linear because you weren't "acclimated to the required amount of work" why do you think you are ready for graduate study? It's about more than having enough prior education to study graduate-level material; you need to be able to handle an immense workload. You view yourself as more precocious than you really are by doing well in graduate classes, too, because they are not necessarily difficult. –  2rs2ts Apr 11 at 18:11
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@user14018: then you need to go back and make sure you understand calculus and ODEs before you attempt to progress. Calculus is straightforward and simple compared to the type of stuff you are talking about studying in grad school. Furthermore calculus is used for almost everything else. –  Michael Martinez Apr 11 at 19:08
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An undergraduate degree is an opportunity to learn more mathematics. How could learning more mathematics possibly be harmful to your intended research career? –  David Richerby Apr 11 at 20:56

8 Answers 8

I know of one case in which someone went straight from high school into a graduate program, so it is theoretically possible if you can find a university that agrees. However, it's almost unheard of, and it would be a bad idea for just about everyone (even setting aside the question of whether undergraduate education provides any benefits beyond preparation for graduate school).

What I assume is that you'd like to do important research and are looking for preparation that will help you accomplish this goal. The minimal standards for a Ph.D. are actually pretty low, and you can squeak by without knowing much or doing impressive research, but this sort of dissertation will set you up for career failure. Instead, you need to enter a strong Ph.D. program, ideally one of the top ones, and really excel.

It sounds like you've completed enough courses for a minimal undergraduate degree in mathematics, but probably not enough to be competitive at a top grad program. The applicants at these programs have generally taken a more extensive and rigorous collection of undergraduate courses (e.g., abstract algebra, topology, complex analysis), and it's not uncommon to have taken more than two grad courses, sometimes many more and in a few cases starting in high school. Plus it's common to have undergraduate research experience or other substantial independent work, such as a senior thesis.

Basically, imagine yourself in four years. That's who you're going to be competing against for admission, the other extraordinary high school students who went to college and became even more knowledgeable and impressive.

If you could find a university that would admit you to graduate school now, you'd probably end up going to a second tier school, spending some time catching up on courses, and then writing a good thesis but not living up to your potential. Your career would get off to an inauspicious start, you'd have trouble on the job market, and in the long term you'd likely end up with a less research-oriented job than you might have had otherwise.

Of course I could be wrong about this: you might have exceptional luck or be the next Terry Tao. But diving into graduate school at a young age with minimal preparation is an incredibly risky strategy.

This is something that's really hard to discern from college catalogs. They usually focus on the minimal requirements to complete each degree and move on to the next level, but that's not how people do things in practice. Instead, the vast majority of future mathematicians spend at least three years on their undergraduate studies, and generally four (in the U.S.). The operative question isn't how quickly you can complete the degree requirements, but rather how much you can achieve along the way. Preparing well for grad school is far more demanding than completing an undergraduate degree.

So what I'd recommend is that you pick an undergraduate program in which you'll have really outstanding fellow students, so that you learn from and are pushed by them. Being part of a cohort like this is probably the single biggest factor in success, and your goal should be to find amazing people to hang around with and discuss mathematics. You should then take whatever undergraduate courses are needed to fix up your background (if any) and move on to graduate courses. You should also look for research opportunities during the summer or perhaps the school year, but it's not worth obsessing over this. No program will guarantee anything, but at the very least you'll surely have some success applying to summer REUs, and it's reasonable to hope for even more than that. Perhaps you'll finish in three years, and perhaps you'll decide to stay for four, but either way you'll be far better prepared for graduate school than you are now.

It sounds like you're in a great position to do well, with a remarkably strong mathematics background. However, I'm convinced that the right question is how to get as much as possible from your undergraduate studies, rather than how to skip by them as quickly as you can.

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"[Y]ou might have exceptional luck or be the next Terry Tao." Notably, Terry Tao earned a bachelor's degree before going to grad school. –  Mark Meckes Apr 11 at 7:33
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+1 for Preparing well for grad school is far more demanding than completing an undergraduate degree. –  jwg Apr 11 at 13:45
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@MichaelMartinez: Mathematics. Aleksandr Khazanov went straight from high school to grad school at Penn State. (His career ended in tragedy when he disappeared in 2001, but he was an extraordinarily impressive young mathematician.) –  Anonymous Mathematician Apr 11 at 18:54
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@MichaelMartinez: That may be, but I haven't known many graduate students who were you. –  Mark Meckes Apr 14 at 18:50
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@MichaelMartinez: Good for you. For the record, of the departments I'm familiar with, the one with the lightest list of specific requirements for a BS in math is Stanford. –  Mark Meckes Apr 15 at 18:38

I am not familiar with math grad programs (your situation would be highly unusual in a CS grad program), but I'd like to question two statements you make about the non-viability of an undergraduate degree:

First of all, I am not guaranteed any research opportunities, which view as highly restrictive on my growth as a mathematician.

If you're good enough to take advanced material, grasp it and even approach an actual research question, then you're functionally equivalent to a grad student and many professors would be happy to take you on. The antecedent is important though: the fact that you've taken courses doesn't mean you're ready to do research, but this is precisely what REU programs are for, and faculty often get extra money to support students for UG research.

