Topology and Geometry: I sketch the content of an independent study I co-taught with Honore Mavinga in the Spring 2012 semester. I am willing to do independent studies in manifold theory, differential geometry, Lie Algebra, representation theory,... there are really many topics you could talk me into here. The idea here is not that I teach the course as I would an ordinary course. The student is expected to read ahead and beside, to fill in cracks, to prove theorems, to excel. This is the purpose of the independent study.

In the case of this independent study, the student came into this course already having completed advanced calculus as well as about a half-semester of manifold theory. Past some point it is hard to continue in manifold theory without developing some maturity in abstract topology. The purpose of this course was two-fold. First, initiate the student in topological theory. Second, continue the study of manifolds with the aim of seeing at least one nontrivial global result.

Dr. Mavinga taught a student from Willard's Topology text which is available as a Dover book. Topics included: definition of topology, continuity, homeomorphism, connectedness, compact sets, filters, law of exponents, basic homotopy and various theorems of abstract point-set topology.

I taught a student from Conlon's Differentiable Manifolds, second edition. We reviewed basic global definitions of manifolds such as coordinate charts, atlas, maximal atlas, transition functions etc. We also studied the tangent bundle and how to understand vector fields as sections of the tangent bundle. An overview of topological manifolds as laid out in the early chapters of Conlon was given. More attention was given to flows and foliations. In particular, we sought to understand Frobenius Theorem. In summary, we covered a reasonable selection of topics from Chapters 1-5 during the semester.

After the semester's end the student returned once more to read further in Conlon. We also read from Burns and Gidea. Our main focus was Riemannian geometry. Basic theorems about metrics and geodesics were considered. We derived some of the standard formulas for the Riemann and Ricci curvature tensors. At the conclusion of our work we calculated the structure equations of Cartan. Our next step is to continue the study to homogeneous spaces, fiber bundles and connections.

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Last Modified: 8-17-12