Are Mathematics graduate programs open to admitting a student out of high school with graduate work? 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.
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.