Question for those doing or having done a PhD in Mathematics



I'm just done with my master degree in mathematics, and I'm currently searching for a PhD.

When I'm looking on the Internet at the general advices for doing a doctorate (so not particularly for mathematics), it is always mentioned that I should prepare a topic for my PhD, in order to present it to my potential supervisor.

But althought I got my master degree with honors, I still feel like that I don't know enough in mathematic in order to propose a topic making sense.

Furthermore, the few friends I have who are currently writing a PhD, said to me that they candidated without having a clear topic, and that they discussed it with the supervisor only after having been accepted.

I wanted to know how it was for you? Did you already had a topic when you candidated?

Thank you in advance!