pazuzu

Of course it isn't impossible to get into a top program, people do it every year. However you are underestimating how good the applicants are that are accepted. These programs are incredibly competitive and often self-selective. Don't think you will have much luck getting in by playing a numbers game. All accepted students have taken the basic and advanced courses, have good GRE scores, and have demonstrated research ability. There may be a 5% acceptance rate (I don't know the exact number), but I know people with much stronger backgrounds than you who were rejected from tier 1 schools. Your transcript looks strange because you have to know the pre-reqs and you have no proof that you so. Let me speak as an adcom member for a second. Math is built on foundations. For example not having linear algebra, but doing well in courses which use, it implies an ad hoc knowledge of the subject. Your record does not show you have mastered linear algebra. Secondly, not having the calc sequence means you might understand some calculus, but I cannot trust (pay) you to teach a calc one class to a bunch of undergrads. Further, not knowing your school, if they let you take classes without prerequisites those classes must not go very deep into the advanced topics. This last paragraph is meant to give you an idea of what the adcoms are going to say. This isn't personal, I am to trying to attack you or belittle your knowledge. I can, however, say in absolute terms that your lack of the basic classes- even if you have shown skills in the more advanced topics- will be a big problem. So you know here is what my school says are basic requirements, and this is a fairly standard statement on most math grad programs. Applicants for admission to either PhD program are expected to have preparation comparable to the undergraduate major at MY SCHOOL in Mathematics or in Applied Mathematics. These majors consist of 2 full years of lower-division work (covering calculus, linear algebra, differential equations, and multivariable calculus), followed by 8 one-semester courses including real analysis, complex analysis, abstract algebra, and linear algebra. These eight courses may include some mathematically based courses in other departments, e.g., physics, engineering, computer science, or economics. You have covered some of the advanced topics which is good. However, I don't know in other fields, but in mathematics it is difficult to substitute the foundation classes. Not having the calc sequence is going to be a deal breaker especially if you do not have research experience to beef up that part of the application. There are very few non-math majors that apply to do pure math. We have too much background required. It will This is the most important point: adcoms in top schools (really any school) DO NOT HAVE to forgive you for performing poorly in undergrad because there are many other applicants who do not have the problems you do. Does that mean everyone with a blemish on their record is rejected? Of course not, but it makes it much harder to find acceptance and darn near impossible to get into a top university. Recent failures are even worse because it shows you haven't changed your habits. Why should a grad school give you resources for 5+ years if you haven't shown them that you can excel? Remember they don't owe you a graduate education, they will take the candidate with the lowest risk. Let me say again, you are severely underestimating the quality of your competition at the top schools. Honestly unless you are of similar ability, probably not. You really should explicitly email some of the programs you were rejected from and ask what they wanted to see. That information would be illuminating for us all. Maybe they are just being nice, or they may say we would love to have you if you filled in x,y,z gaps. You wont know unless you ask. The reason I keep suggesting a post-bac is to get the basics out of the way- or at least take the basics somewhere. There are some masters which let you take some undergrad classes, but these are for filling in gaps at the 400 level (think algebra/analysis) absolutely not for 100 level calc classes. I know how frustrating this process can be, and I wish you success, but you need to be realistic about your background and how you will appear to an adcom.

I was speaking in absolutes, undergraduates can certainly solve open problems. My issue is that you want to solve a problem that will guarantee you admission to a tier 1 school, but this presents a catch-22. If you are as good as Galois, you are going to have done enough to get into a tier 1 school. If you aren't as good as Galois you aren't going to solve a singular problem which will allow you to get in anywhere you want. This goes into the following... There is no magic way to get into graduate school. You are highly highly unlikely to find and solve an "instant admission" problem. You are looking for a problem which professional mathematicians know about and cannot solve, but an undergraduate comes along and solves. I am not going to argue that such a situation is impossible- it isn't- but such a situation is incredibly rare. If you do solve an open problem this will be part of your application material. It will make you look better, but almost all applicants in tier 1 programs have solved some open problem. Graduate programs at all levels want you to demonstrate what your research potential is. This can be done in many ways (REU, directed study, or even publishing something), but a tier 1 (and even any PhD) program is going to expect some record of research. It sounds like you are looking for a quick and easy fix. Mathematics does not move forward this way, save for a few brilliant and rare minds. This whole situation feels like a student who is failing going into the final exam and asking if they get an A on the final if they can pass the course. If you are failing the class you very likely aren't going to get an A on the final, and if you could get an A on the final there is no reason for you to be failing the class. Sure, but what did you research statement say? You don't need to know exactly what you want to do, but you should have (and communicate) a much better idea than this. Your math background is lacking. Any school which funds students by TA positions is going to expect you to have completed the calculus sequence. I have no idea how your school let you do, for example, complex analysis before calculus or multivariate calculus, but that will be something that needs to be explained to any program. A bare minimum of the basics is going to include calc 1-3, linear algebra, and differential equations. This is why your transcript is so strange, how did you do topology, measure theory, and differential geometry without calculus? How did you do topology without linear algebra? I guarantee you every admission committee is going to be highly skeptical of such a transcript. For reference, for my funded masters program I applied with 18 courses (6 basic courses like calculus, and 12 upper level courses) in mathematics from undergraduate school ( a tiny liberal arts school in the midwest). I did have to study at a math program abroad to pick up so many math courses. This may be a bit high, but is not uncommon in top programs. Hopefully you can understand these concerns, but as the poster above me said, your current application package is not good enough to go to a top program, and you would have a hard time at some of the low ranked PhD programs. If you can explain the strange lack of basic math courses, and can demonstrate research potential you may be able to gain admission into a masters program. In terms of funding though, it is cutthroat. There are a lot of people with more classes, more research, and better grades looking for the same positions you are. If you want to be a mathematician that is great, and you can get there. However, the path you want to take will very likely not work for you. I would really recommend post-bac or trying for a masters.

