Geometry

Geometry

The Geometry Group of the Mathematics Department at UCSB has Differential Geometry as its core part, and includes two important related fields: Mathematical Physics, and part of Algebraic Geometry in the department.

The core part, Differential Geometry, covers Riemannian Geometry, Global Analysis and Geometric Analysis. A central topic in Riemannian geometry is the interplay between curvature and topology of Riemannian manifolds and spaces. A well-known example is the classical Bonnet-Myers theorem which states that a complete Riemannian manifold of uniformly positive Ricci curvature must be compact and have a finite fundamental group. Global analysis, on the other hand, studies analytic structures on manifolds and explores their relations with geometric and topological invariants. For example, the celebrated Atiyah-Singer index theorem establishes the relation between the index of elliptic operators-an analytic quantity, and characteristic classes of the underlying manifold which are topological invariants. Finally, geometric analysis combines geometric tools with analytic tools such as PDE, geometric measure theory and functional analysis in geometric contexts to study geometric and topological problems which are often nonlinear. An important example is Hamilton's Ricci flow. Recently, spectacular results in geometry and topology were achieved by employing the Ricci flow. These include Perelman's seminal work on the Poincare Conjecture and the Geometrization Conjecture for 3-manifolds. The research of the Geometry Group covers diverse topics in Riemannian geometry, Global analysis and Geometric Analysis, such as manifolds with lower bounds on the Ricci curvature, minimal surfaces in Riemannian manifolds, Einstein manifolds, the index theory and the eta invariants, Ricci flow, pseudo-holomorphic curves in symplectic geometry, and Seiberg-Witten invariants in the theory of the topology of 4-dimensional manifolds.

The research of the Geometry Group in Mathematical Physics covers various topics such as knot and link homologies, gauge theory, Chern-Simons theory, Calabi-Yau spaces, D-branes, mirror symmetry, the positive mass theorem in general relativity, and constant mean curvature foliations on asymptotically flat manifolds.

The research of the Geometry Group in Algebraic Geometry covers various topics such as mirror symmetry, Calabi-Yau spaces, the minimal models, moduli spaces, and the Kahler-Ricci flow.

Interactions between the various directions of Riemannian Geometry, Global Analysis, Geometric Analysis, Mathemtical Physics and Algebraic Geometry play an important role in the research of the Geometric Group. Interactions with other groups of the Mathematics Department, the Physics Department and KITP play an equally important role.

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