Expository Work

Except for my linear algebra book these are all somewhat unfinished handouts that I try to update from time to time. In the near future I'll have a new handout on undergraduate differential geometry.

• A growing essay on manifold theory, de Rham cohomology and more.
• Counting methods in combinatorics and handout on max flow min cut
  
• A book on Linear Algebra, to appear in UTM, Springer-Verlag, 2012.
• Errata (updated 8.22.12) and additional proofs for 2^nd edition of Riemannian Geometry, New York: Springer-Verlag, GTM 171, 2006.
• An elementary exposition on warped products (updated 2.17.10).
• An exposition on Bochner-Lichnerowicz-Weitzenböck Formulas (12.1.11).
• The Ricci and Bianchi Identities (12.14.11).
• Rough Lin Alg Notes
  

Research

My research is on various aspects of Riemannian Geometry. Recently Chenxu He, Will Wylie and I completed four papers on Einstein metrics that are also warped product. The papers on the Gromoll-Meyer sphere are not up to date. We are in the process of writing one unified paper that includes both the general principles as well as the actual construction. The revised paper should be more palatable as we hope to make the deformation preserve non-negative sectional curvature. We are also simplifying several of the underlying general principles.

• Warped Product Einstein Manifolds with Chenxu He and William Wylie
• Warped Product Rigidity with Chenxu He and William Wylie
• Warped product Einstein metrics over spaces with constant scalar curvature with Chenxu He and William Wylie
  
• On the classification of warped product Einstein metrics with Chenxu He and William Wylie
• Principles for Deforming Nonnegative Curvature (updated 2.17.10) with Fred Wilhelm
• On the Set of Images Modulo Viewpoint and Contrast Changes with Ganesh Sundaramoorthi and Stefano Soatto
• An Exotic Sphere with Positive Sectional Curvature (will be updated this spring) with Fred Wilhelm. (Fred’s beamer presentation)

PApers on mathscinet        Preprints on Arxiv