23 mathematical challenges -- from the Pentagon!

More precisely, from Benjamin Mann of the Defense Sciences Office.
http://www.darpa.mil/dso/personnel/23_math_chall_b_mann.pdf
The following is essentially all the text, plus a few comments of my own in
square brackets.

1
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.
2
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.
3
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.
4
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.
5
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?
6
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?
7
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. [He's thinking of crypto, I'll bet.]
8
Beyond Convex Optimization
Can linear algebra be replaced by algebraic geometry in a systematic way?
9
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?
[This list of 23 problems is evidently pretty recent.]
10
Algorithmic Origami and Biology
Build a stronger mathematical theory for isometric and rigid embedding that
can give insight into protein folding.
11
Optimal Nanostructures
Develop new mathematics for constructing optimal globally symmetric
structures by following simple local rules via the process of nanoscale
self-assembly.
12
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.
13
Creating a Game Theory that Scales
What new scalable mathematics is needed to replace the traditional PDE
approach to differential games?
14
An Information Theory for Virus Evolution
Why not?
15
The Geometry of Genome Space
What notion of distance is needed to incorporate biological utility?
16
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.
17
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?
[A lot of important stuff originated in number theory -- VERY important
stuff.]
18
Arithmetic Langlands, Topology, and Geometry
What is the role of homotopy theory in the classical, geometric, and quantum
Langlands programs?
19
Settle the Riemann Hypothesis
[I'll let you know when I have done so.]
The Holy Grail of number theory.
20
Computation at Scale
How can we develop asymptotics for a world with massively many degrees of
freedom?
21
Settle the Hodge Conjecture
[I'll get on it as soon as I settle the Riemann Hypothesis]
This conjecture in algebraic geometry is a metaphor for transforming
transcendental computations into algebraic ones.
22
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"?
[Jack Saffarti would know.]
23
What are the Fundamental Laws of Biology?
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.


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