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Future of mathematics

  1. Jun 29, 2005 #1

    mathwonk

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    future direction of mathematics

    In regard to the question of where math is going and what is the important recent work and most important work needed, I am of course not qualified to say, not having a global enough grasp of what is being done. that being said, i still have an opinion.


    for a more qualified view, see the lecture from 1900 by hilbert at

    http://aleph0.clarku.edu/~djoyce/hilbert/problems.html

    and the commemorative one attempting to keep things going in 2000 at UCLA.

    I will make the obvious remark that since the "riemann hypothesis" occurs as problem 8 on hilbert's list, and remnains unsoved today, it is still very important. one may also consult the list of Clay institute problems, including this and the so called "hodge conjecture", for other questions that at least could be worth a lot of money to the solver.

    In my own opinion, one of the most important phenomena has been the reconnecting of mathematics wihth physics, with the consequent energizing of both subjects. Witten and the people doing quantum gravity have made big advances possible in enumerative algebraic geometry at least.

    string theory also benefits in reverse from knowledge about riemann surfaces. this interaction between physics and analysis was key to riemann's confidence in the correctness of the results he ahd insufficient proofs for. Physical insight still stands today as as confidence booster for results that the mathematicians only achieve a foundation for sometimes much later.

    I would also say the work unleashed by wiles, and the outstanding problems generalizing his results on modularity of representations is important. i am even less qualified here.

    my own work is in a specialized area begun by riemann, of understanding the relationships between curves and abelian varieties, and their moduli spaces. as such it keeps me interested in the developments surrounding moduli spaces coming from quantum phenomena however, e.g. cohomology of moduli spaces, as well as singularity theory especially non isolated singularities, and vanishing cycles. Higgs bundles in higher dimensions also seem to play a role in problems of interest to me but are hard to compute with effectively when one gets away from curves.

    anyone else?
     
    Last edited: Jun 29, 2005
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  3. Jun 29, 2005 #2

    mathwonk

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    c'mon hurkyl, matt, whats on your mind. i would appreciate it and im sure many others would, enjoy it too.

    or anyone. we all know we are not hilbert here. perhaps i was a little cowardly, imitating others, but i said what i thought, within my limitations.
     
  4. Jun 29, 2005 #3
    Whatever happend to that guy at Purdue who claimed to solve Reimann?
     
  5. Jun 29, 2005 #4

    mathwonk

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    well this is a fun one. this guy has a checkered history. he was denied grant money they say for a while as a questionable researcher. then he busted a big problem (bieberbach conjecture"), and got famous, so all bets are off on him. could be a bust, could be a great breakthrough. you might try reading it. or we could make it a project here!

    he seems to be after the cash, so in my opinion that is a minus. i.e. it pushes one to rush things. but he ahs suceeded before???!! so what do you think? what fun!


    "
    webpage from purdue:
    WEST LAFAYETTE, Ind. – A Purdue University mathematician claims to have proven the Riemann hypothesis, often dubbed the greatest unsolved problem in mathematics.

    Louis De Branges de Bourcia, or de Branges (de BRONZH) as he prefers to be called, has posted a 124-page paper detailing his attempt at a proof on his university Web page. While mathematicians ordinarily announce their work at formal conferences or in scientific journals, the spirited competition to prove the hypothesis – which carries a $1 million prize for whoever accomplishes it first – has encouraged de Branges to announce his work as soon as it was completed.

    "I invite other mathematicians to examine my efforts," said de Branges, who is the Edward C. Elliott Distinguished Professor of Mathematics in Purdue's School of Science. "While I will eventually submit my proof for formal publication, due to the circumstances I felt it necessary to post the work on the Internet immediately."
     
  6. Jun 29, 2005 #5

    mathwonk

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    well i will look stupid if he is right, but he sounded like a nutcase to me, and here is the fruit of websearch; (even if he is extremely bright): (no wonder peopel questioned him before he proved the bieb. conj)

    "Louis de Branges attempts to clarify his "proof" of the Riemann Hypothesis

    "...explains the mathematical motivation for his Riemann Hypothesis proof and reveals that he proved the Bieberbach conjecture so that he could get funding to work on the Riemann Hypothesis."

    In June 2004, Louis de Branges announced another proof of the Riemann Hypothesis. "However", cautions Eric Weisstein's Mathworld, "both the 23-page preprint cited in the release (which is actually from 2003) and a longer preprint from 2004 on de Branges's home page seem to lack an actual proof. Furthermore, a counterexample to de Branges's approach due to Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing."
     
  7. Jun 29, 2005 #6

    mathwonk

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    i think it quite wonderful however that apparent nutcases occasionally prove big theorems, just to keep us modest and honest. i.e it ain't the source of the argument, it is just the argument itself.
     
    Last edited: Jun 29, 2005
  8. Jun 30, 2005 #7
    "Famous mathematician" - talk about an oxymoron!
     
  9. Jun 30, 2005 #8

    mathwonk

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    what!!? i thought we were like rock stars!

    (no wonder there were no groupies, i thought they just got lost.)
     
  10. Jun 30, 2005 #9
    to add to your insights, mathwonk, even in the new approach (well not so new) to tackle the riemann zeta function zeros hypothesis is concerned with physics, specifically the pattern of the points which are conjuctred to lie on one line they try to give them a spectral interpratation.
    and they are more numerous ways by physicists, but im not sure about their rigour.
    here is good and insightful page on number theory and physics:
    http://www.maths.ex.ac.uk/~mwatkins/zeta/physics.htm

    ....for any extra online reading.
     
  11. Jun 30, 2005 #10
  12. Jun 30, 2005 #11

    matt grime

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  13. Jun 30, 2005 #12
    there was a transcription of an address that steve smale gave somewhere in the mathematical intelligencer. it had ~20 open problems in all different areas of math. hold on 1 sec i'll go find it.... it's called 'mathematical problems for the next century' in vol 20 #2. VI arnold wrote on behalf of the international mathematical union to a bunch of mathematicians asking them to provide their best open problems, & that list became smale's lecture.
     
  14. Jun 30, 2005 #13

    mathwonk

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    a google search on "mathematical problems for the next century" bnrought it up first.

    www6.cityu.edu.hk/ma/people/smale/pap104.pdf

    the lecture looks excellent.
     
  15. Jun 30, 2005 #14
    yeah it looks much better than the clay math inst list. there's a lot more stuff on there. i think the following two problems should also be included on any "to-do" list in in the 21st century:
    -- whether there is a separable, infinite-dimensional banach space on which every linear operator has an invariant subspace (aka the invariant-subspace problem)
    -- whether every finite group H occurs as the Galois group of a finite Galois extension of the field of rational numbers. shafarevich proved this for solvable H & others (like hilbert) proved other special cases but the general problem is still unsolved
     
  16. Jun 30, 2005 #15

    mathwonk

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    those do sound good. (i enjoyed including the proof of the abelian case of galois groups in my beginning grad alg course when i taught it, to shpw the power of the decomposition theorem for finite abelian groups, and also dirchlets theorem of course from number theory.)

    by the way shafarevich himself found an error in his proof for the solvable case, although he suggested a fix. the case of nilpotent groups seems agreed to be true though, by his method, since serre has written it up.
     
  17. Jun 30, 2005 #16

    matt grime

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    the origins of much mathematics is in physics. the coming decades may see mathematics, after a period of development in isolation, start to help the physicists. some of the links between lie algebras and topological groups can even be used to talk about magnetic monopoles (work of, amongst others, paul martin on schur weyl duality and hecke algebras. named because he's given a seminar i was in)

    it will be interesting to see where the langlands program gets to as well
     
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