You are likely familiar with algebra as being a (perhaps tedious) exercise in solving equations. The structures of addition and multiplication, and the way they intertwine, allow us explicitly to extract information (solve) from relationships (equations).
Abstract algebra is the rigorous study binary operations, that is, functions which take two inputs and one output. You are already familiar with some binary operations (addition and multiplication of integers, for example), but it turns out there are many many more (addition and multiplication of matrices, composition of functions, mixing colors, applying symmetries, permutations and card shuffles, the list goes on). In studying the abstract properties of binary operations and their interactions, we will discover that they all share many very strong underlying structural properties, which allows us to extract information for given relationships (i.e., solve equations), in many different contexts. This leads to applications in cryptology, geometry, logic and even philosophy which we may glance at if time allows.
Along the way we will gain experience in proof writing and mathematical exposition and communication, and get first hand exposure to the abstract axiomatic approach pervasive within all branches of theoretical mathematics.
Instructor: Gabriel Dorfsman-Hopkins
GSI: Foster Tom (ftom[_at_]berkeley.edu)
Homework Grader: TBA
Class Hours:T,Th 9:30-11:00 AM in EVANS 740 (remote and pre-recorded 1/18-1/28, posted below)
Instructor Office Hours: Wednesday 9:30-11:00 and Thursday 11:00-12:30 or by appointment. On Zoom the first two weeks (or as long as remote instruction continues).
GSI Office Hours:MTThF 10:00-12:00 on Zoom until further notice.
Textbook:Abstract Algebra, 3rd Edition, by Dummit and Foote [DF].
Secondary Text:Introduction to Abstract Algebra by Paulin [P].
Takehome Test 1: Assigned Friday 2/18 and due Monday 2/21
Takehome Test 2: Assigned Friday 4/1 and due Monday 4/4
Takehome Final: Assigned Monday 5/9 and due Friday 5/13
Syllabus: Can be found here
Homework: Due (almost) every Friday at Noon on Gradescope
Schedule: Below
Assignments will be posted here on Fridays. Solutions to selected exercises will be posted when they are returned.
Homework: 50%
Takehome 1: 15%
Takehome 2: 15%
Takehome Final: 20%
Due to the surge in the COVID-19 pandemic, at least the first two weeks of the course the course will be entirely remote and asynchronous. While we are remote I will post 2 recorded lectures a week, corresponding to our scheduled Tuesday, Thursday lectures, but they will be available to watch at your convenience. I will record them on Zoom and post a link on the course website.
Even once we return in person, I would like to maintain a safe an flexible classroom, equitable for those who are sick or have loved ones to take care of. To facilitate this I will be recording all in person lectures on my GoPro, and posting those to bCourses. If you aren't feeling well, please stay home. There will be no penalty, no assumptions made, and no questions asked.
That being said, mathematics is not a spectator sport, and watching someone do abstract algebra is much like reading the New York Times backwards. All the information may be there, but it will take some unscrambling to make sense of it. For this reason I will have exercises embedded in the lectures. While remote I will suggest you pause the video and work out some of the details, and while in person we will take a moment to wrestle out some details in groups. There will also be many written exercises collected as homework assignments, or takehome exams.
There will be homework collected almost every week on Fridays at Noon. They will be assigned the week before they are due. These assignments will be proofs, as well as computations and explorations of examples. It is preferable that they are typed up using LaTeX, or a similar mathematical typesetting language, but handwritten is ok as well. Just make sure to submit \textit{high quality} scans.
There will be four takehome tests. These will look similar to the homework assignments but differ in the following important ways. First, they will be assigned on Friday and due the following Monday at 5PM (except the final which will be available during finals week), and subsequently will be a bit shorter. Second, you must work on them yourself. You may use the course texts ([DF] and [P]), as well as your course notes. You may not use the internet or your peers. They are in theory cumulative, but in practice will reflect material most recently covered. The currentl schedule for these takehomes is posted at the top of this page.
I will be curating a channel on discord, and will send out invitations over email. There you will be able to ask questions about homework and lectures, as well as have informal meetings on an audio/video channel. I will monitor the channel and will try to answer questions promptly, but you should feel free to answer questions posted there as well. I hope this is a space for group study and discussion, and to serve as a sort of \textit{open office door} during remote learning. If you need a new invite link to the discord channel, contact me and I will create one!
I find these notes on proof writing to be amusing and informative.
This may be adjusted as we move through the quarter. I will post an announcement and send out an email if a change to the schedule is made.
Week 1 (1/17-1/21):