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. –  you Mar 12 '12 at 19:36
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... –  Steven Stadnicki Mar 12 '12 at 20:46
Ask yourself which course do you find most interesting? That will give you some directions at least. –  FiniteA Mar 13 '12 at 2:18
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? –  Josué Mar 13 '12 at 3:18
@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. –  JavaMan Aug 8 '12 at 20:06

The truth is that not everyone gets to actually pick their desired research field and many graduate phd students often end up writing their dissertation in an area or specialization of mathematics that they did not plan. This is usually due to the fact that one must find an advisor that one can work with and s/he may want you to work on something that entails doing a lot of mathematics which may be foreign to you.

After I passed my qualifying exams, I talked to several analysts in our department and I choose my research field, Functional Analysis, out of my interest to learn more analysis and because I enjoyed learning mathematics from my advisor.

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If you're not sure which field to study, why not go with the one you enjoy the most, or that feels the most natural to you?

I sort of naturally became interested in Algebraic Topology for two reasons. First, because as a field it appeals to the way I think: abstractly and with pictures. Second, because I needed something to answer a question I had, and Algebraic Topology was that "thing." It started in highschool, before I even knew the word "Topology" existed.

By last year of high school I had already taken the good grade 12 math courses, so I was bored and started trying to think up my own math problems. Many of them didn't really make sense (I realize in retrospect), but there is one in particular that really stuck:
I knew about this thing called the "Möbius strip," which is this one-sided surface with only one edge. I had also heard about this "Klein bottle" thing, which was a one-sided surface with no edge, but if you cut out a disk it has the same surface/edge properties as the Möbius strip. I wondered if you could "continuously transform" the Möbius strip into the Klein bottle, and began drawing sequences of pictures in an attempt to find such a transformation.

As you can imagine (or deduce) I soon became stuck, and moved on to something else. A few months later I arrived at University and decided to ask someone who actually knew what they were doing. His response was "Hmm, good question! Maybe you can find the answer in here:" and lent me Massey's "A Basic Course in Algebraic Topology." After many hours deciphering the contents of Chapter 1, I concluded a few facts:

1) If you attach a disk to the boundary of the Möbius strip, you get the projective plane (mindblow). The Klein bottle is the "connected sum" of two projective planes (mindblow).

2) The projective plane and Klein bottle are compact, connected 2-dimensional manifolds without boundary, and are both non-orientable.

3) You can assign a number (the "Euler characteristic" $\chi$) to compact connected 2-manifold w.o. boundary in such a way that $\chi(M\#N)=\chi(M)+\chi(N)-2$ for any such $M$ and $N$, and if $M$ and $N$ are both orientable (or both non-orientable) then $\chi(M)=\chi(N)$ iff they are homeomorphic. (mindblow)

4) The Klein bottle and projective plane have different numbers: $\chi(P^2)=1$, but $\chi(K)=2\chi(P^2)-2=0$

When I was finally able to confidently say "Therefore these objects I defined cannot be homeomorphic" I felt the power and mystery of Algebraic Topology (as cheesy as that sounds). Although my understanding of the material and of mathematics in general was still quite primitive, I understood one underlying principle: Alg.Top. takes seemingly intractable geometric problems and reduces them to relatively tame algebraic problems, sometimes as simple as comparing two numbers.

Today my current research topic is exotic smooth structures on manifolds, and even here the Algebraic side plays a huge role. In the simplest case of a sphere $S^n$ of dimension $n$, the set of smooth structures (a "smooth" thing) is very closely related to the "stable homotopy group" $\Pi_n$ (a "topology+algebra" thing) via the so called "J-homomorphism" (a "smooth+algebra" thing). The setting is MUCH ore complicated, but the principle remains the same: find a way to express geometric concepts algebraically, where things are more straightforward.

