Second master and phd in math after phd in mechanical engineering I'm currently finishing a PhD in mechanical engineering at a top 20 european university. However, during my PhD I realized that woking or even conducting research as a mechanical engineer is not something I'd like to do. My primary interests are in maths (partial differential equations) and theoretical physics (quantum mechanics, special and general relativity). These are topics that I've always found interesting and wanted to learn. 

I want to get a master in math because what I'm missing is the mathematics needed to understand the theoretical physics problems. Moreover, I've always enjoyed (the small amount of) math and always wished to do more. I used a (probably not worth mentioning) small amount of applied math in my PhD, which is about numerical simulations using the finite element method in a self written code. Other than that I have no mathematical preparation but I have a master in mech. eng. with a good final grade.

I would like to restudy but starting over with a bachelor degree seems a bit too much and it would probably be too expensive. My questions are:

1) Can I get admitted into a masters program in mathematics at a good european University (preferably in Germany or France since these countries have very low tuition fees). It would be ok (as most probably needed) if I had to catch up by tacking additional bachelor courses, no matter how many.

2) What are my chances of being accepted for a PhD in mathematics if the master goes well?

3) How negative will it be for an academic career to have a second master and (possibly second PhD) degree?
My story is somewhat similar to yours. I did statistics, then switched to math as I found I love it. Here is my pathway in France. I didn't restudy the whole bachelor. I applied for (and got accepted to) the 3rd year undergraduate at a French uni. After that, I applied for (and got accepted easily to) the 1st year of master (aka M1) in math at a top French uni (as I got good marks). If my mark is good enough, I can apply for the 2nd year of master (aka M2) (in France, 1st year and 2nd year of master are seperated, you have to apply for both seperately).

So, for your questions:


I would highly recommend you to retake the 3rd year undergraduate (aka Licence 3) in math, and it should be sufficient for you to go further. As you have a technical background (and even better, at PhD level), it's not difficult for you to get admitted directly. The chance would be lower if you apply directly for the master in math because the math in France is highly theoritical, they would prefer someone with a math degree for the master degree in math. Of course, if you can show that you have done sufficient advanced math (especially topology, integration, measure theory, which are a must if you want to do PDE), then no problem. However, due to French-style math, I still recommend you to start from the 3rd year of bachelor rather than 1st year of master.


2) If your mark is good, you can apply for many masters at top uni (like Paris 6, Paris 11, which are strong in applied math, including PDE) or "Grandes Ecoles". Also your chance to be admitted to a PhD in math should be good as well. Applied math is, at least easier to be admitted than pure math.

3) Even though you have two degrees, they are quite closely related. So I don't see any problems with that. Actually I think you even have some advantages to others.