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.