This proposal explores some of the main research areas where algebraic geometry interacts with quantum field theory and string theory including: the moduli of super Riemann surfaces and the related issues of superstring measure and superstring perturbation theory; the geometric Langlands program; heterotic string phenomenology; F theory; and quantum sheaf cohomology. The proposal also considers some issues in moduli spaces of curves and abelian varieties, and some more exploratory math/physics connections such as amplitudes, Grassmannians, and twistors; and some conjectures regarding the 6-dimensional conformal field theory and its mathematical consequences.
The significance of this project is in developing the connections between mathematics (mostly algebraic geometry) and high energy physics (mostly string theory). Each of these areas involves a mixture of issues from math and from physics, and most will be explored in teams involving both mathematicians and physicists. The first research area in particular is expected to have a major impact on the (mathematical) foundations of perturbative superstring theory, while the second addresses one of the major open problems in mathematics using new tools inspired, at least in part, by high energy physics ideas. The PI proposes also to continue a wide range of broad impact community and educational activities, including: the development and guidance of a series of conferences emphasizing the interactions of physics and mathematics, and of other channels for the dissemination of new knowledge concerning interactions of mathematics and high energy physics; curriculum development at the graduate and undergraduate level; writing a strings-for-mathematicians text; extensive work with graduate and undergraduate students and evaluation of the math major at Penn; membership of national bodies such as the AMS Committee on the Profession, and editorship of several journals and book series.
This award is co-funded by DMS and PHY.