# Can I jump straight into a PHD program?

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Hi, I am a senior in High school. In only 4 months I have managed to self teach (from paperback books, PDF files and online courses uploaded to various platforms such as youtube)the following during my free time:

Calculus 1,2,3,

Differential Equations

Linear Algebra

Abstract Algebra

Discrete Mathematics

Analysis 1,2

Calculus of Variations (Optimization Calculus)

Elementary Number theory

Concise Introduction to Mathematical Logic

It is clear to me that I am gifted. I can only imagine once I dedicate myself full time to the study of math. I am not clear what exactly to study. I am interested in Complexity Theory, Number Theory and Artificial Intelligence. I have an extremely good memory. I would say 70% photographic. I am more than happy to hold a Skype session to demonstrate my abilities. I would like to jump straight into a PHD program.

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Hi Dan98,

It's possible and I know of a handful of professionals who did so working today (enumerated somewhere else here on SFN), and personally had some acquaintances who did similar. It is very difficult and universities are reluctant to accept this, unless it's a regarded university where the faculty have strong pull (say, a fields medal winner at Cambridge).

I would advise you find someone with your interests, ideally also in some administrative position, and apply to their university. I imagine they may be sympathetic to your situation and allow you to test out of most of the degree requirements, letting you finish within a year or two. You should also study for and take the mathematics GRE; it's, at least I know in the US, used to sift through competition in graduate school applications.

Alternatively, but similar to the first point, you can apply somewhere that tries to make the undergraduate experience research focused rather than course focused. The best place I know for this is the UCSB College of Creative Studies, which lets you study independently under and advisor and get credit accordingly, without requiring sitting a course. I have a friend who goes there who seems to enjoy it; as opposed to normal UCSB students, CCS students can take any course without formal prerequisites. More to the point, they are paired with researchers and practitioners to pursue their interests accordingly. From my understanding, the Worcester Polytechnic Institute is similar to this, but a good bit more structured.

