New Frontiers of Algebraic Geometry

Algebraic geometry is one of the most advanced and multifaceted areas of modern mathematics. The objects of study are algebraic varieties: shapes defined by systems of polynomial equations. Polynomial constraints on variables are commonplace both in mathematics and in applications ranging from physics, where varieties are used to describe configuration spaces (and even the shape of the universe!), to computer science and statistics.

The Principal Investigator's research focuses on describing algebraic varieties in a concise coordinate-free way that reveals hidden essential features of their geometry. The projects connect many different areas: birational geometry, derived categories, arithmetic geometry, mathematical physics, toric geometry, deformation theory, and transformation groups.

The research program contributes to the development of a diverse, globally competitive STEM workforce through recruiting, training, and supervising of graduate and undergraduate students, including those from underrepresented groups, and providing resources for their research, teaching, and professional development. In particular, the project includes two projects designed for summer REUs in Algebraic Geometry.

The Principal Investigator will organize and lecture at conferences, professional development events, and summer schools for graduate students and junior researchers including AGNES (Algebraic Geometry Northeastern Series), and schools in Latin America attended by U.S. graduate students. The research will advance inter-university and international cooperation.

This research program includes collaborations with high energy physicists as well as software development of interest to mathematical cryptography (arithmetic of elliptic curves). The Principal Investigator will continue to develop graduate and undergraduate courses, including courses designed to introduce the broader public to mathematics and science.

This project builds on the earlier work of the Principal Investigator (PI) on a broad range of foundational topics in algebraic geometry, including tropical compactifications, birational geometry of moduli spaces of curves and surfaces, compactifications of moduli spaces and derived categories of moduli spaces. The PI will advance modern algebraic geometry on several important frontiers: (1) Applications of derived categories to geometry, cohomology and K-theory of algebraic varieties, including moduli spaces of algebraic curves and vector bundles, toric varieties and Mori Dream Spaces. (2) A new interface between birational and arithmetic geometry exploring effective cones of algebraic surfaces via arithmetic geometry of curves. (3) A novel geometric interpretation and generalization of leading singularities of scattering amplitudes of elementary particles. (4) Development of kinematics, scattering equations and propagation of their solutions. (5) A new class of polyhedral approximations of geometric varieties via spherical tropicalization with the potential to develop new compactification techniques. (6) A new approach to parametrizing moduli spaces of surfaces near the stable limit using deformations of exceptional collections of vector bundles and semi-orthogonal decompositions for Q-Gorenstein degenerations.

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.