# Mathematical Sciences Research Institute

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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 over 19 years ago May 11, 2003 almost 20 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)

## Show List of Speakers

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
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Schedule, Notes/Handouts & Videos
Show Schedule, Notes/Handouts & Videos
Aug 11, 2003
Monday
 08:00 AM - 09:00 AM Morning Tea 09:00 AM - 10:00 AM Welcome, Orientation, Announcements, Resigstration Andrew Lucas 10:00 AM - 10:30 AM Break 10:30 AM - 12:00 PM Holonomy 1 Robert Bryant (Duke University) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Complex Geometry 1 Zhiqin Lu (University of California, Irvine) 03:00 PM - 03:30 PM Afternoon Tea 03:30 PM - 04:30 PM Gauge Theory 1 Richard Thomas (Imperial College, London)
Aug 12, 2003
Tuesday
 08:00 AM - 09:00 AM Morning Tea 09:00 AM - 10:00 AM Holonomy 2 Robert Bryant (Duke University) 10:00 AM - 10:30 AM Break 10:30 AM - 12:00 PM Calibrations 1 H. Blaine Lawson (State University of New York, Stony Brook) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Complex Geometry 2 Zhiqin Lu (University of California, Irvine) 03:00 PM - 03:30 PM Afternoon Tea 03:30 PM - 04:30 PM Gauge Theory 2 Richard Thomas (Imperial College, London)
Aug 13, 2003
Wednesday
 08:00 AM - 09:00 AM Morning Tea 09:00 AM - 10:00 AM Holonomy 3 Robert Bryant (Duke University) 10:00 AM - 10:30 AM Break 10:30 AM - 12:00 PM Calibrations 2 H. Blaine Lawson (State University of New York, Stony Brook) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Complex Geometry 3 Zhiqin Lu (University of California, Irvine) 03:00 PM - 03:30 PM Afternoon Tea 03:30 PM - 04:30 PM Gauge Theory 3 Charles Thomas
Aug 14, 2003
Thursday
 08:00 AM - 09:00 AM Morning Tea 09:00 AM - 10:00 AM Holonomy 4 Robert Bryant (Duke University) 10:00 AM - 10:30 AM Break 10:30 AM - 12:00 PM Calibrations 3 H. Blaine Lawson (State University of New York, Stony Brook) 12:00 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Complex Geometry 4 Zhiqin Lu (University of California, Irvine) 03:00 PM - 03:30 PM Afternoon Tea 03:30 PM - 04:30 PM Gauge Theory 4 Charles Thomas
Aug 15, 2003
Friday
 08:00 AM - 09:00 AM Morning Tea 09:00 AM - 10:00 AM Kahler metrics and holomorphic foliations Gang Tian (Princeton University) 10:00 AM - 10:30 AM Break 10:30 AM - 11:30 AM Calibrated submanifolds with boundary: compact special Legendrian surfaces in $$S^5$$ Sung Wang 11:30 AM - 02:00 PM Lunch 02:00 PM - 03:00 PM Representations of surface groups and Hermitian symetric spaces Oscar Garcia-Prada (Consejo Superior de Investigaciones Científicas (CSIC)) 03:00 PM - 03:30 PM Afternoon Tea 03:30 PM - 04:30 PM The geometric complexity of special Lagrangian cones Mark Haskins (Duke University)
Aug 16, 2003
Saturday
 08:00 AM - 09:00 AM Morning Tea 09:00 AM - 10:00 AM Closed ${\rm G}_2$ structures Robert Bryant (Duke University) 10:00 AM - 10:30 AM Break 10:30 AM - 11:30 AM Crepant resolutions of Calabi-Yau orbifolds Anda Degeratu (Universität Stuttgart) 11:30 AM - 02:00 PM Lunch 02:00 PM - 03:00 PM Second order families of special Lagrangian 4-folds Eleny-Nicoleta Ionel (Stanford University) 03:00 PM - 03:30 PM Afternoon Tea 03:30 PM - 04:30 PM Deformations of Sturctures on Manifolds Spiro Karigiannis (University of Waterloo)
Aug 18, 2003
Monday
 08:00 AM - 09:00 AM Morning Tea 09:00 AM - 10:00 AM The geometry of representations of surface groups in $$SL(n,R)$$ François Labourie (Universite de Nice Sophia Antipolis) 10:00 AM - 10:30 AM Break 10:30 AM - 11:30 AM Instantos and Lagrangians on manifolds with vector cross Naichung Leung 11:30 AM - 02:00 PM Lunch 02:00 PM - 03:00 PM Flat Lagrangian submanifolds in $$C^n$$ and $$CP^n$$ xiaowei wang (Rutgers University) 03:00 PM - 03:30 PM Afternoon Tea 03:30 PM - 04:30 PM Constructing $${\rm G}_2$$-metrics from nearly K\"ahler manifolds Shengli Kong
Aug 19, 2003
Tuesday
 08:00 AM - 09:00 AM Morning Tea 09:00 AM - 10:00 AM On a nonlinear Dirac equation and constant mean curvature surfaces Bernd Ammann (Universität Regensburg) 10:00 AM - 10:30 AM Break 10:30 AM - 11:30 AM Examples of associative, co-associative, and Cayley submanifolds Weiqing Gu 11:30 AM - 02:00 PM Lunch 02:00 PM - 03:00 PM Volume minimizing cycles in compact Lie groups Xiaobo Liu 03:00 PM - 03:30 PM Afternoon Tea 03:30 PM - 04:30 PM Constant scalar curvature metrics and the stability of Constant scalar curvature metrics and the stability of algebraic varieties Richard Thomas (Imperial College, London)
Aug 20, 2003
Wednesday
 08:00 AM - 09:00 AM Morning Tea 09:00 AM - 10:00 AM Minimal Lagrangian tori in CP^2 come in real families of every dimension Emma Carberry 10:00 AM - 10:30 AM Break 10:30 AM - 11:30 AM Generalized Hodge metrics and BCOV torsion Zhiqin Lu (University of California, Irvine)