It’s the ninth prime number. An
Eisenstein prime with no imaginary part and real part of the form 3n − 1. The first prime P for which unique
factorization of cyclotomic integers based on the Pth root of unity breaks
down.

It’s also the number of mathematical
challenges listed here that could land you a spot in the history books (if you
solve one – or more – of them).

Discovering novel mathematics will
enable the development of new tools to change the way the DoD approaches
analysis, modeling and prediction, new materials and physical and biological
sciences. The 23 Mathematical Challenges
program involves individual researchers and small teams who are addressing one
or more of the following 23 mathematical challenges.

I bet John Forbes Nash would be interested in this for a number of
reasons. If these challenges are
successfully met, they could provide revolutionary new techniques to meet the
long-term needs of the DoD.

I have just three words for you:

**BRING IT ON**
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.

*Information for this blog post provided by DARPA*

*———-*

*Jessica L. Tozer is a blogger for DoDLive and Armed With Science. She is an Army veteran and an avid science fiction fan, both of which contribute to her enthusiasm for technology in the military*.

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