Can a Modern Mathematician Achieve Universality in Their Field?

[Functor97]

Gauss, Poincare, Euler, Hilbert, Riemman, Cantor, Godel all stand out as universal mathematicians. I define universal as either contributing to a vast number of fields of mathematics in quite distinct areas or someone who has changed the entire face of mathematics as it stands. That list is by no means complete, but those names do stand out.

Is it possible for a modern mathematician to accomplish similar feats? Is it possible to understand vast regions of modern mathematics, from geometric analysis, through algebra, number theory and fundamental set theory, and what's more contribute to them? Or have times changed? Should a mathematician specialise in an area, and remain in that area after say their Phd?

 

[pmsrw3]

Actually, there was a time when one could be not only a universal mathematician, but a universal scientist. I would say no, it's no longer possible. There is just so much more to know now, and our lives haven't gotten that much longer.

 

[micromass]

It's certainly possible to specialize in more than one area. But making meaningful contributions to all areas of mathematics?? No, I don't think that's possible. It's not even possible to know everything about your own specialization...

 

[Functor97]

The thing is, i can't decide on which area i find the most beautiful. I love differential geometry, number theory, topology and analysis, there are just so many areas!
How did you guys choose?

 

[micromass]

The thing is, i can't decide on which area i find the most beautiful. I love differential geometry, number theory, topology and analysis, there are just so many areas!
How did you guys choose?

What year are you in?? What courses did you already took in the subjects?

The thing is that you don't have free choice in your subjects. That means that you'll need to find a professor who has an open place for you and who wants to work with you. These places are hard to find in fields like topology. You'll need to find a math subject that is popular right now.

I tried to do research in topology, and I had even found the professor to work with. But in the end, it turned out that there was no money available, so I couldn't work in topology. I had to pick a quite different field to work in. Picking a field that is popular is a consideration that you must make.

A good thing is to talk to professors and tell them what you're interested in (and why), they might propose something.
You have an interest in phycis too, so doing something with differential geometry in physics isn't too farfetched, for example.

Also, you said you liked all these fields, but I remind you that research involves years and years of study in one particular field. Are you ready for that?? I, for example, like number theory too, but I don't like it enough to spend years and years researching it. Take as many courses as you can and see what you like most, then you can choose.

 

[Functor97]

What year are you in?? What courses did you already took in the subjects?

The thing is that you don't have free choice in your subjects. That means that you'll need to find a professor who has an open place for you and who wants to work with you. These places are hard to find in fields like topology. You'll need to find a math subject that is popular right now.

I tried to do research in topology, and I had even found the professor to work with. But in the end, it turned out that there was no money available, so I couldn't work in topology. I had to pick a quite different field to work in. Picking a field that is popular is a consideration that you must make.

A good thing is to talk to professors and tell them what you're interested in (and why), they might propose something.
You have an interest in phycis too, so doing something with differential geometry in physics isn't too farfetched, for example.

Also, you said you liked all these fields, but I remind you that research involves years and years of study in one particular field. Are you ready for that?? I, for example, like number theory too, but I don't like it enough to spend years and years researching it. Take as many courses as you can and see what you like most, then you can choose.

I had heard that pure mathematics was not as subject to popular trends as say theoretical physics. I had heard it was much more self directed and open to personal interests.
I would probably prefer some field of geometry or topology, but i find number theory so beautiful, and i would love to work on fields in relation to the riemann hypothesis. I understand this is not very realistic, but i can't help wishing i could. Alain Connes for instance works in non commutative geometry, but also seems well known for analysing works in number theory.

 

[micromass]

I had heard that pure mathematics was not as subject to popular trends as say theoretical physics. I had heard it was much more self directed and open to personal interests.

Oh, it certainly is not as subject to popular trends than physics. But trends are there nonetheless. If you study some obscure field, then it'll be hard for you to find funding and to find journals to publish in. This is the case of topology: it is much more difficult for topologists to publish than it was before.

Once you found a place, you have a lot of freedom. But you'll need to find a place first!

