PhD in Mathematics - Geometry in Santa Barbara United States | University of California, Santa Barbara

The Geometry Group within UCSB's Mathematics Department focuses primarily on Differential Geometry, while also encompassing two key interconnected disciplines: Mathematical Physics and aspects of Algebraic Geometry.

At its foundation, Differential Geometry explores Riemannian Geometry, Global Analysis, and Geometric Analysis. Riemannian geometry particularly examines how curvature and topology interact within Riemannian manifolds and spaces. A classic illustration is the Bonnet-Myers theorem, proving that complete Riemannian manifolds with uniformly positive Ricci curvature are necessarily compact with finite fundamental groups. Meanwhile, Global Analysis investigates analytic structures on manifolds and their connections to geometric and topological properties. The renowned Atiyah-Singer index theorem, for instance, links the index of elliptic operators (an analytic measure) to characteristic classes (topological invariants) of the manifold. Geometric Analysis merges geometric methods with analytic techniques like PDEs, geometric measure theory, and functional analysis to tackle nonlinear geometric and topological challenges. A prime example is Hamilton's Ricci flow, which has led to groundbreaking advances, including Perelman's resolution of the Poincare Conjecture and the Geometrization Conjecture for 3-manifolds. The Geometry Group's research spans various topics in these areas, such as Ricci curvature-bounded manifolds, minimal surfaces in Riemannian spaces, Einstein manifolds, index theory and eta invariants, Ricci flow dynamics, pseudo-holomorphic curves in symplectic geometry, and Seiberg-Witten invariants in four-dimensional manifold topology.