Extracted from   www.fbo.gov/EPSData/ODA/Synopses/4965/BAA07%2D68/BAA07%2D68%2Edoc
formating modified; posted 10 Sep 07

Broad Agency Announcement (BAA 07-68)
for
Defense Sciences Office (DSO)

DARPA/DSO SOL, DARPA Mathematical Challenges, BAA 07-68; BAA CLOSING DATE: 9/8/08; TECHNICAL POC: Dr. Benjamin Mann, DARPA/DSO, Ph: (571) 218-4246, Email: BAA07-68@darpa.mil; CFDA#: 12.910;
URL: http://www.darpa.mil/dso/solicitations/solicit.htm;
Website Submission: http://www.sainc.com/dsobaa/

I. Funding Opportunity Description

DARPA is soliciting innovative research proposals in the area of DARPA Mathematical Challenges, with the goal of dramatically revolutionizing mathematics and thereby strengthening the scientific and technological capabilities of DoD. To do so, the agency has identified twenty-three mathematical challenges, listed below, which were announced at DARPA Tech 2007.

DARPA seeks innovative proposals addressing these Mathematical Challenges. Proposals should offer high potential for major mathematical breakthroughs associated to one or more of these challenges. Responses to multiple challenges should be addressed individually in separate proposals. Submissions that merely promise incremental improvements over the existing state of the art will be deemed unresponsive.

  • Mathematical Challenge One: The Mathematics of the Brain
      Develop a mathematical theory to build a functional model of the brain that is mathematically consistent and
      predictive rather than merely biologically inspired.

  • Mathematical Challenge Two: The Dynamics of Networks
      Develop the high-dimensional mathematics needed to accurately model and predict behavior in large-scale
      distributed networks that evolve over time occurring in communication, biology, and the social sciences.

  • Mathematical Challenge Three: Capture and Harness Stochasticity in Nature
      Address Mumford's call for new mathematics for the 21st century. Develop methods that capture persistence
      in stochastic environments.

  • Mathematical Challenge Four: 21st Century Fluids
      Classical fluid dynamics and the Navier-Stokes Equation were extraordinarily successful in obtaining quantitative
      understanding of shock waves, turbulence, and solitons, but new methods are needed to tackle complex fluids
      such as foams, suspensions, gels, and liquid crystals.

  • Mathematical Challenge Five: Biological Quantum Field Theory
      Quantum and statistical methods have had great success modeling virus evolution. Can such techniques be
      used to model more complex systems such as bacteria? Can these techniques be used to control pathogen
      evolution?

  • Mathematical Challenge Six: 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?

  • Mathematical Challenge Seven: Occam's Razor in Many Dimensions
      As data collection increases can we do more with less by finding lower bounds for sensing complexity in
      systems? This is related to questions about entropy maximization algorithms.

  • Mathematical Challenge Eight: Beyond Convex Optimization
      Can linear algebra be replaced by algebraic geometry in a systematic way?

  • Mathematical Challenge Nine:
     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?

  • Mathematical Challenge Ten: Algorithmic Origami and Biology
      Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding.

  • Mathematical Challenge Eleven: Optimal Nanostructures
      Develop new mathematics for constructing optimal globally symmetric structures by following simple local
      rules via the process of nanoscale self-assembly.

  • Mathematical Challenge Twelve: The Mathematics of Quantum Computing, Algorithms, and Entanglement
      In the last century we learned how quantum phenomena shape our world. In the coming century we need to
      develop the mathematics required to control the quantum world.

  • Mathematical Challenge Thirteen: Creating a Game Theory that Scales
      What new scalable mathematics is needed to replace the traditional Partial Differential Equations (PDE)
      approach to differential games?

  • Mathematical Challenge Fourteen: An Information Theory for Virus Evolution
      Can Shannon's theory shed light on this fundamental area of biology?

  • Mathematical Challenge Fifteen: The Geometry of Genome Space
      What notion of distance is needed to incorporate biological utility?

  • Mathematical Challenge Sixteen: What are the Symmetries and Action Principles for Biology?
      Extend our understanding of symmetries and action principles in biology along the lines of classical
      thermodynamics, to include important biological concepts such as robustness, modularity, evolvability,
      and variability.

  • Mathematical Challenge Seventeen: Geometric Langlands and Quantum Physics
      How does the Langlands program, which originated in number theory and representation theory, explain the
      fundamental symmetries of physics? And vice versa?

  • Mathematical Challenge Eighteen: Arithmetic Langlands, Topology, and Geometry
      What is the role of homotopy theory in the classical, geometric, and quantum Langlands programs?

  • Mathematical Challenge Nineteen: Settle the Riemann Hypothesis
      The Holy Grail of number theory.

  • Mathematical Challenge Twenty: Computation at Scale
      How can we develop asymptotics for a world with massively many degrees of freedom?

  • Mathematical Challenge Twenty-one: Settle the Hodge Conjecture
      This conjecture in algebraic geometry is a metaphor for transforming transcendental computations
      into algebraic ones.

  • Mathematical Challenge Twenty-two: Settle the Smooth Poincare Conjecture in Dimension 4
      What are the implications for space-time and cosmology? And might the answer unlock the secret of
      "dark energy"?

  • Mathematical Challenge Twenty-three: What are the Fundamental Laws of Biology?
      Dr. Tether's question will remain front and center in the next 100 years. I place this challenge last as finding
      these laws will undoubtedly require the mathematics developed in answering several of the questions listed
      above.

    Please Note: White Papers and Full Proposals may be submitted and received at any time until the final BAA deadline of 4:00PM ET, September 8, 2008.

    VII. Agency Contacts

    The Technical POC for this effort is Dr. Benjamin Mann, Phone: (571) 218-4246, E-mail: benjamin.mann@darpa.mil.

    DARPA/DSO
    ATTN: BAA 07-68, Dr. Benjamin Mann
    3701 North Fairfax Drive
    Arlington, VA 22203-1714