[ExI] DARPA Mathematical Challenges
Jef Allbright
jef at jefallbright.net
Fri Oct 3 13:04:17 UTC 2008
DARPA has announced a set of Mathematical Challenges "with the goal of
dramatically revolutionizing mathematics and thereby strengthening the
scientific and technological capabilities of DoD,"
<
https://www.fbo.gov/download/9bc/9bce380aafb19f9ad3bda188bfc1ab20/DARPABAA0865.doc
>
It's interesting how closely so many of these dovetail with topics of
interest to the Extropy list.
 Jef
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.
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