Math 242-1:   TOPOLOGY OF SURFACES

Spring 2008 2:10 - 3:00 MWF        Room SC 1431                            

Instructor: Prof. Bruce Hughes
Office: SC 1528 Phone: 2-6660
Electronic mail: bruce.DOThughesAT@vanderbilt.DOTedu
Web page: http://math.vanderbilt.edu/~hughescb/math242.html

Office Hours: Monday, Wednesday, Thursday  3:10-4:00, or by appointment.

Overview: The three main topics covered in this undergraduate mathematics course are:

Point set topology concerns local properties of spaces needed to discuss such fundamental notions as continuity, connectedness and compactness. This is foundational material for many branches of mathematics.

Surface theory is about spheres, tori, Moebius bands, Klein bottles, projective planes and more. We'll prove a classification theorem and learn how to distinguish one surface from another.

Knot theory is about loops of string in 3-dimensional space. We'll prove that knots exist, define some of the modern polynomial invariants that distinguish knots, and find ourselves at the frontiers of research thinking about unanswered questions.

This course is highly recommended if you are mathematically talented and if one of the following fits:

Warning: This is a rigorous proof-based mathematics course. You will be required to understand theorems and their proofs, and discover and write proofs of your own.

Prerequisites include a completion of our calculus sequence  and Linear Algebra (preferably MATH 175 and MATH 204, or MATH 205ab). You should know the basics of mathematical logic, sets, functions and proofs.

Textbook: Topology NOW! by Robert Messer & Philip Straffin, The Mathematical Association of America (2006).

Mid-Term Exam: A mid-term exam will be given on a day to be announced. There may also be a take-home component to this exam.

Final Exam: A comprehensive take-home final examination will be given. It will be distributed on the last day of class (April 22) and is due by 5 p.m. on Wednesday, April 30.

Homework: Homework is assigned daily. Some assignments will be collected and graded, others will be discussed in class by students and instructor. Students are encouraged to work together on the homework assignments. However, a student should not present as their own work solutions to which they did not make a substantial contribution. Collaborators in a solution should be acknowledged. Avoid looking up solutions in books. If you do use a book, be sure to site the source (including page number).

Grades: Your final grade will be determined from a total of 1000 possible points as follows:

  1. Mid-term exam, 150 points
  2. Final exam, 150 points
  3. Homework (including class participation), 700 points
Attendance: Attendance is expected for each class meeting. More than two absences will be considered "excessive" on the mid semester progress reports. See the College of Arts and Science policy on class attendance in the University Bulletin.

Syllabus: We will cover material from chapters 1 - 4 and 7 the text book.

Honor System: Vanderbilt's Honor Code governs all work in this course. Students are encouraged to work together on the homework assignments. However, a student should not present as their own work solutions to which they did not make a substantial contribution. Collaborators in a solution should be acknowledged. Avoid looking up solutions in books. If you do use a book, be sure to site the source (including page number).

Web Resources: The American Mathematical Society maintains a very useful page for undergraduate mathematics majors at http://www.ams.org/outreach/undergrad.html  It includes information on summer programs (Research Experiences for Undergraduates), semester programs, graduate studies in mathematics, clubs, undergraduate journals, competitions, careers, jobs and much more.

Reading (available in the Stevenson Library)

  1. Topology: A First Course by James R. Munkres. Call number QA611 .M82 2000
  2. Knots and Surfaces by N. D. Gilbert and T. Porter. Call number QA 612.2 .G55 1994
  3. The Knot Book by Colin C. Adams.  Call number QA 612.2 .A33 1994
  4. Topology of Surfaces by L. Christine Kinsey. Call number QA 611 .K47 1993
  5. Knot Theory & Its Applications by Kunio Murasugi. Call number QA 612.2 .M8613
  6. Knot Theory by Charles Livingston. Call number QA 612.2 .L585 1993
  7. Basic Topology by M. A. Armstrong. Call number QA611 .A68 1983
  8. History and Science of Knots by J. C. Turner and P. van de Griend (eds.). Call number VM 533 .H57 1996
  9. How Surfaces Intersect in Space: An Introduction to Topology (2nd edition) by J. Scott Carter. Call Number QA 611 .C33 1993
  10. Algebraic topology: an Introduction by William S. Massey. Call number QA612 .M37 1977
  11. Intuitive Topology by V. V. Prasolov. Call Number QA 611.17 .S2813
  12. Knots and Surfaces: A Guide to Discovering Mathematics by David W. Farmer and Theodore B. Stanford. Call Number ON ORDER
  13. Knots and Applications by Louis H. Kauffman (ed.). Call number QC 20.7  .K56 1995
  14. Knots and Physics by Louis H. Kauffman. Call number QC 20.7  .K38 1993
  15. New Scientific Applications of Geometry and Topology by DeWitt L. Summers (ed.). Call number QA 641 .N42 1992
  16. A First Course in Geometric  Topology and Differential Geometry by Ethan Bloch. Call number QA 611 .B55 1997
  17. Proofs and Refutations: the logic of mathematical discovery by  Imre Lakatos. Call number QA8.4 .L34
  18. Geometry and the Imagination by D. Hilbert and S. Cohn-Vossen. Call number QA685 .H515
  19. The Interface of Knots and Physics by Loius H. Kauffman (ed.) Call number QC 20.7 .K56 I58 1996
  20. When Topology Meets Chemistry: a topological look at molecular chirality by Erica Flapan. Call number QD455.3 .T65 F53 2000.
Topology Web sites
 Mathematics and Knots  from the Centre for the Popularisation of Mathematics at the University of Wales, Bangor
 The KnotPlot Site

 

Updated 7 January 2008