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"?