Sets, Groups and Knots

Math 101 / Tu Th 10:00-11:30 / 216 Science Center
Harvard University - Fall 2016

Instructor: Curtis T McMullen (

Content of the course. This course provides an introduction to conceptual and axiomatic mathematics, the writing of proofs, and mathematical culture, with sets, groups and knots as the main topics.
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.

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.
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
    • The axiom of choice
  • Group Theory
    • Groups, subgroups and quotient groups
    • Symmetries of Platonic solids
    • The symmetric group
    • Groups actions; counting orbits
    • Free groups
    • The Cayley graph; group presentations
  • Knot Theory
    • Do knots exist?
    • Reidemeister moves
    • Types of knots
    • Knot invariants
    • The group of loops
    • Skein relations
    • The Alexander and Jones polynomials
Readings and Lectures. Assigned material should be read before coming to class.

Lectures may go beyond the reading, and not every topic in the reading will be covered in class. Students are responsible for all topics covered in the readings and lectures.

The use of laptops and cell phones during class is not permitted.

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

Homework. Weekly homework assignments will be due at the start of class on Tuesdays. Late homework and homework sent by email will not be accepted.

Collaboration between students on homework is encouraged, but you must write your own solutions, understand them and give credit to your collaborators.

Only materials from this course can be referred to for the homework; e.g. searching for solutions on the web is not permitted.

Midterm. There will be two hour-long in-class midterms, on TUESDAYs, Oct.4th and Nov. 8th.

Final. There will be a take-home final, to be completed in the period Dec 5-7. Collaboration on the final is not permitted. You may refer to your graded homeworks, course notes and the texts for the course, but no other materials. Internet resources CANNOT be consulted during the final, except for the course homepage.

Grading. Grades will be based on regular weekly homework (30%), the midterms (30%) and the final (40%). The two lowest homework grades will be dropped. This course is not offered pass-fail.

Other proof-based courses. Students who are interested in proof-based mathematics for further work in statistics or economics are usually directed to Math 112, Real Analysis. This course is suitable for students with an interest in abstract mathematics besides analysis, but in no way required prior to Math 112.

  • Sep 1 (Th). First class
  • Oct 4 (Tu). Midterm
  • Nov 8 (Tu). Midterm
  • Nov 24 (Th). Thanksgiving break
  • Dec 1 (Th). Last class
  • Dec 5-7 (M-W). Take-home final
  • Sep 1 (Th). First class
  • Oct 4 (Tu). Midterm
  • Nov 8 (Tu). Midterm
  • Nov 24 (Th). Thanksgiving break
  • Dec 1 (Th). Last class
  • Dec 5-7 (M-W). Take-home final

Course home page: