A central theme in the PI's research is the connections between partial differential equations (PDEs) such as Einstein's equations in general relativity, and the algebraic or geometric properties of the spaces on which these equations are studied. Much research has been focused on the study of PDEs on smooth spaces, where the local geometry has a very simple model, however singularities can arise naturally in nature, the most well known being black holes. The current research project focuses on the study of PDEs on such singular spaces. This study will have applications in algebraic geometry, where singular spaces play a central role in modern research, as well as in the theory of partial differential equations. At the same time, singular spaces occur naturally as limits of families of smooth spaces, which could collapse in certain directions for instance. In this way the proposed research will shed new light on the behavior of such families. Aside from pursuing these research projects, the PI will continue training PhD students and postdoctoral researchers. In addition the PI will also co-organize a yearly summer workshop for undergraduate students aimed at conveying ideas in geometry to them which do not typically appear in the undergraduate curriculum. The PI will also co-organize a yearly bridge program aimed at helping incoming graduate students with diverse backgrounds get up to speed, to ensure their success.
The objective of the project is to investigate singularities in Kahler geometry from different points of view. On the one hand, singularities can arise in the limit of a sequence of smooth spaces, and it is important to understand the structure of such limit spaces. The PI will study the question of when the limit space can be identified with a singular Kahler space, building on work of Donaldson-Sun as well as Liu and the PI. Of particular interest are non-collapsed limit spaces of Kahler manifolds with Ricci curvature bounds, however the PI will also investigate the much less understood collapsed limit spaces. In a related direction, the PI will study canonical metrics, such as Kahler-Einstein metrics, on singular complex varieties, in particular the behavior of such metrics near the singular set. A good understanding of the geometry of such singular metrics would lead to applications of differential geometric techniques to the algebraic geometry of singular varieties. Finally, in relation to the collapsing theory of Kahler manifolds, the PI will study the construction of canonical metrics on fibrations that are almost collapsed.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
