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 to 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.
Due to the COVID-19 outbreak shutting down the university, the class has moved online as of March 11th. You will find links to the online lectures below. Also all exams including the final have been moved to takehome formats.
Instructor: Gabriel Dorfsman-Hopkins
GSI: Ben Castle (bcastle[_at_]berkeley.edu)
Homework Grader: Lei Guo (lguo8[_at_]berkeley.edu)
Email: gabrieldh[_at_]berkeley.edu
Office:Evans 895
Class Hours:MWF 1:00-2:00
Classroom: Cory 247
Instructor Office Hours: Mondays 2-3 and Wednesdays 12-1 and 2-3 or by appointment
GSI Office Hours: TBA
Textbook:Abstract Algebra, 3rd Edition, by Dummit and Foote [DF].
Secondary Text:Introduction to Abstract Algebra by Paulin [P].
Takehome Test 1: Assigned Friday 2/21 and due Monday 2/24
Takehome Test 2: Assigned Friday 4/24 and due Monday 4/27
Midterm Exam: Friday 3/20 in class.
Final Exam: Tuesday 5/12, 8AM-11AM location to be determined
Syllabus: Can be found here
Homework: Due (almost) every Friday in class.
Schedule: Below
Assignments will be posted here on Fridays. Solutions to selected exercises will be posted when they are returned.
The GSI has uptdated their office hours. Here is a message from them below:
Hello all,
Due to logistical reasons from doing office hours at home, I need to make a couple changes to my office hour times. Starting this week, I will change my Monday and Wednesday hours as follows:
Mon: 2-4 (instead of 1-3)
Tue: 2-5
Wed: 9:30-11:30 (instead of 12-2)
Thu: 12-3.
As shown above, the Tuesday and Thursday times are unchanged. Please also remember that you can always email me questions, or set up a separate appointment with me if the above times don't work.
Ben
I posted solutions to HW7. Problem 1 was quite technical and difficult, so I recorded a video going through the solution in detail. This is supplimentary material, not a required lecture, but for those of you who wrestled with this and want to see it worked out carefully, here you go: VIDEO
It looks like we will be online for the rest of the semester. Therefore all homework will now be collected via gradescope. Let me know if you have any trouble sumbitting.
As you likely already know, our courses have been moved online until after spring break at the soonest. This effects many things, not least of which is the upcoming midterm. As you might expect, it will be adjusted to be a takehome assigment not unlike the first takehome. Since it was supposed to be on Friday 3/20, I will make it available on 3/20. Although it shouldn't take very long (the original exam was only supposed to take one hour), I understand that many of you will be travelling home, have other assignments changed last minute, and be dealing with many other course corrections and adjustments (or god forbid sickness among friends and family). Therefore I will have it loosely due Monday 3/23 via Gradescope, but that due date will be flexible. Please don't hesitate to contact me if you need more time or have issues at all. Thank you all for your patience during this crazyness. I hope we can still have an enjoyable rest of the semester but I do understand that priorities may be pretty skewed for the time being, and if your focusses lie elsewhere I empathize and understand. As I said, do not hesistate to contact me with any concerns.
The GSI will still be holding regular office hours online via zoom. I have copy-pasted a message from him below:
I will have office hours at the normal times: Mon 1-3, Tue 2-5, Wed 12-2, Thu 12-3. I will be in my usual room, so I can still use the board over video. My Zoom ID is 945-628-9579. Any time you want to talk to me during my office hour times, type that code into Zoom and you should be added to the meeting.
My hope is that we can operate the office hours organically (as they have been in person), rather than via appoint times. However this is new to everyone, so I am willing to change that if it seems not to work. In particular, to start:
If multiple people are online I will try to limit the amount I talk to any one person at a time to about 15-20 minutes -- though if I have to switch to someone else I can still get back to you later in the session. You can also type to me over Zoom if you need to talk to me at a particular time (e.g. if you have a class coming up). Ideally, it should be possible in appropriate situations for several people to be active in the conversation at any given time (as is often the case in person). So, I would like to keep the sessions public, so anyone with similar questions can listen in while I'm talking about a particular topic. Thus, if you want to talk, just join the meeting (and if you like, message me about what/when you want to talk about), and I will get to you in whatever order everyone has joined.
If you prefer to talk in private, you can always email me, and we can try to arrange a time to do a private Zoom call.
See you next week, and stay healthy!
Ben
Finally, below are links to the online lectures. I will post them as they come out.
The outlook for the next few weeks. First, due to the ongoing spread of COVID-19, we should be prepared for the university to shut down. If that happens, I will record and share the lectures via email, so look out for that. We may also make the in class midterm a takehome in this case. I will keep you informed via email.
That being said, some folks may prefer not coming into campus during the next couple of weeks. If so, how do you keep up with the class? Unfortunately, our classroom doesn't have the option to webcast a video of the lecture (just audio and slides, which we don't use). That being said, I am following the book rather closely, so you can read Dummit and Foote to keep up. In that case, what is the outlook for the next couple of weeks of lectures? I have a schedule posted below, but we are falling a bit behind. I think the plan will be:
What about homework? If you aren't going to able to be in class tomorrow to turn in the homework here's what you should do. Email it to me and our grader (Lei Guo), the subject line should be MATH 113 HW SUBMISSION. If you type up your homework this should be easy. If you don't type your homework, PLEASE DON'T SEND A CAMERA PHONE PICTURE! Those are impossible to print, read, and grade. Scan it. That being said, there are apps on your phone which turn your phone camera into a scaner quite nicely. I use TurboScan which is free, has good edge detection, and prints well.
Finally, let's have HW7 due on Monday March 16 rather than Friday the 13th. Since there is a midterm on Friday the 20th there will not be homework due that week so it won't cram up.
I will update a plan for post spring break and update the schedule when its all hashed out. We will likely not get to direct and semidirect products until after the break. And then probably immediately follow that up with ring theory.
Here I will post announcements, as well as solutions and statistics to homework, takehomes and exams.
Homework: 25%
Takehome (lower score): 15%
Takehome (higher score): 20%
Midterm: 15%
Final: 25%
Grades are curved, so that your course grade will reflect your performance relative to the rest of the class rather than a fixed percentage scheme.
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. Therefore I plan to lecture enough to introduce the necessary concepts, but spend much of the time working through examples and exercises with your help. This will include time to work through examples and exercises in class. For this reason, it is a good idea to be a bit ahead in the reading, and ready to toy with some problems in class. There will also be many written exercises collected as homework assignments, as well as takehome and in class exams.
There will be written homework collected almost every week on Fridays. They will be assigned a week before they are due, and mostly be problems from the book, but keep in mind that some of the problems may require material discussed on the Monday before they are due as well. These assignments will be proofs, as well as computations. 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.
There will be two take home 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 on Monday. Second, you must work on them yourself. Third, they will consist primarily of proof writing. 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.
There will be one midterm and one final. The midterm will be the day before spring break, and the final during finals week. These will both be cumulative, and will include both computation and examples.
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 (1/20-1/24):