If you have some more applied interests and have an idea of an actual project (with some industrial/sustainable application), even in AI or complexity theory, then you ought to take a look at the ThielFellowship. They provide two years, $50k a year in funding, and heavy mentorship and connections to help you with your project; the only requirement is that you not hold any higher degree. A few of friends were participants, some who had even pure interests in mathematics and physics. If you have had no previous experience with proof-based mathematics, then I have no idea how you got such a good handle in only four months. Congrats if you did so, I wish you luck, but... I am a bit skeptical.. Addendum: I ought to add, that there are regularly more than a handful of students who have covered most of the mathematics undergraduate curriculum in high school. Most visibly some of the IMO gold medallists and Intel STS research winners. From the handful that I've known, about all of these people do a typical four year degree. Lots end up going to schools with very strong math departments, say Harvard, and retake some courses that they've studied their freshman year, and spend the rest of their time taking graduate courses. So, you certainly aren't alone, and the usual path taken is a normal undergraduate degree. Edited by Sato ##### Link to comment ##### Share on other sites Hi, I am a senior in High school. In only 4 months I have managed to self teach (from paperback books, PDF files and online courses uploaded to various platforms such as youtube)the following during my free time: Calculus 1,2,3, Differential Equations Linear Algebra Abstract Algebra Discrete Mathematics Analysis 1,2 Calculus of Variations (Optimization Calculus) Elementary Number theory Concise Introduction to Mathematical Logic It is clear to me that I am gifted. I can only imagine once I dedicate myself full time to the study of math. I am not clear what exactly to study. I am interested in Complexity Theory, Number Theory and Artificial Intelligence. I have an extremely good memory. I would say 70% photographic. I am more than happy to hold a Skype session to demonstrate my abilities. I would like to jump straight into a PHD program. I'd suggest that you go through an undergraduate program. There are several reasons for this. First, there's epistemology: how do you know whether you really have mastered it? Is it possible you've misjudged your own ability? Second, there's credentials. Many places are unlikely to simply accept that you've learned these topics in 4 months; they would want you to show you've learned it. Taking these courses is one form of those credentials. Another would be talking to math professors. Third, there's mindset. Just because you've read from the books doesn't mean you have the right mindset for a Ph.D. program. Consider this paragraph, taken from Terry Tao's (a very famous mathematician) blog: One can roughly divide mathematical education into three stages: 1. The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years. 2. The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner (e.g. re-doing calculus by using epsilons and deltas all over the place). The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. This stage usually occupies the later undergraduate and early graduate years. 3. The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond. If all you've done is read from books, you are likely still in early stage 2, if not late stage 1; the start of the transfer between stage 2 and stage 3 usually coincides with starting graduate school. All in all: no offense, but it's unlikely that you've simply learned the material in 4 months, and extremely unlikely you've learned the mindset. You should talk to professors at some nearby university; they'll be able to ask you more specific questions and advise you better, but jumping right into a Ph.D. program is unlikely. ##### Link to comment ##### Share on other sites The rules and requirements will vary place to place, but in general it is possible to get onto a PhD programme without an undergraduate degree. Funding could be a big issue though. On top of that you would need to convince a supervisor to accept you and convince the admissions team in the department that you stand a good chance of completing the programme. The best way to do that is to have a good undergraduate degree and maybe a masters degree also. So, while it may be possible, your best best is to go down the standard route. ##### Link to comment ##### Share on other sites Hi, I am a senior in High school. In only 4 months I have managed to self teach (from paperback books, PDF files and online courses uploaded to various platforms such as youtube)the following during my free time: Calculus 1,2,3, Differential Equations Linear Algebra Abstract Algebra Discrete Mathematics Analysis 1,2 Calculus of Variations (Optimization Calculus) Elementary Number theory Concise Introduction to Mathematical Logic It is clear to me that I am gifted. I can only imagine once I dedicate myself full time to the study of math. I am not clear what exactly to study. I am interested in Complexity Theory, Number Theory and Artificial Intelligence. I have an extremely good memory. I would say 70% photographic. I am more than happy to hold a Skype session to demonstrate my abilities. I would like to jump straight into a PHD program. why do you want to miss out the fun of being an undergraduate student? ##### Link to comment ##### Share on other sites why do you want to miss out the fun of being an undergraduate student? Please, utilize vulgar forms of speech elsewhere. ##### Link to comment ##### Share on other sites Hi, I am a senior in High school. In only 4 months I have managed to self teach (from paperback books, PDF files and online courses uploaded to various platforms such as youtube)the following during my free time: Calculus 1,2,3, Differential Equations Linear Algebra Abstract Algebra Discrete Mathematics Analysis 1,2 Calculus of Variations (Optimization Calculus) Elementary Number theory Concise Introduction to Mathematical Logic It is clear to me that I am gifted. I can only imagine once I dedicate myself full time to the study of math. I am not clear what exactly to study. I am interested in Complexity Theory, Number Theory and Artificial Intelligence. I have an extremely good memory. I would say 70% photographic. I am more than happy to hold a Skype session to demonstrate my abilities. I would like