Spring 2013, MWF 12-1, Science Center 222
My name: Andrew Cotton-Clay (please call me Andy)
Office: Science Center 527
Office Hours: Wed 1:15-2:15 and Thu 2:45-3:45 or by appointment.
E-mail: acotton at math
TF: Michael Jemison, jemison at college
Syllabus: Math 132 Syllabus
Course iSite: Math 132 iSite
Due Feb 8 (tex)
Due Feb 15 (tex)
Due Feb 22 (tex)
Due Mar 1 (tex)
Midterm Practice Problems (solutions)
Material for midterm: [GP] Chapter 1 plus the Regular Value Theorem /
Transversality Theorem (without boundary) from 2.3. That is, material we
covered before manifolds with boundary. From Appendix 1 on Sard's Theorem,
you should know definitions and statements but not proofs.
Due Mar 29
Due Apr 5
Due Apr 12 (or Apr 15 when I get back
if you prefer)
Due Apr 22
Due Apr 29
Mon 1/28: Section 1.1 of [GP]. See also our figure
eight curve, its
resolution in R^3, and our cross cap. If you take the expressions and enter them in an
actual copy of Mathematica, you'll be able to rotate the figures around.
Here's a slightly
less bulbous cross cap viewed from a better angle. For our cross cap,
the fourth coordinate in the resolution is Sin[2t]. To see this in
Mathematica, you can make it a color. Namely, try
ParametricPlot3D[{(2-Cos[2t])*Sin[s]*Cos[t], (2-Cos[2t])*Sin[s]*Sin[t],
(2-Cos[2t])*(1-Cos[s])}, {s,0,Pi}, {t,0,2*Pi}, ColorFunctionScaling ->
False, ColorFunction -> Function[{x,y,z,u,v}, Hue[(1+Sin[2v])/4+1/4]]].
Here I've rescaled and shifted the fourth coordinate a bit to get it to
play nicely with Mathematica's Hue function.
Wed 1/30: Proof that continuous partials implies differentiable. Section
1.2 of [GP] and section 1.3 through the proof of the local immersion
theorem. Namely, derivatives, tangent spaces, inverse function theorem
statement, immersions, and standard form for immersions.
Fri 2/1: Section 1.3 of [GP]: Recap of local immersion theorem; examples
of immersions; definition of proper; definition and theorem regarding
embeddings. Also a generalization of the embedding theorem.
Mon 2/4: Section 1.4 of [GP]: Submersions; local submersion theorem;
regular and critical values; preimage theorem; examples of the preimage
theorem; cutting out smooth submanifolds with independent functions to R.
Wed 2/6: More on section 1.4 of [GP]. O(2) as cut out by equations. Lie
groups and GL_n(R). O(n) is smooth of dimension n(n-1)/2. Stack of records
theorem (see [GP, Problem 1.4.7] or Milnor, page 8). Hasty sketch of the
Fundamental Theorem of Algebra (see Milnor, pp. 8-9; [GP] gives a
different proof later on).
Mon 2/11: Section 1.5 and part of 1.6 of [GP]: Transversality of a map and
a submanifold, and of two submanifolds. Homotopy of smooth maps and
composition of homotopies. Preliminaries for a stability theorem.
Wed 2/13: Rest of 1.6, part of 1.7 of [GP]: Stability theorem. Statement
of Sard's theorem and immediate corollaries. Application to homotopy of
maps from S^1 to S^2.
Fri 2/15: Rest of 1.7 of [GP]: Morse functions. The hessian is
well-defined up to a change of basis at critical points. Beginning of the
application of Sard's theorem to the existence of Morse functions.
Wed 2/20: Appendix 1 of [GP]: Measure zero sets on manifolds. Fubini's
theorem for measure zero sets.
Fri 2/22: Appendix 1 of [GP]: Proof of Sard's theorem. General application
to the "regular value theorem" (see the Transversality Theorem in section
2.3 of [GP] for the more general version which we'll cover later).
Mon 2/25: Finished the application of Sard's theorem to the existence of
Morse functions (1.7); Whitney embedding theorem, compact case (1.8).
Wed 2/27: Partitions of unity. Whitney embedding theorem. Abstract
manifolds. (1.8)
Fri 3/1: Manifolds with boundary (2.1).
Mon 3/4: Retraction and Brouwer fixed point theorems (2.2), transversality
theorem, epsilon neighborhood theorem (statement) (2.3).