CRN 12109: Final Projects

• The final project will account for 25% of your course grade.

• Groups of three students each will work on one of the final projects.

• Deliverables consist of a complete written solution (target length: five pages) and a 15-minute presentation. (There are some starred projects with no written report.) The paper does not need to be typeset if the handwriting is legible.

• The projects will be presented during the final exam period on Thursday, December 12, 16:00--18:45. The accompanying papers are due before the start of the presentations.

• The student group will be graded as a group. All group members must contribute to both the written solution and the presentation in equal parts. If members of a group feel that one member is not contributing in a meaningful way, they can ask me to remove the particular student from their group.

• The group will be graded foremost on the mathematical correctness and mathematical clarity of their presentation and their written report. Other criteria include the completeness of the written report, the quality of the group presentation, making effective use of the allotted time, and staying within the time frame of 15 minutes for the oral presentation.

• Projects will be assigned on Tuesday, November 12.

• Topics:

1. Alyssa, Jorge, William: Schroeder-Bernstein Lemma (Exercise 1.4.13)
2. Blanca, Carolyn, Joseph: Dirichlet’s and Abel’s Tests (Exercises 2.7.12-2.7.14)
3. Angel, Diana, Xena: A comparison of the Root and Ratio tests* (W. Rudin, Principles of Mathematical Analysis)
4. Double Series and the Cauchy product* (E. Hairer and G. Wanner, Analysis by Its History)
5. Prove that every non-empty open set of real numbers is the union of at most countably many disjoint open intervals.
6. Alejandra M., Clarissa, Djuna: Perfect Sets* (Section 3.4, 1st part)
7. Alejandra E., Brenda, Marisol: Connected Sets (Section 3.4, 2nd part)
8. Sets of Discontinuity (Section 4.6)
9. Carlos, Luis, Suhail: The Euler-Mascheroni Constant
10. Annette, Ariel, Justin: A Continuous Nowhere Differentiable Function (Section 5.4)
11. Hsin Yuan, Paul, Thomas: Uniform Convergence I* (Section 6.2, pp. 154-157)
12. Jonathan, Jose: Uniform Convergence II* (Section 6.2, pp. 157-160, including Theorem 6.2.6)
13. The Cantor Function (Exercise 6.2.13)
14. Arzela-Ascoli Theorem (Exercises 6.2.15, 6.2.16)