Final exam group 21, Thursday 12/18/03, 8-11am. I realize this is the last day of finals, and NO, you can not take the final early, so don't even ask. To repeat: there's still time to drop the class, if you can't bear to wait for this final.

Aug 26	A smooth function on X is one that extends to an open set. Definition of topologies, diffeomorphism, smooth manifold. Rather a lot of definitions today (it won't stay like this!), as I wanted to get to the definition of smooth manifold.
Aug 28	Lightning review of multivariable calculus: the derivative of a smooth function between vector spaces V and W, at some point of V, is naturally a linear map from V to W. The chain rule. Used this to define the tangent space to a smooth manifold, and checked the independence of chart used.
Sep 2	There exists a smooth function that is 0 for x<0 and 1 for x>1. Definition of the tangent bundle. Definition of the derivative of a smooth map.
Sep 4	Tangent spaces to k-manifolds are indeed k-dimensional. Statement of inverse function theorem on R^n, and the version it implies for manifolds.
Sep 9	Definition of immersion, and a first pass at the local immersion theorem.
Sep 11	"T" is a functor from the category of pointed manifolds to the category of vector spaces. The local immersion theorem.
Sep 16	The local submersion theorem, and the preimage theorem. Regular points, regular values. Definition of "product" in the categorical sense.
Sep 23	The group O(m,n) is a manifold. Sard's theorem. The map Hermitians -> characteristic poly is regular whenever H has distinct eigenvalues. In those cases, the fibers are flag manifolds.
Sep 25	Flag manifolds are complex. Enough Morse theory for Tara Holm's talk.
Sep 30	A survey of manifolds in 0,1,2,3,4 dimensions.
Oct 2	A function plus a random functional is a Morse function (p43). If a family of maps is transverse to something, then almost any given map in the family is also transverse to that something (p68, without boundaries).
Oct 7	General position lemma (p74, #4). Conormal bundles, which are locally diffeomorphic to a neighborhood of the manifold (using an inner product).
Oct 9	For compact X in V, there exists epsilon so N_epsilon X -> V is 1:1 and a submersion (p69). Using this, we get the transversality homotopy theorem (p70), without boundary. Partitions of unity (p54). Definition of manifolds-with-boundary.
Oct 14	The Brouwer fixed-point theorem. Transversality theorem with boundary and with extension. Mod 2 intersection numbers are invariant under homotopy, defined even for nontransverse intersections, and cobordism invariant not just homotopy invariant.
Oct 16	Cobordism, mod 2 degree, 1/2 fundamental theorem of algebra.
Oct 21	Winding numbers, orientations.
Oct 23	Degree, winding number, rest of FTAlgebra. Lefschetz number, Euler characteristic (especially of surfaces). A hint of its relation to degree.

Coming up next: more on Lefschetz maps and numbers.