NSF-BSF: Geometric Inequalities Involving Monge Ampere Operators

Many mathematical and scientific questions concern how the shape of an object influences what can occur inside it or along its boundary. This project, supported under the US-Israel Binational Science Foundation (BSF) Collaborative Research Opportunities, will study that relationship for an important family of equations known as Monge–Ampère equations, which arise in geometry, analysis, and mathematical physics. The investigators will seek general principles describing how these equations change when shapes are enlarged, combined, symmetrized, or replaced by related shapes. The project will also support graduate student training through research, seminars, and collaboration among mathematicians in the United States and Israel. These activities will advance fundamental mathematical knowledge and help prepare students for careers requiring advanced analytical and problem-solving skills. The project will investigate geometric and analytic properties of principal eigenvalues associated with Monge–Ampère operators. The primary focus will be the real setting, where the investigators will study Brunn–Minkowski, isoperimetric, Rogers–Shephard, and reverse Brunn–Minkowski-type inequalities, as well as duality and affine-invariant eigenvalue products for convex bodies. The proposed methods will draw on convex geometry, harmonic and geometric analysis, nonlinear partial differential equations, symmetrization, shadow systems, and the asymptotic theory of normed spaces. The investigators will also pursue selected foundational questions for complex, quaternionic, and octonionic Monge–Ampère operators, including existence, uniqueness, regularity, variational formulations, and extensions of geometric inequalities. The anticipated results will provide a more unified framework connecting convex geometry, nonlinear partial differential equations, and geometric functional analysis. 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.