Prerequisites for Master's Degree in Pure Mathematics (and eventually PhD)

Question

This will be my first post on the Academia SE so apologies if there are some conventions I break!

I am an undergraduate student of mathematics at a university in the UK whose emphasis lies a lot more in statistics and applied mathematics than it does in pure mathematics, there being only 2 modules in the entire degree program that focus only partially on higher algebra, and only 4 weeks of the entire 3 year program that have anything to do with number theory. I am in this position because my A-Levels did not go as well as I had expected as I had a few issues and did not achieve the grades needed to go to a more "pure based" university. I am deeply interested in the area of Algebraic Number Theory and intend to go on to study a master's degree in pure mathematics and eventually/hopefully a PhD.

My question is this; what should I/can I do to "boost" my application to a master's degree in pure mathematics if my undergraduate degree has a relatively low pure mathematics content?

I will be applying for an undergraduate research bursary from the London Mathematical Society to fund an 8 week research project, be it expository or otherwise, in the field of Algebraic Number Theory.

Thank you in advance for any information you can give me.

Answer 1

Although I am doing a PhD in the field of Computer Science, I of course cannot give specific advise on what you should do about mathematics MSc, but here are some of my observations on the general case.

In the last two years, I have applied a lot of PhD programmes across Europe. I have been accepted to one recently. And that is probably one out of fifty if not more.

As far as I have observed, the priority is not a student with "a lot of knowledge", but a student "who can study hard on the given task." This observation is based on the profiles of accepted students instead of me (stalker alert). Some of them have lower GPA, and almost none of them have publications. I think what they somehow proved that they can study harder and be more productive.

The interesting part that matches with your question is, both their theses and their stated area of interests were less related to the research area. Surprisingly, someone who took his degree from mathematics was accepted to a research group that focuses primarily on application rather than theory.

If we switch from PhD to master's degree, I don't really believe having a relative background should have more weight than being a hard-worker (of course I am not talking about another discipline like psychology) in the process of decision.

If I were you, I would direct the exactly same question to a professor from the department you want to apply. Questions like

How can I convince you that I am good enough to pursue a master's degree?

or

Are there any specific criteria you are seeking such as having a relatively strong background on pure mathematics?

would do no harm. I think being deeply interested as a master applicant in a research topic puts you one or two steps further. Of course you somehow have to show that you are actually deeply interested (this is the hard part that I could not figure out how).

In short, do not over-try to "boost" your application and do not hesitate to contact one of the faculty members.

Answer 2

For Algebraic Number Theory you can try reading up the topics of group theory, ring theory, field theory, Galois theory. Perhaps write up some notes using LaTeX and solve some of the exercises in books. Showing the notes/exercises to the professors should be quite impressive.