|To apply for Funding you must register by:||April 01, 2003 about 10 years ago|
|Parent Program:||Differential Geometry|
- Background material: Complex manifolds, Hermitian differential geometry, Dolbeault cohomology, vanishing theorems, deformation theory, moduli.
- Applications: Kahler-Einstein metrics, Abelian varieties and integrable systems, enumeration problems (counting rational curves), the geometry of moduli spaces.
- Background material: Basic exterior algebra, norms, calibrations, the fundamental lemma of calibration theory, minimizing cycles, basics of geometric measure theory, singularities, regularity and compactness, moduli spaces.
- Important examples and applications: Complex subvarieties and Wirtinger's theorem; special Lagrangian cycles and mirror symmetry; complex Lagrangian cycles and integrable systems; associative, co-associative, and Cayley cycles and string theory.
- Background material. Riemannian holonomy, de Rham splitting, examples (locally symmetric), Berger's classiffcation, the holonomy principle (parallel forms and spinor fields), explicit examples (Kahler, hyperKahler, and exceptional constructions). Local nature of the problem.
- Construction techniques: Calabi-Yau spaces, HyperKahler spaces, reduction, constructions of compact G2 and Spin(7) examples.
- Moduli: Refinements of the de Rham complex, vanishing theorems, deformations and relative deformation problems.
- Background Material: Basic differential geometry, vector bundles, Hermitian metrics, connections, curvature, Chern-Weil theory, Yang-Mills equation.
- Applications: Anti-self-dual instantons and string theory, properties of their moduli spaces, Seiberg-Witten invariants, gauge theory invariants, holomorphic Casson invariants, relation to calibrated cycles.
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