Prof. Ilya Kofman |
Office: 607 Mathematics, phone: 854-3210
Email: ikofman@math.columbia.edu Web site: http://www.math.columbia.edu/~ikofman/ |
Course Time and Place: 2:40pm - 3:55pm Monday and Wednesday, 312 Mathematics Building
Text: Knots and Surfaces by David W. Farmer and Theodore B. Stanford.
Optional supplementary text: The Knot Book by Colin C. Adams. W. H. Freeman and Company, 2001 (ISBN 0716742195). It can be purchased through AddALL.
Material Covered: An introduction to graphs, surfaces, and knots. There are no prerequisites.
Homework: Assignments will be announced in class and then posted on this website. Any changes will be announced in class. Possibly a small subset of any homework assignment will be graded. Homework is due Wednesdays at 4pm in the box outside Mathematics 410. Late homework will not be accepted. I highly recommend working jointly on homework problems with fellow students, but in the end you must hand in your own work.
Grading: The course grade will be determined by the best of three averages computed for each student:
So the homework is always worth 10-20%, the midterm 25-35% and the final 45-55%.
Help: My office hours are right after class in my office 607 Mathematics. The Barnard help room, 333 Milbank, is open with people to answer questions all day long on weekdays, but some topics we cover require specialized knowledge. Check the schedule for the TA's for this course: Sonja Mapes, Philip Ording, Cristina Caputo.
Optimal Method of Study: (1.) Come to class. (2.) Read the relevant sections after class. (3.) Do the homework. Leave time to think--do not put homework off until it is due! (4.) Compare your solutions with other students to improve what you hand in. (5.) Come to office hours or the help room with any remaining questions.
Goals: The primary goal of this course is to introduce you to a modern branch of mathematics - topology. By studying unfamiliar examples and patterns, you will get a feel for how mathematical results are discovered. I hope you enjoy exploring beautiful mathematical ideas, but if not don't let on!
The schedule below may change as the course progresses.
Week | Topic | Reading | Homework | Due (4pm) |
January 21 | Introduction, graphs | 1.1 - 1.3, Six degrees of separation | 1.1.2, 1.1.4, 1.2.1, 1.2.2 | Jan. 28 |
January 26 | Euler's formula I | 1.4 - 1.6 | 1.3.2, 1.5.1, 1.5.4, 1.5.5, 1.6.1, 1.6.6, 1.6.7 | Feb. 4 |
February 2 | Circuits, dual graphs, coloring | 1.7 - 1.11 | 1.7.4, 1.7.9, 1.8.3, 1.8.5, 1.8.6, 2.1.6, 2.1.8, 2.1.9 | Feb. 11 |
February 9 | Surfaces | 2.1 - 2.3 | 1.8.5, 1.10.4, 2.3.2, 2.3.4 | Feb. 18 |
February 16 | Euler's formula II | 2.4 - 2.6 | 2.4.4, 2.4.5, 2.4.6, 2.5.4, 2.5.6, 2.7.1 | March 1 |
February 23 | Classification of surfaces I | 2.7 - 2.9 | ||
March 1 | Classification of surfaces II | 2.10 - 2.11, ZIP proof | 2.8.1, 2.9.5, 2.9.6, 2.9.7, 2.10.2, 2.10.4 | March 8 |
March 8 | Review, Midterm Exam March 10 | Review, past exams, exam solutions | none | |
March 15 | SPRING BREAK | none | ||
March 22 | Knots and links | 3.1 - 3.3 | 3.1.1, 3.2.3, 3.3.3, 3.3.4, 3.3.9, 3.3.13 | March 31 |
March 29 | Elementary invariants | 3.4 - 3.6 | 3.4.3, 3.5.3, 3.5.9, 3.5.12 | April 14 |
April 5 | Linking number, tricoloring | 3.7 - 3.8 | 3.7.2, 3.7.9, 3.8.1, 3.8.2 | April 14 |
April 12 | Jones polynomial | class notes, Adams book 6.1, website | Compute Jones polynomials for the five 6-crossing knots (p.84) by any of our methods. Show all work! (Answers in Adams Appendix.) | April 21 |
April 19 | Alternating knots | class notes, Adams book 6.2 | Run through the argument that the crossing number of an alternating diagram is a knot invariant for the three 6-crossing knots. (Click for Jones polynomials: 6.1, 6.2, 6.3). Also checkerboard color a big doodle. | April 28 |
April 26 | Seifert surfaces | class notes, Adams chapter 4 | Adams Exercises 4.20 (Fig. 4.58), 4.22 (Fig. 4.61), 4.24 | May 4 |
May 3 | Review for Final Exam | Review | none | |
May 12 | Final Exam 1:10pm - 4pm |