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: Jeremy Meza (jdmeza[_at_]berkeley.edu)
Homework Grader: Kunaal Sundara (kunaalsundara[_at_]berkeley.edu)
Class Hours:Prerecorded. Scheduled for T,Th 5:00-6:30
Instructor Office Hours: T,Th 5:00-6:30 or by appointment
GSI Office Hours: M,W 3-6pm and T,Th 9-11am
Textbook:Abstract Algebra, 3rd Edition, by Dummit and Foote [DF].
Secondary Text:Introduction to Abstract Algebra by Paulin [P].
Takehome Test 1: Assigned Friday 2/19 and due Monday 2/22
Takehome Test 2: Assigned Friday 3/19 and due Monday 3/22
Takehome Test 3: Assigned Friday 4/23 and due Monday 4/26
Takehome Test 4: Assigned Monday 5/10 and due Friday 5/14
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: 40%
Takehome 1: 15%
Takehome 2: 15%
Takehome 3: 15%
Takehome 4: 15%
Due to the ongoing COVID-19 pandemic, the course will be entirely remote and asynchronous. I will post 2 recorded lectures a week on this website, 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 above.
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 where I will suggest you pause the video and work out some of the details. There will also be many written exercises collected as homework assignments, or takehome exams.
There will be written homework collected almost every week on Fridays at NOON. 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. Feel free to work in groups, but each student must write up their results separately. These will be turned in and graded on Gradescope.
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 (1/19-1/22):