This proposal concerns geometric and analytic problems which arise from geometry and physics. The PI will continue his study on existence problem and singularity formation of the Einstein equation and the Yang-Mills equation. For the Einstein equation, the PI will focus mainly in dimension 4 or in the Kahler case. He will also study related Ricci flow and its singularity development and its interaction with classifying projective manifolds in algebraic geometry. Corresponding problems for self-dual metrics in dimension 4 will be also studied. For the Yang-Mills equation, the PI will focus on how to compactify spaces of its self-dual solutions and their applications to constructing new invariants. The PI found before that the Yang-Mills fields forms singularity along classical minimal surfaces like soap bubbles or subvarieties. He likes to explore this further and particularly, the interaction between self-dual Yang-Mills fields and calibrated geometry. The PI also intends to continue his study on problems in symplectic geometry, including deforming symplectic surfaces in 4-manifolds and constructing new deformation invariants.
The Einstein and the Yang-Mills equations have played a fundamental role in our study of physics and geometry and topology in last few decades. Perelman's solution for the Poincare conjecture is an excellent example. Important and central problems include studying when one can solve those equations, what properties of those solutions found, how they develop singular behaviors. It is also important to understand the connection between these solutions and other branches of mathematics, such as, algebraic geometry and differential topology. The resolution of these problems will provide new profound understanding geometry of underlying spaces. The problems involved in this research project were also inspired by the study of the string theory in physics. Through this research project, the PI also intends to develop new tools for studying curved spaces, symplectic geometry and provide new mathematical foundation for some physical theories.
Differential equations arise naturally from geometry and physics. The Einstein and the Yang-Mills equations are of this sort. They have played a fundamental role in our study of physics and geometry and topology in last few decades. A good example is the Poincare conjecture by Perelman using Ricci flow introduced by R. Hamilton more than 30 years ago. However, the equations which interest us are nonlinear and may develop singular solutions. A basic problem is to understand how those solutions behave near their singularity. During the period supported by this grant, the PI studied those singular solutions and developed new theories for the Einstein equation on spaces with extra structures, such as low dimensional spaces and Kahler spaces. He also applied such theories to studying geometry of underlying spaces and obtained breakthrough results. First he extended his previous works with others on singular solutions of the Einstein equation on Kahler spaces to conic cases. Using this extension, he was able to solve a long-standing conjecture on the existence of Kahler-Einstein metrics and geometric stability which can be traced back to classical geometric invariants studied by many prominent mathematicians, e.g., D. Hilbert. Secondly, the PI continued to study the Analytic Minimal Model Program through Ricci flow. This program was initiated and described by him together with his collaborators. It is amount to studying singularity formation of Ricci flow on Kahler spaces. The Pi and his collaborators have made very important progresses on this program and revealed the interaction between singularity theory of Ricci flow and algebraic geometry. Thirdly, the PI and his collaborator introduced two new geometric flows. They provide useful tools in geometry. One flow is closely related to the renormalization flow coupled with B-fields in theoretic physics. It can be used to analyze generalized Kahler spaces which appear naturally in studying mirrow symmetry in physics. Another flow preserves symplectic structures and can be used to studying symplectic spaces, particularly, classifying symplectic 4-spaces. Fourthly, we continue to construct new invariants in symplectic geometry which extend the Gromov-Witten invariants. Those new invariants are constructed by coupling the Cauchy-Riemann equation with the Yang-Mills on surfaces. They may be applied to building a profound connection between symplectic geometry and complex geometry. The PI hopes to provide new mathematical foundations for physical theories through his study. The PI and his collaborator have recently published a book on the Geometrization Conjecture of 3-spaces. The book is designed for graduate students and young mathematicians. In last few years, based on his research, he taught several courses on geometry for first-year or second-year graduate students. He also organized a working seminar on geometry for graduate students and postdoctors. The goal of this working seminar is to help students and postdocs to gain research experiences and broaden their knowledge in mathematics. He was also involved in running differential and symplectic geometry seminars at Princeton. He had advised many students and been a mentor for a number of postodctors in last five years. In summary, the PI has made very substantial progresses on his proposed project and developed new tools which may be used to attack core problems in geometry. The PI had 12 graduate students who had gotten PhD degree during the period supported by this NSF grant. Currently, he has 12 graduate students. He was also involved in organizing mathematical activities, such as, summer schools and conferences, in order to promote circulation of mathematical works and exchanges among mathematicians.
