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Tomorrow's hot topic

  1. May 16, 2006 #1

    J77

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    What do you think the hot topics in maths will be in the future?

    I can only speak from an applied point of view, but perhaps someone else could come in from the pure angle...

    In the 90s, I think the hot topic was ''Chaos Theory'' ( http://en.wikipedia.org/wiki/Chaos_theory ) - even banded around by economists.

    Over recent years, there has been lots attention for networks, from neurons through to the national grid - inlcuding the idea of "Small world networks" ( http://en.wikipedia.org/wiki/Small-world_network )

    Throughout recent years, I also see a lot of biomaths going on but haven't really seen it lead anywhere yet - there always seems to be funding around but the gulf between the biologists extremely large systems and the mathmos simplification seems to be there still (same could be said for chamical reactions).

    Perhaps one field that will take off could be that of animal locomotion. How do animals move? Why do we find it so easy to balance? What are the control mechanisms? All leading to the developements of advanced robotics... ( http://en.wikipedia.org/wiki/Animal_locomotion )
     
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  3. May 16, 2006 #2

    matt grime

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    Here are some possibilities (putting off starting work today)....

    1. Ergodic Number Theory: a development of the methods of Green and Tao who used ergodic results to prove that there are arbitrarily long arithmetic progressions of primes.

    2. The theory of Motives. Some people might actually figure out what Grothendieck thought these would be, and then we might make some progress in some of the big questions in algebra, geometry and number theory combined.

    3. Non-commutative rings in geometry. Traditionally geometry has avoided dealing with objects with 'bad behaviour', such as singularities. For low dimensional cases we can resolve these with commutative methods, and know a reasonable amount about special cases (K3, Kahler, some flips and flops) but recent work shows that we might have to use non-commutative (skew-group) rings to produce smooth resolutions (for gorenstein singularities for instance). People I respect are of the opinion that the synthesis of methods in geometry and algebra might be able to prove hard conjectures on derived equivalences. An example being that if G is a group and N the normalizer of its Sylow subgroup P, then if P is abelian G and N have more things in common than we can currently prove they should have.

    Out of curiosity does anyone know what the Langlands Program is up to at the moment?
     
    Last edited: May 16, 2006
  4. May 16, 2006 #3

    Curious3141

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    I'd like to work in biomaths - it's the best use I can find for my medical degree. :smile:
     
  5. May 17, 2006 #4

    J77

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    Any specific area in mind?
     
  6. May 17, 2006 #5

    Curious3141

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    I've always been fascinated by applying mathematical models to epidemic transmission. During the SARS outbreak, I attended a joint conference of mathematicians and virologists on the variety of mathematical modelling methods one could use to model the transmission of disease.

    It was hardly an introduction, since pretty much everyone else out there was a professional mathematician, but I learned that modelling an epidemic from existing data could tell us about the basic reproductive number as well as predict time to extinction of the epidemic. I learned about the analytical modelling first with a system model using partial differential equations, then using Markov chains and finally cellular automata and discrete temporal steps.

    You know what? I found the mathematics a lot more fascinating than the virology. :biggrin:

    I need to do something like this for a living, if I am to keep my sanity.
     
  7. May 17, 2006 #6
    I just read an article on biocomputers, sounds quite interesting.
     
  8. Jan 18, 2007 #7

    J77

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    There are quite a lot of ODE models out there - so-called SIR (I think standing for susceptible, infected, recovered) plus varients of, eg. SIRS.

    Also, people are starting to add delayed terms to the SIR models - eg. to represent incubation or maturation periods.

    I messed around with some of these equations a few weeks back but couldn't find anything too exciting - just oscillatory bifurcations.
     
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