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I wonder what you all think are some questions that our current understanding of math can't really answer but which, in the reasonably near future (Let's call this "before you retire"), we will be able to answer? This is a deliberately broad question. You can speculate on particular conjectures, on developing interrelations between disciplines, on methods, applications or anything you want. Try to describe a little bit what sort of program you see leading up to these advances.

The only area I can really attest to is that of complex systems (large networks of interacting agents, in general)). Right now, this isn't really a mathematical discipline. Non-linear dynamics and applied probability sometimes touch on questions of complex systems, but right now nearly all study in the field is of a holistic and decidedly non-rigorous sort. NOTE: If you're an applied probabilist, don't think I'm forgetting about you - it's just that there's generally a huge gulf between our ability to simulate these systems very approximately and our ability to define their behavior rigorously.

I believe that, within the next 20 years, we'll see a modest revolution in this field. We'll be able to bring in tools from the fields of probability and statistical mechanics and create new tools such that complex systems will allow us to understand certain collective human networks (genetic networks, contagion, traffic, urban growth are probably some of the most achievable) on the same level that we understand systems of particles today (or even that we understood systems of particles 50 years ago). In its early stages, this probably still won't be rigorous, we'll rely on hybrid techniques from machine learning, stochastic processes and domain-specific techniques to deal with each sort of problem. By the end of a couple more decades, however, we'll have made leaps forward in our ability to rigorously describe, understand and predict complex systems.

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closed as not constructive by Harry Gindi, Andy Putman, Qiaochu Yuan, Charles Siegel, Robin Chapman May 12 '10 at 6:17

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

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It's not very nice to have such a question from an anonymous author. You ask others to open their thoughts but hide yourself. –  Wadim Zudilin May 11 '10 at 23:48
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This question doesn't really have any kind of reasonable answer. It's just an invitation to speculate! I'm thus voting to close. –  Andy Putman May 12 '10 at 0:03
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I don't see it as being unfair in any way. I mean, I'm not asking for secret fantasies or social security numbers, just speculation (wild speculation is good too), not necessarily with justification, of what you see/hope for the future of mathematics. –  DoubleJay May 12 '10 at 0:04
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This question is way too open-ended and speculative for my taste –  Deane Yang May 12 '10 at 0:33
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I think it is interesting to reflect on where mathematics might be heading, and perhaps where it should be heading. Furthermore, there are many more interesting and diverse voices participating on MO who might make such reflections interesting and informative than there would be in a corresponding discussion on someone's blog. (I say this just because an unspecified blog is the usual place that people suggest such questions should be moved to.) –  Emerton May 12 '10 at 3:55

1 Answer 1

What we all think are some questions that our current understanding of math can't really answer but which, in the reasonably near future, we will be able to answer? Many MO highvoted questions which remain unaswered should fit your criteria. An example is this question.

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This answer should perhaps be a comment to the original question ? –  ogerard May 12 '10 at 5:55
    
Could be, but I used a different comment. –  Wadim Zudilin May 12 '10 at 7:00

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