to jump straight into a PHD program. I'd also look at this from a competition perspective. A phd is an academic job. They would compare your application against others. Whilst you claim to understand it other applicants will have actually passed exams on it and applied the knowledge to project work. The second one is a big thing. One it shows if you can think outside the box and manipulate the concepts so they can be practically applied, secondly it shows you understand the material and have a taste in what these projects are like giving you a further true understanding of if you actually want to do it. I've been at Imperial NHS trust for 5 years and will be starting postgrad in physics and engineering in medicine at UCL in sept and I have seen my fair share of people who have read stuff in their spare time, think they understand and have even less of a clue actually applying it. You will not be able to match up against a graduate who has this experience. This is why young people are accepted to university early if they are gifted but not accepted straight to a phd program from high school. ##### Link to comment ##### Share on other sites I would strongly advise against trying to jump into a PhD. What is often not understood by many is that undergraduate degrees are not only about what you learn but how you are taught to think. This is a process that is very very difficult to self teach. ##### Link to comment ##### Share on other sites Please, utilize vulgar forms of speech elsewhere. I do. Now, why not answer the question? ##### Link to comment ##### Share on other sites It is clear to me that I am gifted. You've recognized your strengths, but what are your weaknesses? I think determining what you are weak in would help you to answer this question; at the very least it might humble you a bit. I knew a few students in your position when I was a senior (except they took classes at a community college at night and over the summer) who went on to pursue undergraduate degrees and should be getting them this year. Edited by andrewcellini ##### Link to comment ##### Share on other sites ...what you learn but how you are taught to think. This is a process that is very very difficult to self teach. ...think they understand and have even less of a clue actually applying it. You will not be able to match up against a graduate who has this experience. This is why young people are accepted to university early if they are gifted but not accepted straight to a phd program from high school. Klaynos, physica, your assertions that some substantial thinking skills are taught in college, along with the subject knowledge, is reasonable. What's uncanny is the idea that it can't and shouldn't without it, without considering individual cases, the hypothetical case of Dan98. I happen to think that Dan98 is not who he says he is and is trying to rile people up, maybe a sockpuppet of another recent member. But saying this is a massive dismissal of anyone who is or has found their way out of a situation like it. I think of SFN threads as repositories of information, different from encyclopedias, so I posted some suggestions that might help anyway. Maybe it would be more constructive if you were more specific on the thinking skills that'd be taught. Physica described applying the work and developing more complex projects, which is good and expected, but Klaynos you saying thinking is a very difficult process to learn is not helpful in itself, besides to attach an unfalsifiable quality to college. ##### Link to comment ##### Share on other sites Sato: An example of the "thinking" skills is the ability to think of math in terms of structures, rather than elements. One example: Someone in high school tends to think of basic multiplication as "You take these two elements, and this is what you get as the output". In other words, it's element-focused. Someone who's taken group theory and ring theory starts to think "Here are different possible ways multiplication can work". This starts to get into different rings that can be defined on the same structure. ##### Link to comment ##### Share on other sites I do. Now, why not answer the question? If you were serious then for the following reasons: 1. I do not like people 2. I do not want to waste my most productive years 3. I have no social skills 4. I believe technology is in an exponential trend and thus every year counts, for the serious mathematician/computer scientist anyways 5. I do not like people Hi Dan98, It's possible and I know of a handful of professionals who did so working today (enumerated somewhere else here on SFN), and personally had some acquaintances who did similar. It is very difficult and universities are reluctant to accept this, unless it's a regarded university where the faculty have strong pull (say, a fields medal winner at Cambridge). I would advise you find someone with your interests, ideally also in some administrative position, and apply to their university. I imagine they may be sympathetic to your situation and allow you to test out of most of the degree requirements, letting you finish within a year or two. You should also study for and take the mathematics GRE; it's, at least I know in the US, used to sift through competition in graduate school applications. Alternatively, but similar to the first point, you can apply somewhere that tries to make the undergraduate experience research focused rather than course focused. The best place I know for this is the UCSB College of Creative Studies, which lets you study independently under and advisor and get credit accordingly, without requiring sitting a course. I have a friend who goes there who seems to enjoy it; as opposed to normal UCSB students, CCS students can take any course without formal prerequisites. More to the point, they are paired with researchers and practitioners to pursue their interests accordingly. From my understanding, the Worcester Polytechnic Institute is similar to this, but a good bit more structured. If you have some more applied interests and have an idea of an actual project (with some industrial/sustainable application), even in AI or complexity theory, then you ought to take a look at the ThielFellowship. They provide two years,$50k a year in funding, and heavy mentorship and connections to help you with your project; the only requirement is that you not hold any higher degree. A few of friends were participants, some who had even pure interests in mathematics and physics.

