Math 865 Advanced Topics in Geometry (Fall 2007)
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Lectures
Lectures are by
Jeff Viaclovsky on Tuesdays and Thursdays
at 9:30-10:45 AM in Van Vleck B131.
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Recommended Textbooks
- The Ricci Flow: An Introduction by Chow and Knopf
- Riemannian Geometry by Peter Petersen.
- Differential Geometric Structures by Poor.
- Einstein Manifolds by Besse.
- Hamilton's Ricci Flow by Chow, Lu, and Ni.
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Mailing List
The class mailing list is math865-1-f07 "at" lists.wisc.edu.
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Lectures
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- Lecture 1: Tuesday, September 4
- Riemannian metrics, vectors, and one-forms.
- The musical isomorphisms.
- Inner product on tensor bundles.
- Connections on vector bundles.
- Covariant derivative of tensor fields.
- Gradient and Hessian operators.
- Lecture 2: Thursday, September 6
- Curvature of connections on vector bundles.
- Curvature of the Riemannian connection.
- Sectional curvature, Ricci tensor, and scalar curvature.
- Lecture 3: Tuesday, September 11
- Differential Bianchi Identity.
- Algebraic study of the curvature tensor.
- For n = 3, Ricci determines full curvature tensor.
- Lecture 4: Thursday, September 13
- Orthogonal decomposition of the curvature tensor.
- The curvature operator.
- Curvature in dimension 3.
- Lecture 5: Tuesday, September 18
- Covariant derivatives redux.
- Commuting covariant derivatives.
- Rough Laplacian and gradient.
- Lecture 6: Thursday, September 20
- Commuting Hessian and Laplacian.
- An application to PDE.
- Lecture 7: Tuesday, September 25
- Integration and adjoints.
- Formula for Hodge d and \delta operators.
- Lecture 8: Thursday, September 27
- Bochner and Weitzenbock formulae.
- Lecture 9: Tuesday, October 2
- Manifolds with positive curvature operator.
- Lecture 10: Thursday, October 4
- Isometries and Killing fields.
- Lecture 11: Tuesday, October 9
- Linearization of the Ricci tensor.
- The total scalar curvature functional.
- Lecture 12: Thursday, October 11
- Ricci Flow: short time existence.
- Lecture 13: Tuesday, October 16
- Uniqueness.
- Linear parabolic systems.
- Quasilinear parabolic systems.
- Lecture 14: Tuesday, October 21
- Maximum principles for scalar parabolic equations.
- Lecture 15: Thursday, October 23
- Evolution of scalar curvature under the Ricci flow.
- Einstein metrics.
- Normalized versus unnormalized flow.
- Evolution of scalar curvature under normalized flow.
- Lecture 16: Tuesday, October 28
- Parabolic maximum principles for tensors.
- Evolution of Ricci tensor under Ricci flow.
- Lecture 17: Thursday, November 1
- Evolution of curvature tensor under Ricci flow.
- Lecture 18: Tuesday, November 6
- Evolution of curvature tensor.
- Uhlenbeck method.
- Square of curvature operator.
- Lecture 19: Thursday, November 8
- Lie algebra square.
- Dimension 3.
- Lecture 20: Tuesday, November 13
- Conformal geometry.
- Negative scalar curvature.
- Lecture 21: Thursday, November 15
- The Yamabe Problem.
- Constant curvature.
- Obata Theorem.
- Differential Bianchi for Weyl.
- Lecture 22: Tuesday, November 20
- Conformal flatness.
- Examples.
- Lecture 23: Tuesday, November 27
- Some conformal invariants.
- Weitzenbock formula revisited.
- Lecture 24: Thursday, November 29
- Laplacian of Schouten.
- Yamambe flow.
- Lecture 25: Tuesday, December 4
- Curvature in dimension 4.
- Lecture 26: Thursday, December 6
- Some representation theory in dimension 4.
- Lecture 27: Tuesday, December 11
- TBA.
- Lecture 28: Thursday, December 13
- TBA.