CAREER: New Frontiers in Birational Geometry - Novel Arithmetic Techniques and Applications

While polynomial equations with multiple variables are among the most elementary of equations, requiring only addition and multiplication of variables, the set of points which satisfies a given system of equations carries rich algebraic and geometric structure. The set of points with complex coordinates which satisfies a fixed system of equations gives a geometric object is called an algebraic variety.

The classification of algebraic varieties is the main motivation of the PI's field of birational geometry. On the other hand, arithmetic geometry is the study of polynomial equations whose coefficients are integers. This allows more tools from number theory to be used to study the corresponding varieties.

The field of arithmetic geometry has recently seen an explosion of new techniques leading to spectacular progress. The PI's research on this project will apply these new techniques to problems in birational geometry which were previously out of reach. The educational component of the project involves several initiatives designed to increase access to a career in mathematics.

This includes creating a summer school introducing high school students to mathematical proofs using the Lean proof assistant.

The project centers on interactions between birational and arithmetic geometry. New techniques from arithmetic geometry involving perfectoids and prisms have made it possible to overcome significant difficulties in mixed characteristic birational geometry, mainly related to the failure of cohomology vanishing theorems. These vanishing theorems are the primary method used in inductive arguments throughout birational geometry over the complex numbers, and their replacement in mixed characteristic promises significant advances.

The main overarching objective of the projects will be to develop more powerful ways to replace vanishing theorems in mixed characteristic. The first application will be to study the singularities of integral models of Shimura varieties, which are an important class of varieties in arithmetic geometry. The PI will also investigate generalizing the theory of log canonical centers to mixed characteristic, which will enable him to investigate other problems in birational geometry such as boundedness and moduli of 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.