formating modified; posted 10 Sep 07

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/__

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.

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 21

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"?

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.

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

ATTN: BAA 07-68, Dr. Benjamin Mann

3701 North Fairfax Drive

Arlington, VA 22203-1714