Capstone Course in Mathematics Spring 2015

Facilitators: Shiyu Li (jjl2357 at ...) and Henry Maltby (hwmaltby at ...)
Faculty Sponsor: Yan Zhang
CCNs: 54038 (lower-division), 54317 (upper-division)
Office Hours:

W 3-4pm and Th 11am-12:30pm, 938 Evans
TTh 9:10am-2pm, 1042 Evans

About the course

This course seeks to aid development of mathematical maturity required for further courses at and beyond the undergraduate level. As such, it is intended for students who have completed the lower division major requirements and have begun to work on the upper division courses. We will assume familiarity with many concepts present in a traditional undergraduate core (Math 104, Math 110, Math 113, Math 185). That being said, the only requirements for this course are Math 53, Math 54, and Math 113. We hope students who complete this course will have the fundamental tools to begin taking introductory graduate courses, and we hope to supplement future courses by providing illustrative examples and bridging gaps in intuition.

Lecture will take place 4-5:30 pm on Wednesdays in 251 Dwinelle. A supplementary discussion period will take place either 3-4, 4-5, or 5-6 pm on Monday, depending upon student availability and interest. This discussion period will be focused more greatly on review of lecture from the previous week and assistance on key problems in the assignment. An office hour will be offered in conjunction with the Mathematics Undergraduate Student Association's open hours for the semester.

Students will have problem sets each week to review the material presented in each lecture. The grading scheme will be as follows:

60% problem sets (assigned weekly), graded on:

half completion
half correctness of selected problems

40% participation:

Students must contribute to classroom discussions and attend the 90-minute lecture each week.
Students are expected to do the relevant readings.
Missing more than two lectures may result in a failing grade.

We plan on covering many different topics throughout the semester. Our tentative weekly syllabus is as follows:

Lecture Topics

January 28th: Zorn's Lemma, Set theory, partially ordered sets
February 4th: Algebraic Construction of Vector Spaces
February 11th: A Study of (In)Dependence: Matroids
February 18th: Whirlwind Tour of Algebraic Duals
February 25th: Let's Talk about Category Theory!
March 4th: Polynomial Irreducibility & Galois Extensions
March 11th: Tensor Algebras and Products
March 18th: Algebra Unit Review
April 1st: Point-Set Topology
April 8th: Normed Vector Spaces & Frechet Derivatives
April 15th: Implicit/Inverse Function Theorem, Contractions and Fixed Points
April 22nd: Exact Sequences and Homology
April 29th: Final Review

Materials

Materials from lecture will be posted here after each practice.

Week 1
- Introduction
- Basic Definitions
Week 2
- Vector Spaces
Week 3
- Graph Theory
- Matroids
Week 4
- Duals and Universal Properties
Week 5
- Category Theory
Week 6
- Fundamental Theorem of Galois Theory
Week 7
- Tensor Products
Week 9
- Point-Set Topology

Homework

This page was last updated on April 23rd, 2015.