Math Studies Algebra, Spring 2018

Robot flying a dodecahedral balloon over fields
From symmetry groups to fields
© 2015 Laure Bukh
Used with permission

When:

Mondays, Wednesdays, Fridays 10:30

What:

Algebra is the art of changing the perspective. The change is mainly achieved through abstraction, which strips the irrelevant details and brings the important to the forefront. The extra generality also enables the connections between far-flung mathematical concepts.

The aim of this course is both to introduce the algebraic way of thinking, and to convey the basic language of algebra. That language is the language of groups, rings, modules, fields. We shall see, for example, how the group theory unifies such topics as integer arithmetic, tessellations, solubility of polynomial equations, and counting holes in a pretzel. We shall also learn and use some category theory, which is a higher-level abstraction that unifies different algebraic notions.

The division of mathematics into subfields such as "algebra" is arguably quite artificial. As a believer in the unity of mathematics, I will be making frequent detours into analysis, number theory and geometry.

Resources:

The book for the course is Abstract Algebra, 3rd edition, by Dummit and Foote.

Not all topics that we cover are in the book, and some topics we will cover differently.

Links to additional resources will be posted as the course progresses.

More fun:

The regular office hours for the course are Monday/Wednesday/Friday 11:30am–12:30pm in Wean 6130. I can usually meet by appointment at other times with 24 hours notice, send me an email to make an appointment.

Course activities:

Mastery of any subject requires practice. Hence, there will be regular homeworks. For your own good, you are strongly encouraged to do as much homework as possible individually. Collaboration and use of external sources are permitted, but must be fully acknowledged and cited. All the writing must be done individually. Failure to do so will be treated as cheating. Collaboration may involve only discussion; all the writing must be done individually. The homeworks will be returned one week after they are due.

Students are expected to fully participate in the class. The main advantage of a class over just reading a textbook is the ability to ask questions, propose ideas, and interact in other ways. In particular, discussions during the lectures are encouraged.

In the spring semester, the homework will count for 30% of the grade. There will be a take-home midterm counting for 30% and a take-home final counting for 40%.

Homework must be typeset (preferably using LaTeX) and submitted in PDF via the Canvas site. The filenames must be of the form andrewID_alg_homeworknumber.pdf. Pictures do not have to be typeset; a legible photograph of a hand-drawn picture is acceptable.

The homework must be submitted by 10:30am of the day it is due. For each minute that it is late, the grade will be reduced by 10%.

Course information:

This is an honors course. It is designed to challenge your brain with new and exciting mathematics, not to wear your body down with sleepless nights.

Students with disabilities: If you have a disability and require accommodations for this course, you should contact the Director of Disability Resources and get an accommodations letter. If you have an accommodations letter from the Disability Resources office, I encourage you to discuss your accommodations and needs with me as early as possible. I will work with you to ensure that accommodations are provided as appropriate.

Stress: CMU is sometimes a stressful place. If you or anyone you know is experiencing stress, anxiety or depression I strongly encourage you to seek help and support. The university's Counseling and Psychological Services (CAPS) offers confidential counseling services and an emergency 24/7 hotline.

Lectures:

For reference, here is the schedule of lectures from the first semester.

Homework sets:

  1. Homework 1 in PDF and TeX. Homework 1 solutions.
  2. Homework 2 in PDF and TeX. Homework 2 solutions.
  3. Homework 3 in PDF and TeX. Homework 3 solutions.
  4. Homework 4 in PDF and TeX.