Math 545/6, Winter/Spring 2008 - Homework Assignments

Reminders:

• Homework always includes working out all the exercises in the text for yourself, even though you don't have to turn them in.
• In each Reading Report, please address the following questions.
  • What do you think was the most important idea (or one of the most important ideas) in the reading, and why?
  • What is another idea in the reading that you thought was important or interesting, and why?
  • What questions do you have about the reading, or what parts of it would you most like to see addressed in class?

Current assignments listed here. Previous assignments.

• Week 9 (Sp). Reading report due by noon, Tuesday, 5/27 on Chapter 18. You may omit the sections on the orientation covering, pp. 451-455, and densities, pp. 470-475. I will discuss the former briefly in class. If you are in interested in analysis on manifolds, you should probably read the section on densities sometime.
  Problems to work out for yourself: Problems 6-12, 18-1, 18-2, 18-3, and 18-4.
  Written homework, due Friday, 5/30: The problems below and Problems 16-11, 16-13, & 18-10.
  1) We skipped problems earlier about complex projective space because now we can construct its smooth manifold structure more easily. Consider the action of C \ {0}, the nonzero complex numbers, on Cⁿ⁺¹ \ {0} by multiplication. Show that the orbit space is a topological manifold with a unique smooth structure such that the quotient map is a smooth submersion. We will define the complex projective space CPⁿ to be the orbit space with this smooth structure.
  2) (A special case of Problem 16-10.) Let X be the set of type (1,3) flags in R⁴. Show that Proposition 16.19 may be used to give X the structure of a homogeneous space. What is its dimension?


• Week 10 (Sp). Reading report due by noon, Tuesday, 6/3 on Chapter 19. You may omit the section on the Riemannian density, pp. 492-494. Optional; continue on to read part or all of Chapter 20.
  Problems to work out for yourself: Problems 18-9 (notice it follows easily from 18-10(b)), 18-12, & 18-13.
  If you are interested in analysis on manifolds (PDE's, math physics, etc.), you should read all the problems 19-6 through 19-16 sometime. (Perhaps include them in the Riemannian reading course next year?)
  Written homework, due Friday, 6/6: Problems 18-17, 19-1, 19-2 with addition below, 19-4, & 19-5.
  Addition to 19-2: (c) In R³, let E be the ellipsoid given by (x-1)²/4 + (y-2)²/9 + z² = 1, oriented by the outward normal vector and the standard orientation on R³. Evaluate the integral of the form from 19-2 over E.


• Spring Quarter Final Exam.

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