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Thank you that was very interesting to read. Algebraic Topology does indeed sound like a very cool field. I would try read up more about this field for sure. –  Hardy Mar 12 '12 at 22:54

I would definitely not consider myself a mathematician, but I admit, when I sign up for classes, I want to learn about my professor and TA's before class starts. I'm not sure why but I'm just curious as to what their research interests are (not that I would understand it anyway). Of all my math classes and econ as well, I have never seen one field or research topic show up twice. It seems like there's so many different options out there. I plan on self studying after my undergraduate and really digging into a certain subject. I personally am looking into studying up on Brownian Motion, Stochastic Calculus, Monte Carlo Methods, and other financial mathematics, mostly because of my high interest in finance. It's like a win-win to me, Math $+$ Finance $=$ Awesomeness (to me of course). Take a good look into the other subfields of subfields. Maybe there is some really cool specialty field of probability theory that not many people know about. But again, I'm just an undergrad math major who won't even be going to graduate school. I've just noticed that there's a ton of different options out there. Good luck in your choosing, and I envy your position in going to grad school.

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Of course by the time you see this (or even if you see this) I'm sure you will have made your decision. I thought this was an interesting question though I've always wondered myself. –  TheHopefulActuary Dec 12 '12 at 5:16
+1 from me, why do u not want to go to grad school ? I think as part of my doctorate i might undertake research on Fractional Brownian motion. –  Hardy Feb 19 '13 at 20:01
@Hardy It's not because I don't want to, it's because I have not had a strong undergraduate GPA and I have been told that the GPA I have probably wouldn't get me into a program. So my plan is really to study a lot on my own after undergrad and become an actuary. Which in the process actually involves the Brownian Motion, Stochastic Calculus, Probability, and financial mathematics methods. –  TheHopefulActuary Feb 19 '13 at 21:55
I wish you all the best Mate. –  Hardy Feb 20 '13 at 17:35

I began graduate school wanting to be an analyst. My first year I did the standard introductory algebra, analysis, and topology sequences. In the second semester, the algebra course did a lot of module theory and homological/categorical things, and I was hooked. I work mainly in module theory because I find it interesting seeing what does and doesn't generalize from vector spaces and Abelian groups, and what I can learn about rings from modules. I enjoy what I do. I'm not at a research university so I don't have to worry if my work isn't fashionable or high-powered. I do it for me, in my own time. I didn't get in the business to get rich, or famous, but to pursue what I consider some beautiful ideas, and maybe make a contribution here or there. It's been a good ride so far.

I think mathematics is always establishing new results. In my experience, the basic graduate texts provide an introduction to the various fields. For the most part, you won't find in them the advanced results you need to do serious research. That's where advanced texts, papers, etc. become very valuable.

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I really like this answer. –  t.b. Mar 12 '12 at 21:13
I wish i could switch my research life with u. :-) –  Hardy Mar 12 '12 at 22:35

I agree with Martin's assessment. I seems that most of the "low hanging fruit" has been picked. It is rare to come across a recent result that is both accessible and important (in areas like "Euclidean geometry", "highschool algebra", or "freshmen calculus" etc.). This isn't too surprising given that the areas which are presented to the population as a whole (i.e. non-math majors) have been studied for hundreds of years and thus contain mostly very old results.

Counter-examples to the idea "everything useful is old": Consider fields like numerical analysis and algorithmics (i.e. fields related to computation). Many algorithms and numerical techniques are decades not centuries old. Another good counter-example is statistics. Many statistical methods are relatively "new".

There is certainly another component of inertia. It does take a long time for new results to filter on down from research level mathematics to grad school to undergrad etc. For example, it's taken a couple hundred years for calculus to find its way into most (US) highschools. I imagine over time we'll see more and more statistics show up secondary education.

Is the current generation somehow doomed to studying obscure subfields whose results have limited reach? No. It's only a matter of time some new field emerges with immediate astounding applications. I can't tell you what that might be. I imagine such a field will appear quickly without much warning.