I would probably prefer some field of geometry or topology, but i find number theory so beautiful, and i would love to work on fields in relation to the riemann hypothesis. I understand this is not very realistic, but i can't help wishing i could.

Have you considered algebraic or arithmetic geometry? This combines the world of number theory with the world of geometry. It's a very popular research field too (but quite difficult too).

Also, there is a difference between "I find this beautiful and I love to work on the Riemann hypothesis" and "enjoying the actual research". Take classes on the subjects first, and then see if you like it enough. The more classes you take, the more you'll know what suits you!

Feel free to send me a Private Message if you want to discuss things or if you want to know more!

 

[micromass]

Alain Connes for instance works in non commutative geometry, but also seems well known for analysing works in number theory.

Alain Connes is a genius!
Note that in a PhD, you'll have to focus on a small specialization. So you probably won't have to opportunity to do ground-breaking stuff in topology and differential geometry and number theory. You'll have to choose.
After your PhD, you have much more freedom, and it's not unheard of to start working in a totally different field.

 

[Hurkyl]

The thing is, i can't decide on which area i find the most beautiful. I love differential geometry, number theory, topology and analysis, there are just so many areas!
How did you guys choose?

A lot of what I have studied was because I was curious about non-standard analysis. I spent some time learning enough formal logic to see how it worked. I had the notion of a universe of mathematics constructed out of the natural numbers, and you could replace the natural numbers with the non-standard natural numbers to get the non-standard universe. This prompted an algebraic notion of set theory. My interests thus turned towards category theory, and eventually my quest was effectively achieved when I came across topos theory. Of course, by then, I had almost forgotten why I had started this line of study.

Addendum: this is by no means the only thing I study. It's just something I did study with a reason to do so.

 

[Functor97]

A lot of what I have studied was because I was curious about non-standard analysis. I spent some time learning enough formal logic to see how it worked. I had the notion of a universe of mathematics constructed out of the natural numbers, and you could replace the natural numbers with the non-standard natural numbers to get the non-standard universe. This prompted an algebraic notion of set theory. My interests thus turned towards category theory, and eventually my quest was effectively achieved when I came across topos theory. Of course, by then, I had almost forgotten why I had started this line of study.

Ah, don't remind me of another area of interest! Logic sounds very interesting as it is the field upon which the other fields of mathematics are built.

Another mathematician who comes to mind as a bit of a universailit is michael atiyah, but most of his fields have been geometric.

 

[Anonymous217]

Like you, I'm having trouble finding a particular concentration in Math because I'm interested in so many different fields. However, I'm sure that as I take more and more advanced courses, that list of fields will slowly dwindle, and by the time I'm applying to grad school, I'll at the very least know what general field to apply in.

So, for me, I guess it's a matter of patience. Take whatever math seems most interesting and keep advancing until you find a particular focus. It is a shame, though, that specific things will ultimately sway your choice: professors in the course, the text, the material covered, the courses taken, what is currently being offered, the skill of your university at each field, research positions, etc.

 

[SteveL27]

There are examples of mathematicians becoming prominent in one field, then changing to a radically different field. A famous example is Paul Cohen. In the 60's he worked in analysis. One day he decided to teach himself mathematical logic and take a shot at the Continuum Hypothesis. He ended up proving the independence of CH and also the Axiom of Choice (AC); and invented the technique of forcing, which has allowed set theorists ever since then to cook up models to order of various axiom systems.

So it's always possible that you could specialize in something for years, and then specialize in something else. But in the academic world, I think you are required to specialize. It's too bad they don't value wide and extensive knowledge as much as narrow and specialized knowledge, but that seems to be the system.

 

[Anonymous217]

^ As far as changing fields, here's a relevant link that's pretty useful: http://mathoverflow.net/questions/69937/changing-field-of-study-post-phd

The general idea is to specialize in a particular field, and produce some results. Then after getting tenure, try extending your field by incorporating other subjects into your research. So it's basically a slow transition from field to interdisciplinary to field. I'd assume some people just skip the intermediate step though.