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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?
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?
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
    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
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”?
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.

Nice challenges, very ambitioned. I wonder what the selection creterias are? DARPA is part of the DOD (Department of Defense), so first of all I look how these chalenges can be used for military applications. Mhhh? No idea? No idea how it could be used directly. There are at least two other possibilities.

First alternative Mr. Mann, the author mentioned above is using the all degrees of freedom to announce his very personell wish list (in fact many problems of them would be on my list).
Second alternative the DARPA wants to keep attention to young mathematicians working in the military apparatus.

I think the motivation for the “23 challenges list” is a mixture of both arguments.
In fact I doubt if this initiative helps to solve any of these speculative challenges. Many of them are holy grails (Riemann, Hodge,…) and many mathematicians in the world would be happy to answer one of them….