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Workshop

Von Neumann Symposium on Complex Geometry, Calibrations, and Special Holonomy August 11, 2003 - August 20, 2003
Registration Deadline: August 20, 2003 almost 11 years ago
To apply for Funding you must register by: May 11, 2003 about 11 years ago
Parent Program: Differential Geometry
Organizers Robert Bryant (Co-chair), Simon Donaldson, H. Blaine Lawson, Richard Schoen, and Gang Tian (Co-chair)
Speaker(s)

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Description

TO REGISTER PLEASE SEE: http://www.ams.org/meetings/vonneumann03.html The last day the AMS will accept registrations for this workshop is April 1, 2003. The focus of the symposium will be on an introduction to the subjects of the title, i.e., ideas and tools being developed in differential geometry in response to the challenges posed in modern mathematical physics, particularly string theory. The symposium will present reviews of current mathematical developments including many special talks appropriate for non-experts (especially students and postdoctorals). These will take the form of interrelated minicourses supplemented by individual up-to-date research lectures. The four mini-course topics and speakers are: Complex Geometry (Zhiqin Lu)

  • 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.

Calibrations and Calibrated Cycles (Blaine Lawson)

  • 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.

Special Holonomy (Robert Bryant)

  • 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.

Gauge Theory (Richard Thomas)

  • 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.

This workshop is co-sponsored by the American Mathematical Society. TO REGISTER PLEASE SEE: http://www.ams.org/meetings/vonneumann03.html Deadline: April 1, 2003 There will also be a one-week workshop to be held at the Banff Conference Center before the von Neumann Symposium. The purpose of this workshop will be to allow a more leisurely introduction to the background material of the workshop for the benefit of graduate students and postdoctorals who are interested in attending the Symposium in Berkeley. Please see for details: Preparatory Workshop for the 2003 AMS/MSRI von Neumann Symposium


Schedule
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