Expository Work

  •  Riemannian Geometry, 3rd edition,  New York: Springer-Verlag, GTM 171, 2016.
  • Linear Algebra, UTM, Springer-Verlag, 2012. With additional comments and notes Linear Algebra Notes.
  • A growing essay on manifold theory, de Rham cohomology and more for first year graduate students.
  • Counting methods in combinatorics and handout on max flow min cut.
  • An elementary exposition on warped products (updated 2.17.10).
  • Notes on Differential Geometry for undergraduates.


My research is on various aspects of Riemannian Geometry and recent papers can be found on ArXiv through the link below.
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.

PApers on mathscinet        Preprints on Arxiv