Program Manager: Benjamin Mann, Ph.D.
The Focus Areas in Theoretical Mathematics (FAThM) program aims to foster major theoretical breakthroughs in pure mathematics whose potential for long-term defense implications is high. By supporting closely integrated and concentrated collaborations among small numbers of leading experts, FAThM will pioneer a new approach for conducting focused research to explore fundamental interconnections between key areas of mathematics where critical insights should lead to both new mathematics and innovative applications.
The initial focus area investigates various aspects of the Langlands Programs, a conjectured series of connections or "dualities" relating diverse areas of mathematics:
- Classical (number theory and representation theory).
- Geometric (algebraic geometry, number theory and representation theory).
- Conformal (representation theory, physics, and topology).
The team seeks to extend recent results and plausible conjectures in harmonic analysis involving symmetry groups that arise in physics and their Langlands duals, leading to new mathematical connections with field theory in physics. New representation theory necessary to extend the geometric Langlands conjecture from complex curves to complex surfaces and from compact forms of symmetry groups arising in physics to other real forms is being developed. Homotopical connections to the classical conjectures are also being explored, and exploitation of these mathematical results as new tools for modeling of physical phenomena is being undertaken.
The fundamental mathematics developed in this program is expected to have broad significance in basic science and several avenues of possible long-term defense impact, including quantum algorithms and devices; cryptography; fast structured algorithms for signal/image processing and other DoD-critical applications; and high-density data coding. Additionally, this program is engaging a community of premier theoretical mathematicians and mathematical physicists in DoD-sponsored research, creating a new resource for defense science and fostering infusion of leading-edge theoretical research into national security applications.
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