Would-be mathematicians learn about David Hilbert’s 23 problems way before they even learn how to ask someone for a first date. Presented by Hilbert at the International Congress of Mathematicians held in Paris in 1900, these problems stimulated mathematical research in the 20th century, and all but two have now been resolved in one way or another.

There have been many attempts to repeat Hilbert’s amazing feat, but it is now clear that the field has grown so much in so many directions that no single person can have the required global vision to fill Hilbert’s 21st century shoes.

Many organizations tried but … DARPA?  Yes, this stands for Defense Advanced Research Projects Agency, and the Defense Sciences Office (DSO) within DARPA is in the business of supporting fundamental research by frequently issuing  challenges (solicitations) to scientists and engineers. These challenges are not for the faint of hearts, but first look at its mandate.

“The mission of the DSO is to bridge the gap from fundamental science to applications by identifying and pursuing the most promising ideas within the science and engineering research communities and is committed to transform these ideas into new DoD capabilities.”

In other words, they believe in fundamental science even though they are essentially funded to support the Defense department. Compare to some of our Canadian research councils, which were originally founded to support basic science, but ended up believing in … the automobile of the 21st century.

“DSO places no limit on the range of ideas it pursues. Although distinct in their respective technical objectives, all DSO programs focus on mining “far side” science.”

In contrast, some of Canada’s research councils were founded to support “far side” science, yet chose the route of putting limits on the range of ideas that Canadian scientists want to pursue.

In any case, here are DARPA’s mathematical challenges. They are nothing short of the holy grail of 21st century mathematical sciences. They were put forward in 2007, but –unlike other DARPA 2-year projects– these challenges may take a century or two to solve.

Department of Defense, BAA07-68

Description: DARPA seeks innovative proposals addressing these Mathematical Challenges. Submissions that merely promise incremental improvements over the existing state of the art will be deemed unresponsive.

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 solutions, 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.

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 Partial Differential Equations (PDE) approach to differential games?

14. An Information Theory for Virus Evolution

• Can Shannon’s theory shed light on this fundamental area of biology?

15. The Geometry of Genome Space

• What notion of distance is needed to incorporate biological utility?i.e. how to measure the similarity or difference between two genomes instances.

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

• This conjecture in algebraic geometry is a metaphor for transforming transcendental computations into algebraic ones.
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”?
Smooth Poincaré Conjecture

23. What are the Fundamental Laws of Biology?