If you have had no previous experience with proof-based mathematics, then I have no idea how you got such a good handle in only four months. Congrats if you did so, I wish you luck, but... I am a bit skeptical..

I ought to add, that there are regularly more than a handful of students who have covered most of the mathematics undergraduate curriculum in high school. Most visibly some of the IMO gold medallists and Intel STS research winners. From the handful that I've known, about all of these people do a typical four year degree. Lots end up going to schools with very strong math departments, say Harvard, and retake some courses that they've studied their freshman year, and spend the rest of their time taking graduate courses. So, you certainly aren't alone, and the usual path taken is a normal undergraduate degree.

Thank you.

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1. I do not like people

3. I have no social skills

I think that'd be a good reason to go to school and get your undergraduate degree. You're going to meet people that have interests similar to yours and be able to work together and exchange ideas, as well as talk with your professors about your ideas and get guidance and help for achieving your goals from people who work or have worked in the fields that you're interested in.

2. I do not want to waste my most productive years

What makes you think you're in your most productive years?

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I think that'd be a good reason to go to school and get your undergraduate degree. You're going to meet people that have interests similar to yours and be able to work together and exchange ideas, as well as talk with your professors about your ideas and get guidance and help for achieving your goals from people who work or have worked in the fields that you're interested in.

What makes you think you're in your most productive years?

"No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game." - G. H. Hardy.

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In the case of our department, that standard requirement for entry into the program is a minimum undergraduate or master's GPA of 3.0, a minimum GRE quantitative score of 155, and a minimum GRE analytical score of 4.

If someone does not meet those requirements, it is possible for a PI to grant an exception for a particular student, however they will not receive funding from the program for that student, and thus have to find external funds to both support them, and pay their tuition. As a result, it's a rare occurrence. Applications from qualified students usually outstrip the number of available places, so there would be no reason to take on a student that's going to potentially cost ~$40,000 a year, when you can take a less risky candidate who is supported by the program. As a PI, there's several things I look for in a potential student, and I mean this as positive criticism, but I already know from this thread that I would not consider a candidate such as yourself. 1) A functional lab group requires people who get along. A problematic individual can interfere with the productivity of the whole group and therefore I will overlook someone with good on paper credentials if it looks like they might cause interpersonal problems in the group. If you do not like people, you probably won't fit well into a research lab. 2) A PhD is a significant undertaking that represents the start of a journey, rather than the end. I generally take on students with some life experience and a clear understanding of why they wish to do a PhD, and what their medium to long term goals are. Someone just out of high school does not have that experience. 3) I generally consider students who have some research experience, and more often than not, a publication or two under their belt. This is because I have some demonstration that they know what it takes to produce research. You might find a PI willing to take you on as a graduate student with no undergraduate degree, sure. It would take a pretty compelling case for me to consider someone straight out of high school, and an exceedingly mature attitude. I have seen it happen once, however that student had published their first peer reviewed paper at 14, and had a solid publication record by the time they graduated high school. Unfortunately, it doesn't sound like you have either the experience or the attitude I would consider acceptable to consider skipping undergraduate studies. Best of luck with whichever path you choose. Edited by Arete ##### Link to comment ##### Share on other sites If you were serious then for the following reasons: 1. I do not like people 2. I do not want to waste my most productive years 3. I have no social skills 4. I believe technology is in an exponential trend and thus every year counts, for the serious mathematician/computer scientist anyways 5. I do not like people Thank you. These reasons just strengthen my resolve that you need to do an undergrad. There is more to a phd than simply doing the math. You will have to communicate with others and work with others. You will also have to teach subjects to undergrads as a phd student. Doing undergrad will get you out of your comfort zone and make you a more rounded person. I think few people on this thread look back at their high school selves and think that he had it all figured out back. Also you cannot under estimate exams. Reading and thinking you understand something is very different to aactually passing exams on it. Again we have all known people who sat exams thinking they knew the material back to front only to fail. This is why we have them. Also I hate to break it to you but on a global scale you're not that special. If you're as smart as you think you are you will be able to go to Harvard, MIT, Oxford, Cambridge. There will be plenty of smart people there who will challenge you and teach you things. Again Imperial NHS has let me work with Oxbridge grads and UCL postgrad has also let me bump shoulders with very smart people. They all say the same thing, at high school they thought they were super smart but when they got to university they were fairly average. I used to think I was super smart but seriously, some exposure in a top university really put things into perspective for me. Also you're undergrad isn't wasted, if you breeze though it use your spare time to prep for post grad, start a buisness or invent something. Really smart people do this all the time in undergrad. update: some of my response parrots Arete's must have been writing at the same time. Edited by physica ##### Link to comment ##### Share on other sites I'm a PI who supervises graduate students, and recently served on a graduate admissions committee. In the case of our department, that standard requirement for entry into the program is a minimum undergraduate or master's GPA of 3.0, a minimum GRE quantitative score of 155, and a minimum GRE analytical score of 4. If someone does not meet those requirements, it is possible for a PI to grant an exception for a particular student, however they will not receive funding from the program for that student, and thus have to find external funds to both support them, and pay their tuition. As a result, it's a rare occurrence. Applications from qualified students usually outstrip the number of available places, so there would be no reason to take on a student that's going to potentially cost ~$40,000 a year, when you can take a less risky candidate who is supported by the program.

