The PI proposed research focuses on several basic problems related to the geometry of complex and symplectic manifolds, which can be studied using nonlinear PDEs. In the first project the PI will study a recent conjecture of Donaldson that aims at extending Yau's theorem in Kahler geometry to symplectic four-manifolds, building on his work with Weinkove and Yau. If proved, this conjecture would provide a powerful new tool to construct symplectic forms on compact symplectic four-manifolds, and would have striking applications to symplectic topology. The second project regards the geometry of compact Calabi-Yau manifolds, and specifically the way in which Ricci-flat Kahler metrics on a Calabi-Yau manifold can degenerate when their cohomology class approaches the boundary of the Kahler cone. These degenerations have also been studied by string theorists in connection with mirror symmetry. The PI proposes to continue his study of these degenerations, as well as investigating Ricci-flat metrics on a family of quintic threefolds near a large complex structure limit. The third project falls in the area of canonical metrics on compact Kahler manifolds, such as Kahler-Einstein or constant scalar curvature Kahler metrics. It is believed that the existence of such canonical metrics should be equivalent to the algebraic stability of the manifold. The PI will study this using two natural evolution equations associated to these problems, the Kahler-Ricci flow and the Calabi flow, with the aim of connecting the limiting behaviour of the flows to algebraic stability through the use of natural energy functionals. The final project also involves canonical Kahler metrics, and more specifically the problem of existence of constant scalar curvature Kahler metrics on complex surfaces with ample canonical bundle in cohomology classes that are known to be stable.
Most of the problems that we will consider, for example the Einstein equations, were originally discovered by physicists who were searching for models of the fundamental laws of nature. More recently, geometric aspects closely related to the proposed research have found applications in high energy physics, and are being used to deepen our understanding of the Universe and of elementary particles. The geometric ideas of the PI's research revolve around the problem of finding the optimal shape of a geometric space, the one with the largest possible symmetry, and understanding the possible singularities that form in spaces where such an optimal shape does not exist. Any progress on these questions will not only shed some light on some basic problems in mathematics, but will also have applications in physics and other sciences.
This NSF grant supported the PI's research in the field of geometric analysis. This is the study of problems of geometric nature (i.e. how a certain high-dimensional space is curved), using the tools of analysis and differential equations. The fundamental laws of physics are described, in the language of mathematics, by differential equations. Understanding the behavior of solutions to differential equations is the key to unraveling the mystery of the geometry and structure of the universe. One of the main themes of the research which has been carried out is the geometry of Calabi-Yau spaces, which have become of central importance to the study of realistic string theory models. In a nutshell, string theorists believe that our physical world is not four-dimensional (three spatial directions plus time) but rather ten-dimensional. The remaining six dimensions are extremely small, so that we don’t normally perceive them, but are crucial for understanding elementary particles. These six dimensions together form a (tiny) geometric space, which is a Calabi-Yau space. Physically, these spaces capture essential features of particle physics and cosmology, and understanding their geometry will push string theorists closer to the goal of making contact with observations. Of special interest is the study of degenerations of Calabi-Yau spaces, which is a key ingredient in the theory of mirror symmetry (a mysterious duality between families of Calabi-Yau spaces). Understanding how Calabi-Yau spaces can degenerate leads not only to a broadening of our mathematical knownledge but also to applications in high energy physics. The findings of the research that the PI has done have clarified the possible degenerations of hyperkahler manifolds, which are a special class of Calabi-Yaus, and have shed some light on the mechanisms that underlie mirror symmetry. Another main strand of the research that has been performed is the study of geometric flows, where one starts with a given geometric space and continuously deforms its shape according to its curvature, with the goal of making the space as homogeneous as possible. However, sometimes this is not possible, and the flow develops singularities in finite time. One of the main outcomes of this project is a complete understanding of finite-time singularities of one of the most famous geometric flows, the Ricci flow, on Kahler spaces whenever the volume of the space does not go to zero. The research funded by this grant has resulted in several publications in leading international mathematical journals, dissemination of the research findings at international conferences, workshops and seminars at Universitites, training of graduate students through informal reading seminars and graduate courses, and collaboration with mathematicians at institutions in the US and China. Furthermore, the PI has co-organized two one-day conferences on Complex Geometry and Partial Differential Equations, and a Special Session at a Sectional AMS Meeting in Arkon, OH, where more than half of the speakers were chosen among graduate students, early career mathematicians, and underrepresented groups.
