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Every so often, the Pentagon's blue sky research arm gets slammed, for funding investigations that are a little too down-to-Earth. Then Darpa turns around, and sponsors a new project to "develop a mathematical theory to build a functional model of the brain."

Wired Blogs, By Noah Shachtman, September 26, 2008

It's one of 23 "mathematical challenges" issued today by the agency. In addition to cooking up a theory that predicts how to make a mock-brain, Darpa is looking for mathematicians to:

• Finally solve the 150 year-old Riemann Hypothesis, the "Holy Grail of number theory."

• "Develop the high-dimensional mathematics needed to accurately model and predict behavior" in biology and human interactions.

• Create "an information theory for virus evolution."

• Figure out the "fundamental laws of biology."

[Gregory Piatetsky-Shapiro: Here is the full list of DARPA's 23 Math Challenges]

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.

• 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.

• Address Mumford's call for new mathematics for the 21st century. Develop methods that capture persistence in stochastic environments.

• 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.

• 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?

• 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?

• 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.

• Can linear algebra be replaced by algebraic geometry in a systematic way?

• Can profound theoretical advances in understanding three dimensions be applied to construct and manipulate structures across scales to fabricate novel materials?

• Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding.

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

• 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.

• What new scalable mathematics is needed to replace the traditional Partial Differential Equations (PDE) approach to differential games?

• Can Shannon's theory shed light on this fundamental area of biology?

• What notion of distance is needed to incorporate biological utility?

• 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.

• How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice versa?

• What is the role of homotopy theory in the classical, geometric, and quantum Langlands programs?

• The Holy Grail of number theory.

• How can we develop asymptotics for a world with massively many degrees of freedom?

• This conjecture in algebraic geometry is a metaphor for transforming transcendental computations into algebraic ones.

• What are the implications for space-time and cosmology? And might the answer unlock the secret of "dark energy"?

• This question will remain front and center for the next 100 years. DARPA places this challenge last as finding these laws will undoubtedly require the mathematics developed in answering several of the questions listed above.