New Geometric Structures from String Theory

Our proposed research is aimed towards discovering the mathematical structures underpinning the fundamental nature of matter, the origins of the universe and the quantum structure of spacetime each of which would be a revolution in science, with possible consequences as far-reaching as they were when matter was shown to be composed of atoms. Just as differential geometry was vital for Einstein's theory of gravity, the mathematics we are investigating could be an essential part of a revolution the formulation of string theory, with possible far-reaching consequences. Theoretical physics has long been central in motivating vital new directions in geometry. Recent research in string theory points to the existence of a rich class of remarkable new geometrical structures. The aim of the research is to develop the underlying unifying mathematics of these structures, laying the foundations for new areas of geometric study using tools and ideas from string theory and gauge theory. It is rare to find a new type of geometry that is both beautiful and tractable. Generalized geometry, introduced by Hitchin, is one such new example, and like others is strongly linked with ideas from physics. The overarching focus of our proposal will be the much larger set of ideas of which this is part, including extended geometries, doubled geometries, flux geometries, non-geometric spaces and holographic structures. We anticipate wide ranging applications across mathematical sciences from geometry and topology to algebra and number theory and mathematical physics.

The project is blue skies research in mathematics. As such, the social and economic impact of the research is less immediate and direct than in more applied areas.
Our work will have a significant impact on geometry and topology. There will be important applications to string theory, gauge theory and other areas of mathematical physics. We will bring our work to these communities through talks and articles. We will hold two workshops on the subject of this proposal, and one of the aims of these will be to assess and develop the impact and applications of our work to pure mathematics and to physics.

Fundamental scientific breakthroughs lead to technological advance and cultural transformation. Our proposed research is aimed towards discovering the mathematical structures underpinning fundamental physics. Just as differential geometry was vital for Einstein's theory of gravity, our research could be an essential part of a revolution in science, with possible far-reaching consequences. The timescales for this kind of foundational new knowledge about space, time and matter to translate into technological developments are unpredictable and can be long.

Two aspects of the impact of the research are certain. One is progress on the integration of science into culture, which inspires and attracts people, and by our outreach activities will directly benefit society by educating the public, creating the widest possible base of support for science and motivating young people to study science. Members of the team have given popular talks and appeared in the media. We will aim to identify and exploit opportunities to bring the work of our proposal to the public, through public talks and publications.

The second route to impact is through the direct training of young people, PhD students and RAs in advanced mathematical methods and problem solving, developing independence and creativity. The positive impact on the economy is the supply of highly trained individuals with the skills to contribute at the highest level to modern business and industry.