Sets, Maps and Symmetry Groups

Math 101 / MWF 10-11 / Science Center 216
Harvard University - Fall 1998

Instructor: Curtis T McMullen (

Course Assistants: Required Text:
  • Fraleigh, Abstract Algebra. Addison-Wesley, 1994.
Recommended Texts:
  • Halmos, Naive Set Theory. Springer-Verlag, 1987.
  • Adams, The Knot Book. W.H. Freeman, 1994.
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 Friday, except for Thanksgiving when homework will be due the following Monday. 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 Wednesday, 14 October.

Final. There will be a take-home final, available at 12 pm on Monday, 11 January, and due by 4:00 pm on Friday, 15 January. The final should be picked up and returned to the Math Department Office, 325 Science Center. 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.

16 Sept (W): First class
12 Oct (M): Columbus day -- no class
14 Oct (W): Midterm
19 Oct (M): Last day to add or drop courses
11 Nov (W): Veteran's day -- no class
27 Nov (F): Thanksgiving -- no class
16 Dec (W): Last class
4 Jan (M): Reading period begins
11-15 Jan (M-F): Take-home final
16 Jan (Sat): Exams begin

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