What should I do to get into a PhD in Mathematics in Europe without a thesis in my BSc and MSc? I'm going to apply for a PhD in Mathematics in Europe with my BSc and MSc in Mathematics and Applied Mathematics (respectively), from two top-tier (say, top 10-20) US universities with advanced courses, and good GPA. Neither of my BSc and MSc had a thesis component though, and a capstone project wasn't available to me. 

I do have some research experience, but my research mostly involved computational programming and data analysis, not serious Mathematics. More importantly perhaps, none of my research experiences were independent research, meaning that I was working on a project defined by my supervisor and my work was pretty much directed by my supervisor. 

I'm aware that the norm in Europe is that students write a substantial thesis for their Master's degree, and possibly for their undergrad too, and then they'll apply for PhD. I did want to do a 2-year Master with a substantial research project, but since funding was an issue, I went with the 1-year MSc program that would fund me, and that one didn't have any thesis.

Given this situation, I'm looking for a way to fill that gap in my resume to make myself competitive for a top-tier PhD in Mathematics (or Applied Mathematics) in Europe. 

First - do you think I'd have any chance without doing some sort of independent research before applying to a PhD? 

Second - do you have any suggestion on how I can fill that gap?! I thought of doing an MPhil or an MRes in the UK, but financing is an issue for those programs (I'll apply for scholarships, but chances are high that I won't get one). Is there any sort of program where I can just spend some time doing a self-directed, but supervised, research in Math and get paid for that?

Edit: As for the specific countries in Europe I'm looking for, my top choices are Germany, Switzerland, UK, and France, but I'm open to other options if I find a good opportunity.
First - do you think I'd have any chance without doing some sort of independent research before applying to a PhD? 


Possibly. However, it would likely involve you finding some "angle" on the basis of which you can strike up a relationship with a prospective supervisor, who could promote/sponsor your Ph.D. candidacy.

But there's a deeper issue here:

Why do you want to enter a Ph.D. program?

Hopefully, you have some subject you are interested in studying. If that is not the case, and your motivation is to simply get the title and the piece of paper - then you should seriously reconsider. Working on a PhD without focused motivation is not just exceedingly difficult, but also exasperating. You may well find yourself in the middle of it, two or three years from now, grabbing your head by your hands and asking "What was I thinking? Why in blazes am I even doing this?" - and not having a good answer.

So, do consider some independent research, or even going into industry doing something you are interested in, but with a mind to sketch out relevant mathematical challenges to study more deeply in academia. These are my suggestions for filling the gap.


  my top choices are Germany, Switzerland, UK, and France


Once you find something relevant to research, locate relevant research groups in that field (in those countries or elsewhere), and talk to them.