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I graduated with a bachelor in math/physics with a cgpa of 2.74/4.00 but didn't study many courses of abstract mathematics mostly computational math . I applied for a master's in the same Uni. and about to graduate. I had a little exposure to abstract math (some real analysis, measure theory and functional analysis) during the pre-master courses (seven courses) I scored 3.8/4 as a cgpa

I want to shift towards abstract mathematics. How can I make the shift? I'm trying to apply for good schools abroad in the UK and the US. I'm trying to apply for a master's in abstract mathematics but some of the courses in the programs I'm applying for requires a pre-requisite that I don't have so how can I do it right?

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    Is your country US? Are those grades good or bad? Are they relevant for the question? Did you already apply somewhere? Did you ask those universities if you can somehow make up for the prerequisites? – user111388 May 6 at 8:35
  • No I'm not in the US. 2.74 is equivalent to good and my masters cgpa is equivalent to excellent. Yes, I think they are relevant since I am a math student, I just want to shift from one branch to another. No I haven't applied yet but I'm asking to learn if others can give me an idea. Thanks – A witch May 6 at 8:47
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For application to a US doctoral program in pure math, I think you would be well situated, other things being equal. This would be especially the case if your chosen field is Analysis or something related. However, the path might take a bit longer than if you had studied a broader spectrum of math.

The program you would be entering would have two essential requirements (perhaps a few minor ones). The first is to pass a set of comprehensive examinations. These are intended to assure that you have breadth of knowledge. The second is to write an acceptable dissertation. This is to show you have the ability and insight to delve deeply into a narrow aspect of math. There might be a few other options at a few places, but this is the standard.

One comprehensive exam will probably be in abstract algebra or some sub field. Another might be in Topology. But there will likely be at least three such exams, and one or more of them might be orals. In fact, there might be two levels at some places: written exams and orals.

But a US doctoral program also normally comes with a fair amount of advanced coursework. This is intended primarily to get you through the qualifying exams. Many US students starting a doctoral program won't have a masters, so the coursework is intended to give them the necessary breadth.

You may be ready for the Analysis qualifying exams, but would probably need the coursework for any others. It would take some time. If you are ready for that, it might be worth applying.

Also note that most math doctoral students in US work as TAs. This provides funding for them and assistance in teaching undergraduates. This adds to the time required to earn the degree, of course.

Note also, that in some cases the pure and applied math departments are separate. In others, they are joined in a single department. It might be a consideration in your case.

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From what I have seen in the past, it is not impossible and happens in a good number of cases that students switch from Pure to Applied Maths and vice-versa, as in your situation. At this time of the year, it may be a bit late to apply for Masters in UK as a good part of the offers have already been sent. Nonetheless, it would be good to apply to hone your writing skills and get a grasp of the application process at the universities you are interested.

Regarding your shift to Pure Maths, it may be good to contact the academic staff in charge of the programme you are applying to. In this way you can find out exactly what they are looking for and what "concessions" they could make regarding people missing 1-2 prerequisites that could have actually been informally completed through other courses you'd done.

Additionally, your personal statement should clearly include the motivation behind your switch and back it up by courses you have taken within your credit allowance (I presume the Masters you are about to complete forces you to take applied courses rather than pure). In light of this, it may be wise to talk your switch also to your prospective referees for applications. It could give them a heads-up and might benefit your applications by having more tailored letters of recommendation.

As a side note, the UK and US offer great Mathematics programmes, but these are not the only options. As an EU student, I would be also looking at very good and reputable Maths schools from Germany and France, just to name a few (a lot of the programmes are taught in English). If your long term goal is to enter graduate school, it may be good to check possible supervisors/research groups you'd like to join and check the backgrounds of the PhD students. This would give a glimpse of the knowledge and preparations graduate schools are expecting from applicants. By doing this you may discover that some students (particularly US and some in UK) jump from undergraduate to PhD studies and this could also be your case.

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