Einstein Metrics and Ricci Flows in Dimension 4

Topology is a branch of mathematics that is concerned with the shape of a space. Two-dimensional surfaces are topologically characterized by the number of holes within, a simple notion of complexity. Optimizing geometry helps to better understand the underlying topology; for example, flattening a crumpled sheet of paper or untying a balloon animal makes it easier to count their holes. Ricci flow, a geometric optimization technique, searches for optimal geometries called Einstein manifolds, generalizing flat planes and round spheres. This flow provides an algorithmic approach to decomposing three-dimensional geometries and has been used to characterize topologies in dimension three. While challenges in dimension five have been addressed through systematic methods, the final frontier is dimension four, where Einstein manifolds have been extensively studied in physics. This project aims to further study Einstein four-manifolds while developing a new, four-dimensional-specific theory of Ricci flow. It will bring together researchers from analysis, geometry, topology, and physics. The PI will involve undergraduate and graduate students in the project and integrate research questions into teaching and advising. This project aims to understand and construct 4-dimensional Einstein metrics and Ricci flows. The main difficulties are singularities, where topological surgeries occur, specifically "orbifold" singularities, "cusp" formation, and collapsing. The PI will continue studying these degenerations, their stability, and their links to additional structures such as a Kähler one. Most results about Ricci flows are either 3-dimensional or apply to any n-dimension, with few focusing on the specifics of dimension 4, where most topological questions remain open. Through his extensive study of Einstein 4-manifolds, the PI is familiar with the rich set of 4-dimensional techniques developed over decades. He aims to apply these techniques to study Ricci flows and their singularities, focusing on the topological content of the three types of degenerations: orbifold singularities, cusp formation, and collapsing. 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.