As a PI, there's several things I look for in a potential student, and I mean this as positive criticism, but I already know from this thread that I would not consider a candidate such as yourself.

1) A functional lab group requires people who get along. A problematic individual can interfere with the productivity of the whole group and therefore I will overlook someone with good on paper credentials if it looks like they might cause interpersonal problems in the group. If you do not like people, you probably won't fit well into a research lab.

2) A PhD is a significant undertaking that represents the start of a journey, rather than the end. I generally take on students with some life experience and a clear understanding of why they wish to do a PhD, and what their medium to long term goals are. Someone just out of high school does not have that experience.

3) I generally consider students who have some research experience, and more often than not, a publication or two under their belt. This is because I have some demonstration that they know what it takes to produce research.

You might find a PI willing to take you on as a graduate student with no undergraduate degree, sure. It would take a pretty compelling case for me to consider someone straight out of high school, and an exceedingly mature attitude. I have seen it happen once, however that student had published their first peer reviewed paper at 14, and had a solid publication record by the time they graduated high school. Unfortunately, it doesn't sound like you have either the experience or the attitude I would consider acceptable to consider skipping undergraduate studies. Best of luck with whichever path you choose.

You sound like a jackass that hasn't accomplished anything significant. Good look buried in your own self pity ^^ and who told you I haven't done some research beforehand?

These reasons just strengthen my resolve that you need to do an undergrad. There is more to a phd than simply doing the math. You will have to communicate with others and work with others. You will also have to teach subjects to undergrads as a phd student. Doing undergrad will get you out of your comfort zone and make you a more rounded person. I think few people on this thread look back at their high school selves and think that he had it all figured out back. Also you cannot under estimate exams. Reading and thinking you understand something is very different to aactually passing exams on it. Again we have all known people who sat exams thinking they knew the material back to front only to fail. This is why we have them. Also I hate to break it to you but on a global scale you're not that special. If you're as smart as you think you are you will be able to go to Harvard, MIT, Oxford, Cambridge. There will be plenty of smart people there who will challenge you and teach you things. Again Imperial NHS has let me work with Oxbridge grads and UCL postgrad has also let me bump shoulders with very smart people. They all say the same thing, at high school they thought they were super smart but when they got to university they were fairly average. I used to think I was super smart but seriously, some exposure in a top university really put things into perspective for me. Also you're undergrad isn't wasted, if you breeze though it use your spare time to prep for post grad, start a buisness or invent something. Really smart people do this all the time in undergrad.

update: some of my response parrots Arete's must have been writing at the same time.

