Problems of fundamental physics in the context of string theory and gravity can often be rephrased purely as problems in geometry, the space of vacua of string theory and in particular the spectrum of supersymmetric (BPS) objects on them. On certain "walls" in the moduli space of vacua, a BPS object can decay to less massive BPS objects. Such a wallcrossing phenomenon has an extremely rich and mysterious mathematical structure. It is proposed to study wall-crossings from various points of views including vacua with high number of supersymmetries, applying tools of mirror symmetry and string dualities. Understanding the spectrum of BPS objects is important for calculating black hole entropy and topological string theory. The PI proposes to uncover new geometrical invariants which are necessary for physically characterizing these string vacua, and explore the implications of a recently defined quasi-local mass which satisfies all known consistency conditions. The projects address exciting questions that have important implications for theoretical physics, and for differential and algebraic geometry. Central to the proposal is the training of postdoctoral fellows and graduate students to conduct cutting edge cross-disciplinary research and to be effective in interacting with both the physics and mathematics communities.
This project was devoted to the study of geometrical problems in physical theories such as string theory and general relativity. Within string theories, which are formulated in higher dimensions, the physics of our four dimensional world is determined by the properties of higher dimensional geometries, the Calabi-Yau manifolds. The singularities that can occur in these spaces are classified mathematically and lead to interesting physical theories. Together with my postdoc Mboyo Esole we investigated the geometrical methods of smoothing certain singularities in Calabi-Yau fourfolds, which are relevant for F-theory, a twelve dimensional theory put forward by Cumrun Vafa, which has witnessed revived interest among physicists. Another set of geometrical problems, which are relevant for the interaction with physics, are the problems addressing the variation of geometrical structures attached to manifolds, such as the variation of Hodge structure, which is captured by period integrals of differential forms. In joint work with my postdoc An Huang we developed new methods to put forward differential equations governing these period integrals for more general variation problems, which are characterized by Lie groups. The study of topological string theory, which is another version of string theory with intricate connections to many areas of mathematics offers many surprising insights into mathematical problems and gives hints to how they are connected and how they can be generalized. Together with my postdoc Murad Alim and my student Jie Zhou we developed a differential ring of functions, which can be attached to any Calabi-Yau threefold and which generalizes classical quasi-modular forms. Furthermore, the amplitudes of topological string theory can be expressed as polynomials in the generators of this ring and can be computed. At special values of the parameter spaces, these give generating functions of mathematical invariants. In addition to my research, I wrote two books popularizing mathematics and gave several public talks and colloquia on these. The first book is on the mathematics of String Theory, it’s title is: "The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions" and the second one is about the history of the Harvard mathematics department with the title: "A History in Sum: 150 Years of Mathematics at Harvard (1825-1975)"
