Wednesday, December 26, 2007

DARPA Mathematical challenges

Via Ars Mathematica and Not Even Wrong, a DARPA-issued list of challenges in mathematics for the 21st century. Some that puzzled me:
  • Computational Duality: Duality in mathematics has been a profound tool for theoretical understanding. Can it be extended to develop principled computational techniques where duality and geometry are the basis for novel algorithms?

    I'm not sure what this is trying to say, or whether I'm reading it wrong, because the story of linear programming, primal dual schemes, and the Lagrangian is the story of using "duality and geometry as the basis for novel algorithms"
  • What are the Physical Consequences of Perelman’s Proof of Thurston’s Geometrization Theorem?
    Can profound theoretical advances in understanding three-dimensions be applied to construct and manipulate structures across scales to fabricate novel materials?

    Thurston's geometrization conjecture talks about the structure of 3-manifolds: i.e 3 dimensional surfaces living in higher dimensional spaces ? What kinds of materials could be fabricated using this ?
  • Computation at Scale: How can we develop asymptotics for a world with massively many degrees of freedom?

    This sounds like there's a nice computational question lurking somewhere, but I'm not quite sure where.
Nice to see algorithmic origami and self-assembly also mentioned. I am particularly intrigued by the reference to the geometry of genome spaces.

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