If I remember correctly, the Tripos is a pretty full on masters course and you need to have really stellar grades. I also thought you had to take an entrance examination but the website says you don't.
The other question that I forget to ask is what kind of speciality of pure mathematics? You are going to have different schools with different strengths. For example some schools will be strong in algebra, where another will be strong in analysis or topology. Then you have all the things like functional analysis, the modern geometry field including all the studies of exotic geometries (the kind you'll see in a string theory paper or textbook) amongst other things.
Are you thinking about the possibility of doing a PhD (I ask this because you are studying pure math which lends itself to a PhD by the nature of the field)? If so, you will want to try and get into a department that has a strength in the particular area of interest.
Also the other question is where do you live at the moment? Do you like in the UK or are you somewhere else?
My situation is as follows: I am currently doing a one-year masters in physics and I plan on doing a PhD in physics, but I would first like to do a one year master in math to cover some general areas (partially because I love math for its own sake, and partially because I plan to work on the mathematical side of physics, like for example Penrose, in which case a firm knowledge of pure math is handy). Courses I am thinking of are differential geometry, algebraic geometry, algebraic topology, differential topology, Lie algebras, homological algebras, (non)commutative algebras, category theory; this list is not at all fixed but it should give a sense of what I am looking for. Tripos III seems exactly what I am looking for, since it consists purely out of courses which you can pick freely out of a wide range and the program lasts only one year (taking two years out for math is too much for my taste). I also have good grades and I already got accepted into Tripos III last year (but dove into a physics master instead) so it seems realistic that I can get in this time too. The two downsides is on one hand the cost, but this is not much of a downside: I can get the funds and if it is worth it, it is worth it; more importantly I hear the exams are mainly parrot tests (do correct me if I was misinformed on this one, although I have had it confirmed by multiple sources), which is a big put-off in my view.
I am originally from Western (continental) Europe, now I am one year in Canada, and I am open to any country for the one year master (although I suppose the PhD consideration doesn't really come into play since I am not going to do a PhD in math, but rather in physics). Countries I have for example already looked into are the UK, Germany and Sweden, but I can't seem to find something.
Well you have mentioned quite a number of areas and not just say two or three. For example if you were considering mainly Non-Commutative Geometry I would have suggested trying to take a course by someone like say Alain Connes.
I am not a pure mathematician though, but I imagine that if you wanted to become a mathematical physicist then a place like Cambridge would be pretty good.
In terms of the US, Princeton is another good place for math but again I don't really know anything about the particular departments or other specifics for the areas mentioned above.
I thought though that the Masters program in Mathematics was a couple of years due to them requiring you to basically become a genuine "master" in the field, but it seems like they have a "light" version of this.
I'm also wondering if you can narrow down some of the students or people that have worked with Grothendieck for category theory.
You can obviously read the books of guys like this and the only reason I am suggesting these kinds of names is that people that develop mathematics often have an intuition that surpasses the people that learn from an external source.
If you ever get the chance to work with a group of people that develop something, I'd recommend taking it.
Someone was commenting in an earlier thread that Ed Witten did this with some mathematicians where he went to see them and got a lot of knowledge from the creators themselves.
Obviously you wont be able to do this straight away, but the process of doing this should be one that I think you should have in the back of your mind considering the scope of your endeavor and how deep this goal is.
The only other question I have for you is where you want to do your PhD in physics? Answering this question will probably help you with questions about your Masters as well. The schools, the departments, the supervisors, the stuff they work on, their students, their collaborators, where they are currently teaching, where they got their PhD, and so on.
Since I can't give you an specific advice, the best I can do is to ask questions that get you thinking about the possible answers, and hopefully I have done that in some small way.
I'm resurrecting this thread, as I'm in pretty much the same position as Mr. Vodka. Does anyone know of any good one-year math Master's programs, aside from Cambridge's Part III?
I'm interested in most of the same math areas as is Mr. Vodka, and also especially probability theory. I'm also put off by the idea of learning material in order to parrot it on a test: I'd like to learn it to understand it.