Differential Geometry

Differential geometry is one of the classical, core disciplines of mathematics. Its primary objects of study are smooth manifolds, which are simply the subsets in the Euclidean spaces to which calculus applies. 

Examples include smooth surfaces and their higher dimensional analogues. One main objective is to understand and classify their topological, geometric and analytical structures. The recent classification of three dimensional manifolds is considered as one of the most significant achievements in mathematics.

Differential geometry is deeply connected to many other mathematical areas such as topology, analysis, algebraic geometry and number theory. It also interacts closely with theoretical physics, from general relativity to string theory. It has wide-ranging applications throughout mathematics, science, and engineering as well.

Research topics include

• Low dimensional topology
• Symplectic topology
• Contact topology 
• Poisson geometry
• Riemannian geometry
• Geometric analysis
• General relativity
• Geometric dynamical systems
• Foliations
• Ergodic theory
• Hyperbolic groups
• Trees
• Teichmuller spaces
• Hyperbolic geometry
• Fuchsian and Kleinian groups
• Complex dynamics
• Lie groups and pseudo-groups
• Moving frames
• Cartan connections
• Equivalence and symmetry
• The variational bicomplex
• Exterior differential systems
• Applications to image processing, materials science, and anthropology