Notes

Over the years I have been given the opportunity to lecture on a variety of topics in differential geometry, analysis, and topology. The quality of the notes varies, however they are written with the intent of being read by others. These notes are actively being edited and expanded on so please send me an email if you have any questions, remarks, comments, or corrections you would like to make!

(Differential Geometry and Characteristic Classes) These are a collection of notes that I have written which give detailed explanations on why particular characteristic differential forms always give integer multiples of π. These facts are well known and there are a variety of ways of proving such theorems in the literature. Our approach is to use some modern technology to simplify Chern's original proof of integrality of the euler class and Chern classes. The notes on the Chern forms is, to the best of our knowledge, a new proof of this fact.

(Hodge Theory on Manifolds) These are a complete set of notes written by Sergio Zamora Barrera which are based on a set of lectures I gave at Penn State the topic of which was to prove the elliptic regularity theorem for elliptic differential operators. We prove the theorem for arbitrary bundle coefficients and derive the generalized Hodge theorem which states that for any ellitpic differential operator on a closed manifold, the kernel is finite dimensional. The approach is a mixture of what is in J. Roe "Elliptic operators, topology, and asymptotic methods" and in F. W. Warner "Foundations of Differentiable Manifolds and Lie Groups".