This list of problems was published in 2008 by the USA’s Defence Sciences Office; not surprisingly, they remain unsolved: each problem is a brief description of a whole new direction in mathematics, computer science, mathematical biology and equires raising a new army of researchers. What attracts attention is the obvious geometric nature of many of them. Where will all the geometers come from?
Mathematical Challenge 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.
Mathematical Challenge 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.
Mathematical Challenge 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.
Mathematical Challenge 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.
Mathematical Challenge 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?
Mathematical Challenge 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?
Mathematical Challenge 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.
Mathematical Challenge 8: Beyond Convex Optimization
Can linear algebra be replaced by algebraic geometry in a systematic way?
Mathematical Challenge 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?
Mathematical Challenge 10: Algorithmic Origami and Biology
Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding.
Mathematical Challenge 11: Optimal Nanostructures
Develop new mathematics for constructing optimal globally symmetric structures by following simple local rules via the process of nanoscale self-assembly.
Mathematical Challenge 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.
Mathematical Challenge 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?
Mathematical Challenge 14: An Information Theory for Virus Evolution
Can Shannon’s theory shed light on this fundamental area of biology?
Mathematical Challenge 15: The Geometry of Genome Space
What notion of distance is needed to incorporate biological utility?
Mathematical Challenge 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.
Mathematical Challenge 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?
Mathematical Challenge 18: Arithmetic Langlands, Topology and Geometry
What is the role of homotopy theory in the classical, geometric and quantum Langlands programs?
Mathematical Challenge 19: Settle the Riemann Hypothesis
The Holy Grail of number theory.
Mathematical Challenge 20: Computation at Scale
How can we develop asymptotics for a world with massively many degrees of freedom?
Mathematical Challenge 21: Settle the Hodge Conjecture
This conjecture in algebraic geometry is a metaphor for transforming transcendental computations into algebraic ones.
Mathematical Challenge 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”?
Mathematical Challenge 23: What are the Fundamental Laws of Biology?
This question will remain front and center in the next 100 years. This challenge is placed last, as finding these laws will undoubtedly require the mathematics developed in answering several of the questions listed above.