Is it possible to do research in several branches of mathematics? I'm finishing my undergraduate studies and I've been interested in several branches of math to research in the future, to say: functional analysis, differential equations, dynamical systems, differential and riemannian geometry, (general, differential and algebraic) topology, algebraic geometry, theory of groups and rings, etc. 

My question is if is it possible to learn about all those things and to do research using all? or are there imncompatible areas between them? Because I have seen that almost all of the researchers of my faculty are centered to research in a very specific topic and that makes me believe that is not possible to learn all of the math that i want.

Thanks in advance for your answers!
It is definitely possible to do research in several branches (if you are good enough that is). But the people do so generally took decades to get to where they are. If your plan is to start in all of them at once, you are doomed to fail. 

For the sake of the argument, let me arbitrarily distinguish different levels of knowledge in a topic. 


Textbook knowledge (what you learn in classes; definitions, big theorems, general ideas)
Working knowledge (given a new problem (not a textbook problem, those are designed to be solved), knowing what could be done to solve it and the ability to do so)
State of the art knowledge (what are the current problems in a topic, what are other people doing, what are new ideas, etc.)


Since you just finished undergrad, 1. will likely be the highest level you are at in any topic. Getting there is by far the easy part. It does not seem so in the beginning, but learning a new topic to that level gets easier the more you know.  Level 2 is the minimal level needed to do actual research if you have someone to guide you. Getting you there in a single topic will take you the next few years. Level 3 is what you need to be an independent researcher. Even assuming that you are a dedicated, above average student, getting you there in a single topic will take you your whole Phd, if not a bit longer. The last level is also one that requires upkeep. Expect to spend several hours each week just to keep up with what other people are doing in a single topic.

Now the simple truth about academic careers is that you can't be a PhD student or post-doc forever. You'll need some sort of permanent position to survive, which requires you to be an independent researcher, which requires state of the art knowledge. This is why you see so many people specializing on a single topic, to get to that level and to stay there. 

I don't think that is the only way, but note that you'll need some focus. If you try to work in all topics on your list at once, you will never get anywhere and likely not even finish your PhD before you have to drop out. But if you first focus to get one to the highest level, you might still at some point find that you finally have enough free time to look at some of the others. From there things can develop naturally.

Personally I like to read textbooks on different topics in my free time. I even have some collaborations with other people on topics where I have working knowledge but wouldn't consider myself to be at the state of the art. But the reason that they work with me is that I can contribute ideas from the topic I spent most of my time in and where I am at the state of the art.