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Manifold theory, with its emphasis on global geometry, has become much more important in many branches of both pure and applied mathematics in the last quarter of a century. It is the backbone of any further study in differential geometry, and it has now reached the status of being a standard course in the graduate curriculum. It is good preparation for the geometry part of the Geometry/Topology qualifying exam in our graduate program. Thus, the purpose of this course is to provide the student with the foundations for the study of modern differential geometry. It is assumed that the student has a basic understanding of linear algebra, group theory, advanced calculus, and point set topology and elementary homotopy theory through the fundamental group. I will review some basic topological notions as we need them in the course. I hope to be able to cover the material in Lee's book up through Stokes Theorem on manifolds, but skipping Chapter 4 on Curves and Hypersurfaces in Euclidean Space. Chapters 5 and 6 will be touched on sparringly according to our needs. I also have some old notes that I follow, and part of which I may distribute as supplemental reading. Here are my very old typed notes which have many typos for which I apologize, notes1; notes2; notes3; notes4; notes5 (these are new); notes6; ; notes7; ; notes8; ; notes9; notes10; notes11. Some additional notes: here.
You can get a pdf file of the course Syllabus here.
You can see homework assignments here.
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