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I was wondering how come modern mathematicians do not seem to discover as many theorems as the older mathematicians seem to have done. Have we reached some kind of saturation limit where all commonly needed mathematics has already been discovered or is it just that the standard common textbooks do not get updated with the newly discovered theorems ? Are mathematicians of our generation only left with research topics in super specialized sub-fields ? I am wondering what made you choose your research field, what's so beautiful about it ? I understand that this may be a little personal question and i do not mean to intrude on your privacy. I guess if you could just shed light on the field you are familiar with and why u would or would n't recommend me to conduct research in it then i'd be great full to you. This would help me make an informed decision about picking a field.

PS:I know "commonly needed mathematics" is subjective to interpretation but i was thinking of defining that as anything one is taught in an undergraduate level.

Edit: As per the request my educational level is i have a bachelor's degree in computer science, a graduate diploma in mathematics. I am currently a honours student and would be starting a PHD next year. I have taken mostly non-rigrous undergrad level math courses, mostly because that's all the uni was offering at the time. These courses were on financial maths, Dynamics, ODEs, Mathematical modelling with multiple ODEs, linear algebra, basic Probability, Statistcial modelling, statistical inference, vector calculus, Time series. Out of rigrous fields i have only self studied basic abstract algebra and some basic mathematical analysis. I guess i am in mathematical infancy and being made to choose which seems very scary.

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    On the contrary, I think modern mathematicians discover WAY MORE theorems than classical mathematicians. The difference is that is usually takes an expert to understand a modern theorem, whereas classical theorems are more approachable without heavy machinery.2012-03-12
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    Speaking of 'informed decisions': it would be immensely useful to know what your current educational level is and what you're hoping to go towards; the answers will be vastly different depending on what can be assumed...2012-03-12
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    Ask yourself which course do you find most interesting? That will give you some directions at least.2012-03-13
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    I do not intend to hijack this thread, and Hardy, I apologize if I do, but I have an uncontrollable interest for number theory. However, I have the feeling that this field is so "pure" that it will be seldom useful outside professions dealing with cryptography and the likes. In other words, I am afraid that I will not be able to put bread on the table. Would anyone shed some light on this matter for me?2012-03-13
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    @JosuéMolina, You are most welcome to ask. Contrary to yoru worries i think if u can devise an algorithm to generate large primes easily u might be able to make enough money for an early retirement.2012-03-13
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    @Hardy: In my opinion, the best way you can figure out what you like and what you want to study is to attend as many seminars in your new department as you can. If it seems like a rather large time investment, that's because it is, but seeing the nitty gritty of a particular subject is the only way you'll be able to tell whether you love a subject or not. If you are lost after 10-15 minutes in a seminar, it is not considered rude (at least in my experience) to zone out and do your own work. I should point out that it is also helpful to look into prospective faculty mentors' research areas.2012-08-08

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