Choosing a PhD topic My supervisor and I recently had a long chat about PhD related stuff. He said something to the extent that your chances of employment after finishing your PhD among other factors depends on the topic of your PhD. The reason he mentioned was that for certain fields there aren't that many open positions. He mentioned that there are many open positions in the fields of Algebraic Geometry and also in Number Theory. 
(I will enumerate the questions to make it easier to answer)
The following are my questions: 

(1) Now I am wondering are there any other fields where research is
  comparably active? Concretely, what are the top topic in pure maths
  apart from AG and number theory that are "in"?

(answer below does not address this question)

(2) And in particular: are differential geometry and differential topology
  among them?

(answer below does not address this question either) 

(3) And which are the fields with the least open positions for lecturers?

(answer below does not address this question either)
I am passionate about more than one field hence I would like to choose the optimal topic from among the topics I am passionate about. I am prepared to move to just about anywhere. To make answers more precise and useful for others too maybe you could include geographical information in your answer. 
 A: If I have any advice for pursuing your Ph.D., choose something about which you are passionate and will maintain that passion over a period of 3-5 years.  If being employed after you finish is the main concern, keep this in mind: your success in your work will depend on your ability to learn new things, to finish small projects, to keep your verbal and written commitments, and to communicate with your colleagues.  
I really feel that, while your field of expertise is important, you will be best served by working at something that your truly enjoy and will succeed at doing.  I have seen too many Ph.D. prospects flounder because such prospects treated their Ph.D. as a professional program .  Such people largely end up miserable because a Ph.D. takes too much time and dedication to treat as a hurdle in the way of your career development.  Rather, it is an opportunity to pursue something about which you simply want to do for its own sake, because you love doing it. If you are lucky enough to find the right combination of research opportunity and compatible advisor, then the career success will come by itself.
That all said, there is certainly a variation in opportunities over subfields.  I am not a professional mathematician and therefore do not have a feel for which positions currently have the best/worst opportunities for lecturing.  As far as which subfields in pure math are the "hottest," keep in mind that some of the heat is driven by non-academic considerations.  Many mathematicians I know who are not in academia pursue computational geometry (believe it or not, but there are quite a few opportunities in semiconductors and electronics) and number theory (network security).  Lecturing opportunities in these fields are likely the best because of the employment demand (which goes against what I said above, sadly).  I am in the United States, so there may be different opportunities in other countries.
It sounds like you are interested in differential geometry and topology.  Your advisor is correct: there are few positions in academia.  But as I said above, if you focus on what you really want to do in graduate school and publish great work and make a name for yourself, you will find that, if you have geographical flexibility, you will find good work post-graduation.
A: My advisor told me, twice, before accepting me (although he said that it is just to keep him with a clear conscience) that it is very hard to find a position with a Ph.D. in set theory today, and he remarked that my interest in the extremely pathological properties of models without the axiom of choice would make it even harder.
He added that if you're really good then you won't have an actual problem, but it's difficult to be that good. I am aware of the problems set theorists have in finding a position, especially in Israel nowadays where set theory is in decline.
You need to consider two things, I believe:


*

*Are you willing to move to another country? Or do you want to go back to your home land, or even home town? Perhaps to your current university? If you are willing to uproot it might be easier to find positions in your preferred topic.

*How good do you think you can write about a topic you are less passionate about? I am currently in the midst of choosing the exact questions I will work on in my dissertation, but the process began with my advisor and me compiling a list of five possible topics, and a sixth one which I told him I may be interested in working on (as a topic that did not come up in the conversation). 
Regardless to the fact that all the topics we discussed about are very interesting to me, I kept drifting back to my own idea, and after a long period of two months where I was trying to fight it I gave in and decided that I really have to pursue the things I want to pursue, or else I will give up in the middle of the work.
Of course, understanding all the things I want to understand I am bound to work on the other topics we had originally discussed, but my aim is solve another problem, and that's fine.
A: I will not tell you the answer to any of your questions, since I do not feel comfortable drawing conclusions in broad strokes from the data available to me. But I'll show you some data, and tell you how to find some more, so you can make up your mind for yourself. 
The American Mathematical Society produces an Annual Survey of mathematical sciences, and among it contains information on the fields of studies of new doctoral recipients and their hiring statistics. If you go to this page you will find "Supplemental Table E.3" showing last year's employment statistics of new PhDs by field of specialty. From there you see that 


*

*Applied maths is much more employed over all, with specifically biostatistics/statistics leading the field, this undoubtedly having to do with its industry applications. 

*Of the traditional pure maths, the most popular is grouped under "Algebra/number theory", and in second place is "geometry/topology" (good for you!). 

*You also see that pretty much across the board for the pure fields, around 10% of the students were still seeking employment at the time of the survey. 


(This last point actually brings up something interesting: while your advisor is right that certain fields are more popular than others in terms of having more research interest and more jobs available, be aware that this also means that those fields usually have more people competing for those jobs. On average you are much better off trying to find a subject you enjoy and are good at, instead of finding a subject that has more absolute number of jobs. This is sort of the standard advice that you would get everywhere.)
(I am also slightly surprised to see that the statistics have not improved since the economic bubble of 2008; for the more applied fields it seems the situation has slightly improved, though this is drawing on just 4 data points.) 

In addition to the AMS data, an imperfect proxy for research interest is the number of arXiv postings per period of time. Of course you will need to adjust by typical length of paper, and other field-dependent cultural aspects to make the numbers really meaningful. 
For employment availability, however, a simple way is to browse through the job postings at MathJobs. I do not know if there is a way to quickly filter by fields and such, but at worst you just have to read through every single job posting and categorize them yourself. 
