Benjamin Mann of DARPA has constructed a list of 23 challenges for mathematics for this century. Here are five of the top challenges:

1. The Mathematics of Quantum Computing, Algorithms, and Entanglement(DARPA 15) : “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.”

2. Settle the Riemann Hypothesis(DARPA 19) : “The Holy Grail of number theory.”

3. Geometric Langlands and Quantum Physics(DARPA 17) : “How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice versa?”

4. The Geometry of Genome Space(DARPA 15) : “What notion of distance is needed to incorporate biological utility?”

5. Algorithmic Origami and Biology(DARPA 10) : “Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding.”

See the rest of the 23 here. [PDF]

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