CAREER: Termination of Flips and Cluster Geometry

This project investigates problems in algebraic geometry. Algebraic geometry is the field of mathematics that studies the geometric shape of objects defined by polynomial equations. Often, the shape of these objects, called algebraic varieties, reflects the complexity of the defining equations and vice-versa. For instance, from the defining equations, we can understand if the algebraic variety has singularities, i.e., whether the geometric object has sharp points at which the curvature changes abruptly. These points are known as algebraic singularities. This project aims to develop new tools to understand algebraic singularities and apply these techniques to understand algebraic varieties of positive curvature. As part of the broader impacts of this project, the PI will run online research seminars for graduate students related to the proposed research. The PI will initiate a summer reading program in algebraic geometry for undergraduates. The educational component of this project includes training in algebraic geometry for graduate students. Two summer research schools will be hosted at UCLA by the PI and experts in the field to train the new generation of mathematicians in birational geometry. Fano varieties are considered one of the three building blocks of projective varieties. There has been a lot of progress towards understanding the geometry of Fano varieties in the last few decades. Nevertheless, the understanding of log terminal singularities (the local analog of Fano varieties) is far from being satisfactory. Complexity is an invariant that allows us to understand how far a Fano variety or a log terminal singularity is from being toric, i.e., defined by binomial equations. In this project, the PI will develop techniques to understand Fano varieties and log terminal singularities of small complexity, for instance, complexity zero, one, and two. There are two guiding principles: understanding the connection between the complexity and minimal log discrepancies (in the local setting) and understanding the connection between the complexity and anti-pluricanonical systems of Fano varieties (in the global setting). The PI aims to apply the former to study the termination of flips and the latter to understand the classification of cluster type Fano varieties. 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.