In addition, I am not guaranteed funding opportunities like I would be in the graduate school. This in combination with the fact that I would have to take 50% more credit hours of course work would mean I would end up having to take out more loans than I would like.

You're not wrong here, but there are often scholarship opportunities available, so the situation isn't completely black and white.

A final note: one reason that grad programs might be leery in taking someone without an undergraduate degree is because the UG degree is more than just a credential. It's a proxy for a long sequence of coursework that exposes you broadly to an entire field of study. In almost any discipline, a basic level of breadth is very important for research work: because you need to

  • understand what questions are important and why
  • understand how different questions fit together
  • acquire the ability to see connections among different parts of an area.

Of course you can do an entire undergraduate program via self-study, but then you don't have a "short certificate" proving your command of the material.

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I appreciate your advice. It sounds like I would need to find a professor willing to help me getting into a graduate program. –  user14018 Apr 10 at 21:54

If you can't get into a good graduate program without a bachelor's (and it seems highly unlikely; I have never heard of such a case, even with child prodigies): Pick an undergraduate university that will offer you a full scholarship (I'm sure that if you have good SAT and grades, you will be able to find some such university), graduate as fast as you want to (could easily be done in 2 years at most public universities, with your background, assuming credits transfer which they should, and you might not even have to take summer classes), and aggressively seek out undergraduate research opportunities from day 1, both during the semester and during the summer.

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Better yet, get admission as an undergraduate, and then take the full four years, alternating between graduate math classes, research, and growing up. Life is not a race. Nobody gives a monkey's how fast you graduate. –  JeffE Apr 11 at 1:10
    
I didn't suggest he/she should graduate early. But he seems to have an aversion to doing an undergrad at all, so that was just a suggestion as a sort of compromise. Also, he might be older, in which case graduating quickly might be attractive. –  user2258552 Apr 11 at 19:46

You're not going to get into grad school in Math without going through a Bachelor's first. Just like you won't get into grad school in any of the hard sciences without doing the foundational work first. There's good reason for this: you need the solid background in the fundamentals before you delve into a specialty. The specialties emerged out of the fundamentals, historically, and it makes no sense to try and skip ahead. Also, if you try to do this, you're doing yourself a great disservice, which you would probably realize later on. You'll be approaching your studies without adequate knowledge/experience.

Why would you want to skip undergraduate courses anyway? There's a lot of mathematics to be learned in the undergrad classes, and if you truly love math, then you ought to welcome the chance to learn and explore. You say you didn't do well in calculus and differential equations, but these are foundational for so much mathematics that follows.

So, the main question here is not whether you can go straight into grad school (the answer is no), but what your true goal is: to learn and appreciate math, or simply to get into a field that you think is interesting without truly knowing what's involved.


I went to an engineering school that is well known for the hard sciences and engineering, I've had a lot of exposure to academia, have known, studied and worked with a lot of people in these fields, and have never heard of someone going straight from high school to grad school in physics, math, chemistry or engineering fields.


now that said, once you are enrolled in an undergraduate degree program, you can definitely take grad level classes, particularly once you get the basic prereqs out of the way, once you start showing your interest and abilities, it's pretty easy to take those grad classes even if you haven't finished your B.S. yet. I've had friends who basically worked on both their B.S. and M.S. or Ph.D. at the same time!

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You need a mentor.

Consider your current experiences and gifts as a large and powerful chainsaw. You may find some immediate gratification at cutting through swaths of wood at a pace you like. However, you can burn yourself out, wreck the chainsaw, reduce your number of functioning limbs, etc. (I didn't lose any limbs as an undergraduate, but I got sick of school in my second year and left it in my third. If I had paid more attention to the people who were there to help me, my earlier years might have been more satisfying.) If you give yourself the discipline to handle the routine and mundane, and also the time to have fun and work on things at which you excel, you will have something better than an awesome trajectory: you will have a life of enduring and satisfying achievement. And you can still spend time doing graduate work.

The mentor will have to know you well to tell you what is best for you. (A team of mentors might be better.) I can imagine more disasters for you than successes if you try this without at least one. Also, the Internet is no substitute for a mentor: this kind of life decision, while up to you, can benefit from talking face to face with someone who is interested in your personal as well as your academic success. As commented elsewhere, hardly anyone cares how fast you did something as much as how well you did it.

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Don't focus on the letters you'll get from your first degree.

It is very likely that you will need to get a bachelor’s degree before going enrolling for an advanced degree, especially since you sound like you would do best in a highly selective program. But for now, what you need is a university and more specifically, a math department, that is willing to be flexible, encouraging of advanced work, and give you the chance to prepare yourself for a really top graduate program. You need this whether you are admitted as a masters student or an undergrad. Assuming you want to continue to a PhD after your first degree, you need to focus more on what you will be doing for the next few years, rather than the degree you’ll have at the end. You are better off finding an excellent school that will support an undergrad doing graduate level work (fairly common) than an OK school that will accept a grad student with uncertain credentials (very unusual).

