New Perspectives on Four-Dimensional Geometry

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The mathematical concept of a four-dimensional object allows a more comprehensive study of three-dimensional objects that exist in our physical world by considering its points (encoded as three space coordinates) together with any other relevant quantity, for example its temperature, pressure, or electric conductivity. This simultaneous treatment of space coordinates and other pointwise parameters is both essential and ubiquitous in modern science, from spacetimes in Einstein's theory of General Relativity and D-branes in Particle Physics, to mathematical models used in medical imaging and diagnosis, industrial robotics, urban traffic flows, financial markets, and wireless communications, among many others. The main goal of this project is to advance the geometric understanding of four-dimensional objects as abstract mathematical entities, an approach that is general enough to allow applications to any field that makes use of four-dimensional models. In particular, this project will analyze how rigid or malleable certain four-dimensional shapes are, how that changes under certain natural curvature assumptions, and how to efficiently detect these curvature properties. A defining characteristic of the research to be conducted is the use of cutting-edge techniques recently developed in areas of mathematics not traditionally associated with geometry, bringing new perspectives to a classical subject, in a disruptive attempt to solve some of its most important open questions. This project will also partially support the CUNY Geometric Analysis seminar, and its educational mission in the CUNY graduate program, through the communication of latest research advances in the field; as well as several public outreach activities at CUNY Lehman College, a Hispanic Serving Institution in the Bronx borough of New York City. In more technical terms, this project will pursue various applications of the emerging fields of Convex Algebraic Geometry and Semidefinite Programming to Geometric Analysis and Riemannian Geometry, for instance through the study of semialgebraic sets of curvature operators of four-manifolds with sectional curvature bounds as spectrahedral shadows and limits of spectrahedra. This real algebro-geometric viewpoint on curvature operators of four-manifolds is expected to have several global consequences in the form of explicit algebraic (polynomial) characterizations of sectional curvature bounds, rank rigidity results, and optimization of curvature estimates. Furthermore, the project seeks to detect new topological obstructions to positive curvature in four-manifolds using the aforementioned techniques and trisections of their fundamental groups. Other topics covered include four-dimensional Einstein manifolds, Ricci solitons, and Ricci flow. Most of the research will be conducted in collaboration with doctoral students, early-career mathematicians, and other researchers, partially supporting the training of new specialists in the field.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.