On a global scale there are plenty folk as smart as me and smarter. But guess what? Not you

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Please, utilize vulgar forms of speech elsewhere.

You sound like a jackass that hasn't accomplished anything significant. Good look buried in your own self pity ^^ and who told you I haven't done some research beforehand?

On a global scale there are plenty folk as smart as me and smarter. But guess what? Not you

Sigh.

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Well, short as it is that post clearly demonstrates significant lack in a surprising wide range of areas. I will highlight one thing though. Arete has provided important insight into some aspects of a graduate degree and how it relates to succeed in that endeavor. Considering that being able to derive and utilize information is but a very basic skill (even disregarding the need for social skills for now) it casts severe doubts in the ability of the poster to even succeed in undergrad.

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Well, short as it is that post clearly demonstrates significant lack in a surprising wide range of areas. I will highlight one thing though. Arete has provided important insight into some aspects of a graduate degree and how it relates to succeed in that endeavor. Considering that being able to derive and utilize information is but a very basic skill (even disregarding the need for social skills for now) it casts severe doubts in the ability of the poster to even succeed in undergrad.

It is called aspergers. As far as succeeding in undergrad, not sure if you read but I am trying to skip it altogether.

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Unfortunately, it doesn't sound like you have either the experience or the attitude I would consider acceptable to consider skipping undergraduate studies.

You sound like a jackass that hasn't accomplished anything significant. Good look buried in your own self pity ^^ and who told you I haven't done some research beforehand?

It's now clear you do indeed have the wrong attitude, even if you do have the requisite experience, which is doubtful. Sorry.

Edited by Arete
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I will say this: the skills that Arete is highlighting aren't as relevant for a math Ph.D. program. Unlike many science Ph.D. programs, you don't get a "laboratory" setup. Math is still a collaborative effort, but much more can be done individually, and so the ability to "get along" is reduced. It is still necessary - much of the best work is done in small groups. Most of the papers I read are written by 2 or 3 people.

Let's try a problem that I'd expect a junior in college with experience in linear algebra and abstract algebra to be able to do.

Let [latex]\mathbb{F}_p[/LATEx] denote the field with p elements; let n be a positive integer, and let [LATEX]\mathbb{F}_p^n[/LATEX] denote the n-dimensional vector space over [LATEX]\mathbb{F}_p[/LATEX]. How many bases are there for [LATEX]\mathbb{F}_p^n[/LATEX]?

Edited by uncool
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!

Moderator Note

Dan98,

The people responding to you in this thread are giving you well thought out advice that stems for their own experience in the matter. Is this not what you came here for? Calling someone a jackass for helping you is absolutely inappropriate and will not be tolerated.

With respect, your attitude here and your stubborn determination to skip undergrad without considering the benefits of not doing so shows that you are not ready for a PhD. I commend you on learning as much as you claim to in so short a time, but please understand that undergrad is not just about memorising content, and not doing one puts you at somewhat of a disadvantage. At the very least, you would do well to take on the words of other members here with more grace and humility than you are currently demonstrating.
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!

Moderator Note

Dan98,

The people responding to you in this thread are giving you well thought out advice that stems for their own experience in the matter. Is this not what you came here for? Calling someone a jackass for helping you is absolutely inappropriate and will not be tolerated.

With respect, your attitude here and your stubborn determination to skip undergrad without considering the benefits of not doing so shows that you are not ready for a PhD. I commend you on learning as much as you claim to in so short a time, but please understand that undergrad is not just about memorising content, and not doing one puts you at somewhat of a disadvantage. At the very least, you would do well to take on the words of other members here with more grace and humility than you are currently demonstrating.

A hundred years from now no one will remember you

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