Ph.D. Applications went wrong this year
I am currently a master's student in pure mathematics at a top university in the UK. This year I applied for PhDs across Europe and the US (10 applications), and got rejected from all of them.
I topped in my undergraduate university, which is not very well known, and had good recommendation letters. I think that I was immature in my applications (especially in my statement of purpose), and this is the main reason why they didn't go so well.
I am still interested in doing a Ph.D. in pure math, and so I would like to reapply again next year. However, I am unsure what to do in order to improve my applications. I expect that finishing my Master's with good grades will be good, but I can't just wait a whole year without doing anything.
Tips for reapplying + Filling a gap year
Do you think that doing another Master's will help me improve my applications? My current Master's is just one year, so I think that another Master's will help me get closer to where I need to be in terms of math to start a Ph.D. Plus, it will also give me time to mature a little bit more, and help me fill in this gap year.
However, this feels like a very big commitment. But still, I don't see many more options to continue studying pure math next year otherwise, as there isn't really any possibility of doing research assistantships and the like in pure math. Do you have any other suggestions for how I can fill this gap year?