New Advances on Flat Surfaces

This project focuses on studying flat surfaces and exploring their applications across various fields. Flat surfaces are polygons with identified pairs of parallel edges, where the vertices of a flat surface are glued to form conical singularities. Flat surfaces provide a valuable framework for investigating the intricate connections between geometry, dynamics, and algebraic structures. The understanding of flat surfaces has already led to notable discoveries about space volumes and billiard trajectories. The principal investigator is dedicated to advancing knowledge in this area by continuing his research efforts and achieving new results in the study of flat surfaces. This project will also create many opportunities for students and postdoctoral scholars. Alongside his research, the principal investigator will engage in mentoring students, organizing workshops, and participating in outreach activities. A key focus will be on fostering diversity within the field and preparing the next generation of scientists for future challenges and opportunities. Flat surfaces correspond to differential forms on Riemann surfaces, where the conical singularities of a flat surface correspond to the zeros of a differential. These equivalent yet distinct descriptions make flat surfaces lie at the interface of many fields as a research hotspot. This project will cover various focal points in the study of flat surfaces, including dynamical invariants, intersection theory, residue theory, birational geometry, compactification, Brill-Noether theory, topology, cycle classes, higher differentials, and affine structures. To address the complex challenges associated with these areas, the principal investigator will employ a combination of techniques from algebraic geometry, analytic geometry, dynamics, and enumerative geometry. By utilizing these diverse methodologies, the principal investigator will comprehensively analyze different types of flat surface structures to gain insights into their geometric properties. These flat surface structures will serve as valuable tools for deepening our understanding of fundamental mathematical concepts and uncovering new avenues of research. 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.