Where is mathematics at now?

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Like what are the big conjectures or problems that if proven will yield great results in mathematics? Where is topology and abstract algebra now? Is analysis finished? Is linear algebra finished? Is 2d geometry finished? At the frontier of mathematics, what are the most important questions that mathematicians ask themselves?
 

chiro

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Like what are the big conjectures or problems that if proven will yield great results in mathematics? Where is topology and abstract algebra now? Is analysis finished? Is linear algebra finished? Is 2d geometry finished? At the frontier of mathematics, what are the most important questions that mathematicians ask themselves?
You'd be surprised how many problems there are in mathematics.

Think about the problem in information theory that relates to Kolmogorov Complexity: find a computer program that is the minimal representation that represents some data or process.

Any serious advances in this field would revolutionize this area of mathematics. It would give us a way to characterize a pattern for things that do not have any immediate pattern. It would allow us to develop ideas relating to pseudo-randomness and randomness in ways that we can not currently contemplate.

Also you should be aware that John Wheeler had shifted his public opinion that physics is about information towards his death and other physicists have adopted the same kind of viewpoint.

I see mathematics developing for a very long time in the future to a point where it becomes as important as our normal sensory perceptions. If it develops to a point where we can turn processes that look like junk into a minimalistic program, then this kind of understanding will allow us to see and understand things in a way that our normal sensory perceptions don't let us do (or at least in ways or amounts of time).

That is just one area! Think of all the different applications we have now and imagine what it will be like in ten, one hundred, or even a thousand years. It's pretty exciting!
 
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Mathematics is everywhere! Mathematics has branched off into so many fields and subfields, Each has many open problems. Because of this, any reasonably intelligent and motivated person can make substantial contributions to the mathematical field.

The older problems, however, get more attention. That is why a solution of the Navier-Stokes problem would raise more eyebrows than a solution of some obscure problem in computational geometry.
 
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The big question in topology now seems to me to be the smooth Poincare conjecture in dimension 4. If I'm not mistaken, all the other Poincare conjectures have been solved.

Jacob Lurie appears to have solved one of the big problems in my field, topological quantum field theory (Baez-Dolan hypothesis), so now it will be interesting to see what can be done with it. Will it help us understand quantum gravity, as Baez had hoped?

I think it's okay for some people to work on math for math's sake, but I think that there needs to be a little more work done to try to bridge the gap between pure math and applications. That's what interests me the most, although I am fairly "pure math" in my training and way of thinking. Can we put some of the theories to practical use? And not just for the sake of using math, but because it actually helps?

Here's an interview that appeals to me with my rare combination of topology and electrical engineering background:

http://www.johndcook.com/blog/2010/09/13/applied-topology-and-dante-an-interview-with-robert-ghrist/
 

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