Connection Geometry SeminarWe assume a good introduction to smooth manifolds, bundles, and groups, as found in my book-in-progress that began as the lecture notes for my 2011--12 seminars. Review of necessary material will be provided as needed. Dodson's little book is also a handy reference. There is also an appendix that may be integrated into the book, Manifolds in Fluid Dynamics by J. Ryan. In addition to the exercises in my book, there are also some good ones in Bröcker & Jänich and in Warner, even though some of his notation is archaic, to put it kindly.So far, the main sources are DRP5 and P10. The lectures are LaTeXed by J. Ryan and edited by me. Draft lecture notes are available here, updated as deemed appropriate. Only the most basic elements of categories and functors are assumed (much less than found in 713 or Dodson's little book, for example). Topology at the 725-level is essential. Algebra and analysis prerequisites are about at the level of 513 (M & B) and 640 (Bartle); the most essential things are groups, group actions, and differential analysis in real n-space. Group actions are covered thoroughly in the book. Remarks and comments therein, however, may assume much more: homotopy, (co)homology, nuclear LCTVSs, topological function spaces, etc. The seminar is open to anyone. If you become interested in presenting, please check with me before planning one to avoid duplication or scheduling conflicts (at least). Grad students wishing to obtain credit may enroll in a Math 828/9 by Blue Card (see Terri Griffith). All participants for credit will be required to contribute substantively; see me to discuss/negotiate. |
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Splitting Tangent Bundles Seminar[Parker and Walsh] A seminar on the existence of splittings of the tangent bundle TM of a manifold M. Our first goal is to go through two papers of E. Thomas (paper1, paper2). In order to prepare for this, we first review some necessary algebraic topology, starting with some old notes by Parker. (With many additions and more details, they grew into my book with Dodson.)Parker's interest is that a splitting TM = E p + E q, where p and q are fiber dimensions, is equivalent to the existence of a pseudoRiemannian metric tensor of signature (p,q) on M. A solution is known only when p or q equals 1: a Lorentzian manifold, alias a spacetime in physics. Such a splitting is also a necessary condition for the existence of a codimension p or q foliation of M. Walsh's interest lies in better understanding the topology of the total space of the tangent bundles to the even-dimensional spheres S 2n. These objects are used in the construction of "exotic" metrics of positive scalar curvature. Note that in even dimensions, the tangent bundle to the sphere is nontrivial; in fact, it does not admit even one nonvanishing vector field. Graduate students are encouraged to attend; we will tailor to the audience. |