Hi all. I am an international student from Asia, and I completed my undergraduate degree from a small liberal arts school in the US, not at all known for its math department and with no serious / famous active researchers on the faculty. I applied for the PhD programs this past fall (2023) mainly for number theory / combinatorics.
I have previous research experience (one paper under review at a good journal and another on the way), high grades, a high math GRE (90th percentile), departmental awards, and strong recommendations. I was however unable to gain admission to any T30 school in the USA (using US news rankings) -- the ranking is not very relevant, but more importantly I will not be able to work with my dream supervisors. Furthermore, I do have admission to a few schools in the 30s, and while the departments are quite nice, I am rather unsatisfied and feel certain I can do very well at better institutions.
What was most lacking on my application I think were graduate-level courses, of which I only had 3 in the form of independent studies, given that my liberal arts school only offers undergraduate-level courses. Also lacking I think were recommendations from professors with any name recognition. I am thus considering first pursuing a Masters in Europe before reapplying to PhD programs. The main programs I am considering are Bonn, Cambridge Part III, and the Oxford M.Sc. in Mathematical Sciences.
What thoughts do people have on this plan? Should I just not bother with the Masters, suck it up, and go to the PhD programs I have admission in? If not, does anyone have any recommendations between the 3 Masters programs I mentioned above?
Thanks, and I would really appreciate any advice.