Features
Subject: Darpa's Math Quiz: Model the Brain, Find Biology's Laws, Solve Number Theory 'Holy Grail'
Every so often, the Pentagon's blue sky research arm gets slammed, for
funding investigations that are a little too downtoEarth. 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 mockbrain, Darpa is looking for mathematicians to:
 Finally solve the 150 yearold Riemann Hypothesis, the "Holy Grail of number theory."
 "Develop the highdimensional 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."
The gray area between circuitry and gray matter has become one of the hotter topics in military research. As of last month, Darpa was in latestage negotiations with Malibu's HRL Laboratories to spearhead its Systems of Neuromorphic Adaptive Plastic Scalable Electronics ("SyNAPSE") program. The goal: an electronic chip that mimics the "function, size, and power consumption" of a cat's cortex some time in the next decade.
[Gregory PiatetskyShapiro: 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.
Mathematical Challenge Two: The Dynamics of Networks
 Develop the highdimensional mathematics needed to accurately model and predict behavior in largescale 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 NavierStokes 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 selfassembly.
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 Twentyone: Settle the Hodge Conjecture
 This conjecture in algebraic geometry is a metaphor for transforming transcendental computations into algebraic ones.
Mathematical Challenge Twentytwo: Settle the Smooth Poincare Conjecture in Dimension 4
 What are the implications for spacetime and cosmology? And might the answer unlock the secret of "dark energy"?
Mathematical Challenge Twentythree: What are the Fundamental Laws of Biology?
 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.
 