To answer your other (rather) different question: "How did I choose my subfield and what is beautiful about it?" My subfield is the representation theory of infinite dimensional Lie algebras and vertex operator algebras. Like many other mathematicians I think many random factors played into my choice. I very nearly went into algebraic topology, but while I was trying to decide on topology or algebra, the adviser I had in mind for topology took on another student. Thinking he would be too busy to work with me, I decided on Lie algebras. Although I was already drawn to this area. Lie theory is filled with beautiful connections to all sorts of different areas of math. A Lie group has a group structure and a manifold structure, so in studying Lie groups you get to play with group theory and manifold theory (so essentially modern algebra, differential equations, analysis, geometry -- all rolled up into one area). There also was something very alluring about Lie algebras. They're non-associative and at first quite mysterious. Then after getting to know them better -- they're everywhere! I also loved the connection between infinite dimensional Lie algebras and string theory. This leads to vertex operator algebras which then lead back into sporadic groups and all other manner of interesting (seemly disconnected) mathematics. So I guess in the end it was the connections (often quite surprising connections) to other areas of mathematics which drew me to this area.

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wow, yoru field sounds like a lot of fun. Thank you so much for your answer. it was very insightfull. –  Hardy Mar 12 '12 at 22:14
+1 Some of the most important results in statistics are only around a century old, like the word of Fisher, Haldane and Wright. And numerical analysis has undergone a rebirth in the last 50 years. –  joshin4colours Mar 13 '12 at 16:35

I had a course in Probability Theory, which I thought was very challenging. Then I had a course in Graph Theory which I loved. In the end I had a course in Machine Learning, which I thought was very useful, and this all led me to Bayesian Inference in Complex Networks.

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I always put it this way, which is probably not 100% accurate but gives a meaningful picture:

All math you see in highschool and the first two undergrad years is more than 300 years old, with few exceptions (elementary linear algebra and elementary group theory are more like 150 years old, say). The notation is most often more modern than that of the original mathematicians who made the discoveries, though.

Most math you see at the advanced undergraduate and basic graduate level (say Master's) is from 100 to 50 years old.

Mathematics knowledge is incremental, and so most often to learn each new subject a decent knowledge of previous stuff is required (together with maturity). So, after 12 years of school and 6 of university, one is supposedly ready to start learning what mathematicians are doing nowadays (or in the last few decades, say): this is what you do when you start a Ph.D.

And, as was mentioned in the comment above, there is way more math produced these days than ever before. I would say that I'm fairly sure that more math was created and developed since 1900, say, than altogether before.

To get a glimpse of today's amount of productivity, I would suggest a look at the Arxiv, where much (but not all by any means) of new math is currently posted.

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Firstly thank you very much for your insight. I have one more question to ask you i realize this may be a little personal and if you wish to not answer it i would n't insist. Why did u choose operator Algebra ? To be honest i am not even sure what that is. As a newbie researcher i need to find a field and originally i was going to go for probability theory though i am confused if i am making the right choice. There seems to be so many sub fields how can i make an optimal decision without knowing all the fields. –  Hardy Mar 12 '12 at 20:13
I don't mind answering, but that doesn't mean I can give a very good answer... It kind of happened on its own. At some point you have to move towards your M.Sc. thesis, and you need a supervisor; same with the Ph.D. That mostly settles into a broad area. As for what Operator Algebras is, it is not very easy to explain if you don't have enough background (i.e. beyond Functional Analysis). When you do "linear algebra" in infinite dimension, topological considerations arise, and it becomes a (very nice) mixture of analysis and algebra. –  Martin Argerami Mar 12 '12 at 20:36
Thank you very much, i think i seem destined to follow a similar trajectory, although i am worried if there is something i could do to validate my choice before passing the event horizon. –  Hardy Mar 12 '12 at 20:44
In general, when you get immersed in one area, you basically end up knowing little to nothing about any other area! So, after a Ph.D., very very few people do a major area switch. Minor displacements are common, though, and in the long term some time they involve a major change. –  Martin Argerami Mar 12 '12 at 23:18