To start with, I’d strongly recommend contacting professors at well regarded math departments whose research you find interesting, and let them know your situation. This will mean you are looking at schools as though you were going to grad school even though you will probably be an undergrad. Find professors you think you’d like to work with, and arrange to meet them and meet their other grad students. In person is best, but at least talk on the phone. Apply to a few schools where the math department supports and encourages their grad students, not just a place with a good reputation. I've had a number of friends who have dropped out of PhD programs due to department politics or unsupportive advisors. Start yourself out on the right track.

Many schools have scholarships available, and in some cases, a school will offer a full ride including a living stipend to the most exceptional undergraduates. The math departments and professors may be able to help you out here, too, if they especially want you. Just ask. Don’t limit your options because of a preconceived bias.

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I think a very good options for you is to apply to very competitive (private) US institutions for undergraduate (i.e. Harvard, MIT, Princeton, Stanford, Yale, etc.).

I mention these specifically for a few reasons. They all have very well regarded graduate programs where you will have the flexibility to take very advanced courses. They have extraordinary peers with similar advanced backgrounds (my brother, who was an undergraduate at MIT, took many years of graduate math coursework in high school). Finally they have very good financial aid opportunities. I will say from my own experience (at Yale), that strongly prepared undergraduates are known to jump straight into graduate coursework and research freshman and sophomore year.

The most important thing to do if you are interested in mathematics (or physics) professionally is to interact with great professors and go to a great graduate program. I am extremely skeptical that the top notch of graduate math programs (the past list + Berkeley + University of Michigan + Cambridge) would take you right now. People apply from undergraduate with graduate coursework, research experience, and letters of recommendation from very well known people in the field. The last point is especially important for these top notch programs (math is a fairly small community). By going through undergraduate, you will make these connections early on, find a subfield, position yourself for the very best program in your interested subfield, and get into an excellent program.

Another thing to note is that assuming your "local university" isn't a large, well-regarded state-school (or international equivalent), these courses are simply not up to the level of those you would take at top-tier university. Generally, in my experience taking courses at my local university in high school, theses course are very computational and those that are proof-based aren't nearly as challenging as those I found at Yale.

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Unlike some other respondents, I don't think this is impossible. In fact I have a friend who dropped out of high school for various reasons unrelated to his academic achievement. He spent several years studying math and CS, by himself, to graduate level. He tried in various ways to access a research-based course, culminating in him being offered a place to do a PhD in CS, without having either a Bachelor's or a Master's (which is a usual requirement to start a PhD in Europe).

However, to follow a trajectory like this there are at least four necessary conditions:

  • You must be very good. Someone who looks like they are going to do a passable PhD and publish a couple of papers before stagnating or disappearing into industry is not going to get this opportunity. Someone who has already published peer-reviewed work might be a candidate.
  • You must have some good reason to not want to do an undergraduate course. You aren't going to be supported with this just because you don't feel like sitting in a lecture hall for several years, listening to things you already know. Most working mathematicians, including prodigies and household names, did exactly that. Usually you will be advised to do the BSc and study harder things / do research in your own time. Good reasons might include some disability or personal circumstances that make it extremely hard for you to get through the undergraduate system, or the fact that you are already spending most of your time doing professional level research. Money issues might or might not be seen as valid, depending on whom you ask.
  • You need to show that you are mature enough to be a graduate student. Undergraduate studies involve a certain amount of hand-holding for a reason - many students who have just left high school aren't ready to take full responsibility for their education. Even if you are a genius, a school will be worried about your abilities to perform well in your studies, while taking care of yourself on your own, and not succumbing to mental health issues, addictions, etc. Teaching or TAing undergrads also requires a certain maturity. The exception is if someone is going to take care of you (for example, if you have ASD, and you live with your parents who support you with all the non-mathematical things you need help with).
  • You need to put your case to a professor in a school you want to work at, including convincing her of the things above. This is key. There are various obstacles put in place to stop the over-confident, the under-qualified, and the insane from getting onto advanced math courses. If you apply in the usual way you will almost certainly be rejected.

Your best bet is to approach someone in a field you care about, write to them directly and explain why it is that you know you are capable of a PhD. Most people you approach will reject you out of hand. (Think Ramanujan - statistically you are likely to be less talented than Ramanujan.) If you get lucky, you might find someone who is willing to hear you out. Be aware that anything that gives the impression of a scattershot approach (standard letter template, not being able to explain why you chose that professor or that school) will rule you out very quickly. You should be looking for someone whom you have read and understood several papers by (if you aren't reading and understanding journal papers on a regular basis, why not?). You should be able to show them published or publishable work you have done, related to their own interests. This person, if they end up supervising you, will have to fight to get you in to their department, and then deal with an unusually onerous burden of formalities and paperwork to have you as their student. They aren't going to do this unless there is some compelling reason why they would want to work with you.

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You need to show that you are mature enough to be a graduate student or in my own experience undergraduate studies provides the time and freedom to make mistakes while you are busy growing as a person (assuming typically high-school graduate age or younger). –  mctylr Aug 8 at 16:58

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