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Math 411

General information

Spring 2010
David Perkinson
L316, ext. 7417
Office hours

Course Description

This is a course on manifolds including Stokes’ theorem, de Rham cohomology, and Poincaré duality.

Text Vector Analysis, by Klaus Jänich.

Books on Reserve

  • Foundations of differentiable manifolds and Lie groups, by Frank Warner.
  • Differentiable Manifolds: A First Course, by Lawrence Conlon.
  • Introduction To Smooth Manifolds, by John Lee.

Exams

To be announced.

This Week

  • Monday. Products, coproducts, tensor products.
  • Wednesday. Exterior and symmetric powers.
  • Friday. Dual space.
HW 2, due 2/12

Assignments

Handouts

Sage

Sage is a free open-source mathematical software system. It can plot functions, take derivatives and limits, integrate, and solve equations (among many other things). Check out the Sage website if you are interested. You can use it from your web browser or download it for free onto your own computer (Linux, Mac, or Windows).

Class Summary

Week 1

  • Monday. Course overview. First examples.
  • Wednesday. Projective space. Definition of a manifold.
  • Friday. Mappings.
HW 1, due 2/5: HW1

Week 2

  • Monday. Course overview. First examples.
  • Wednesday. Projective space. Definition of a manifold.
  • Friday. Mappings.
HW 2, due 2/12: HW2

Overview

Here is a rough plan for the semester:

Week 1:Examples of manifolds, especially projective space; definition of the category of manifolds.
Week 2:Tangent space: three equivalent characterizations.
Week 3:Multilinear algebra.
Week 4:Multilinear algebra on manifolds. Orientations.
Week 5:Integration.
Week 6:Integration; manifolds with boundary.
Week 7:Tangent space for manifolds with boundary; the Cartan derivative; Stokes’ theorem.
Week 8:De Rham cohomology.
Week 9:Homotopy invariance theorem; combing the hair on a tennis ball.
Week 10:semi-Riemannian manifolds; the star operator.
Week 11:Poincaré duality.
Week 12:Toric varieties: basic constructions.
Week 13:Toric varieties: cohomology, homogeneous coordinates, embeddings.

Homework and Grades

Your grade will be based on the weekly homework, class participation, and various quizzes/exams. It is important to not miss any of the homework assignments! When I return your homework, I will put numbers next to each problem according to the following scheme:

5 - perfect
4 - minor mistakes
3 - major mistake, right idea
2 - wrong but contains a significant idea
1 - wrong but contains a relevant idea
0 - none of the above

NOTES:

  • Late assigments (i.e., turned in after class on Friday), if complete, may be given half-credit, but it is unlikely you will get written comments on your work.
  • If you miss an assignment or quiz due to an illness, please inform me as soon as possible.
  • Note our final exam date before making airline reservations to leave at the end of the semester.
  • Deadline to add classes or change sections: Friday, February 8.
  • Deadline to drop spring semester classes without the grade of “W” (withdraw): Monday, March 3.
  • Deadline to withdraw from spring semester classes: Monday, April 7.

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