Sets, Groups and Knots

Math 101 / MWF 10-11 / 216 Science Center
Harvard University - Fall 2005
Instructor: Curtis T McMullen (ctm@math.harvard.edu)

Required Texts:
  • Halmos, Naive Set Theory. Springer-Verlag, 1960.
  • Fraleigh, Abstract Algebra (7th ed.) Addison-Wesley, 2003.
  • Adams, The Knot Book. Amer. Math. Soc., 2001.
Prerequisites. An interest in mathematical reasoning. Acquaintance with algebra, geometry and/or calculus is desirable. Students who have already taken Math 25a,b or 55a,b should not take this course for credit.

Topics. We will discuss mathematical proofs, sets and mappings, group theory and knot theory. Some possible topics include:
  • Proofs and Set Theory
    • Methods of proof: induction, contradiction.
    • Sets, maps, functions and relations
    • Cardinality; different sizes of infinity
    • Ordinals
    • The axiom of choice
  • Group Theory
    • Groups, subgroups and quotient groups
    • Symmetries of Platonic solids
    • The symmetric group
    • The Cayley graph; group presentations
    • Groups actions; counting orbits
    • Linear groups
  • Knot Theory
    • Do knots exist?
    • Reidemeister moves
    • Types of knots
    • Knot invariants
    • The group of loops
    • Skein relations
    • The Alexander and Jones polynomials
Reading. Students must read the assigned material before coming to class.

Sections. Attending sections is an essential part of the course, especially for learning to write proofs.

Homework. Weekly homework assignments will be due on Monday, except for Columbus day when homework will be due on Wednesday. Collaboration between students is encouraged, but you must write your own solutions, understand them and give credit to your collaborators.

Late and Early Homework. Late homework will not be accepted. If you need to hand your homework in early , bring it to class and give it to a course assistant.

Midterm. There will be one in-class midterm, on FRIDAY, Nov. 4th.

Final. There will be a take-home final. Collaboration on the final is not permitted, but you may refer to your course notes and the texts for the course.

Grading. Grades will be based on regular weekly homework (30%), the midterm (20%) and the final (50%). The two lowest homework grades will be dropped. This course is not offered pass-fail.

Calendar.
  • Sep 19 (M). First class
  • Oct 10 (M). Columbus day
  • Nov 4 (F). Midterm
  • Nov 11 (F). Veteran's day
  • Nov 24 (F). Thanksgiving break
  • Dec 19 (M). Last class
  • Jan 9-11 (M-W). Take-home final

Course home page: http://math.harvard.edu/~ctm/math101