I am about to finish a PhD in mathematics with a thesis in an area in between differential geometry and PDE. I still have more than half a year, but I want to start thinking about my future. In the last years there were moments where I was feeling quite depressed and frustrated, but overall I have to say that I have enjoyed my time as grad student. I love learning new things and being challenged by some nice problems.
At the moment I don't feel like pursuing a post doc in my field. First of all, I am scared of getting stuck in a sort of "dead end" (it could happen that after doing several postdocs around the world I will not find any permanent position and at that point I will maybe be too old to do something else..). Secondly and most importantly, I don't know if I like math enough. It's a nice and interesting activity, but I'm not sure if I want to dedicate the rest of my life on that and I feel that I don't have enough motivation for fighting for a job in academia.
Working on some topics related with the ecological and climate crisis that we are facing, would probably motivate me more than just dealing with theoretical problems in differential geometry.
My questions: How common is to switch field after a PhD? Could I get a job in some research institution (possibly in Europe) where I could do research maybe on ecology or climate change?