I should also say, my observations are assuming that your application had all of the issues from your initial post. If that is incorrect, let us know.

I am a fourth-year PhD student at a tier 1 mathematics program. My advisor one of the faculty members that reviews graduate applications. I am happy to give some insight, but I do have a few questions for you. It depends on your coursework and current grades. Some post-bac programs are partially funded and can help get you into a better position for applying to graduate school. Some other options (though expensive) could be to spend a year some place like Budapest Semesters in Mathematics. You could definitely get more classes and experience, but it wouldn't be free. I am not currently on the job market, but I have had many friends apply this year. The academic job market for mathematicians isn't great, but you don't need to go to a top school to find a position. Likely you will be doing a (even up to three ) post-docs, but there are jobs out there. Tier 1 schools may make this job easier, but you need to have done meaningful research wherever you go to find a position. So, why do you want to be a professional mathematician? What other positions with a mathematics PhD would be of interest to you (always good to know your options)? Failing several classes even in non-math is a huge red flag- especially so late in your degree. It says something about your ability to complete coursework. There really isn't a way to hide this from your application. The best bet would be to take another year of classes and absolutely rock them all. Remember, the top programs have really good applicants (of course other programs do too!). If you have poor grades and are otherwise equal to the other applicants you are not getting in. You need to be very strong in the other parts of your application to make up the difference. Realize that most tier 1 acceptances go to people with stellar understanding of mathematics and demonstrated research experience- any major flaw will be very difficult to overcome. Not getting rejected out of hand is a good thing, but you would have to ask if you were waitlisted. Have you contacted the schools that rejected you and asked them what you can do to improve your application in the future? You should mention what you study in your free time. They will believe it if it is true and you can demonstrate the knowledge you have gained from self-study. I mentioned relevant papers that I read (not related to my undergrad coursework) in my application. Showing the application committee that you are self-motivated is great, and will really hone your research statement because you will be reading enough to know what you want to work on. However, the next statement really worries me, you should be well past the calculus sequence if you are applying for a PhD (if you are in your last year now- which I am assuming). Can you provide a list of the math classes have you taken? This would really help everyone understand your current standing. You will not get into a math PhD program without skills in at least abstract algebra and real analysis. A tier 1 PhD program will expect much much more. Also, if you do not have a terrific foundation in writing proofs, you will not be able to pass the coursework- let alone do research. It doesn't hurt. You still must meet some minimum standards, but bringing in a full fellowship would be a great boon to your application. You definitely can try for outside funding. Be aware though, the outside funding sources will have the same concerns that the application committee may have with your application. I think so, this is the route I went. I earned my masters (funded) at a good tier 2 school before moving on to my PhD. I would not have gotten into a tier 1 without the research experience my master's thesis afforded me. Math research skills take a long time to develop (I'm still working on it myself!). I definitely was not ready for a PhD right out of undergrad, though others may certainly have been. You mean solve a grand problem in spite of your application to gain admission? This really really really doesn't happen. Math doesn't work as a start-from-nothing-and-get-a-big-result process. Lots of undergrads have some results from REUs or directed studies. These show you have the potential to be a researcher, but even with this experience they are unlikely to be solving massive open problems. I can come back to this question when you have a chance to post your course work, but if I had a big important unsolved problem that I thought could be done by an undergrad, I'd be working on it. The real question is what are you interested in/good at? What kind of math can you stare at for five years and still love (most of the time)? My suggestion would be to read some research papers in fields you are interested in and see what mathematical research looks like- you definitely need to have some idea of what you are getting into. To be quite blunt (and to repeat others above) it seems like you do not have the background necessary for a PhD in math. Let's look at what you have done, but it sounds like a post-bac may be the best place to start if you want to pursue a PhD. I really would also be curious to know why you want a PhD in math. I don't mean this to sound belittling, I am genuinely interested.