189-457B: Honors Algebra 3

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Professor: Henri Darmon

Tutorial leaders: Frédéric Cai and Ludovic Rivet.

Markers: Megan Millet and Connor Campbell.

Classes: MWF 8:35-9:25 AM, in Burnside Hall 1B36.

Office Hours: MW 2:00-3:00 or by appointment, in Burnside Hall 1111.

Tutorials: There will be some tutorials for this course by Frédéric Cai and Ludovic Rivet, as well as expert guidance from Helena Heinonen at the math help desk, as per below:

Frédéric Cai: Wednesday 3:30-4:30 in BH 1104 and Friday 3:30-4:30 in BH 1B36.
Ludovic Rivet: Tuesday 12-1 PM in BH 1104 and Thursday 12-1 PM in BH 920.

Frédéric and Ludovic have kindly volunteered to give extra Tutorial sessions after end of classes, in room BH 1214, as follows:

Frédéric: Thursday Dec. 5, Tuesday Dec. 10, Wednesday Dec. 11, from 12:00-13:00.
Ludovic: Monday Dec. 9, Thursday Dec. 12, Friday Dec. 13, from 12:00-13:00.

I will also be giving extra review sessions on Wednesday in 1B45 and Friday in BH 1104, 8:30-9:30 AM next week. (December 11 and 13.)

There is also a Math Help Desk in BH911 operating Mondays to Fridays from noon to 5:00pm. This is a valuable ressource if you need extra help on the material or assignments, and you are encouraged to make use of it.
One of the former students of 456A, Helena Heinonen, will be there on MW 11-12 and would be happy to help you out with your algebra questions.

Main texts: I recommend the following free on-line course notes as good references for the course:

The notes by Thomas Judson which cover the basic undergraduate material with a computational emphasis which I like, and also contains helpful exercises.
The notes by Paul Garrett which claims to be a first graduate course but seems fairly accessible. (And hey, this is McGill.)

Optional Textbooks:
The classic textbook by Dummit and Foote comes highly recommended, if you do not balk at the hefty price tag.

In a more challenging vein, I also highly recommend the textbook
Eléments d'analyse et d'algèbre (et de théorie des nombres) by Pierre Colmez.
It covers a lot more ground than we will in this course, and would be equally appropriate for the analysis courses that you might be taking concurrently. It is beautifully written and belongs on the bookshelf of any mathematics student who is passionate about the subject.

Syllabus:
The course syllabus is divided roughly into three equal parts:

1. Group theory. Definition and examples. Homomorphisms and normal subgroups. Permutation groups and permutation representations. Matrix groups and linear representations. Theorems of Cayley, Lagrange, and Sylow. Classification of groups of small order. Very brief introduction to the representation theory of finite groups. (We will see this in more depth in Math 457).

2. Rings and Fields. Integral domains, Euclidean rings, unique factorisation domains, polynomial rings, localisation.

3. Modules and Vector spaces. Modules over a ring, vector spaces over a field. Duality. Euclidean spaces. The spectral theorem for self adjoint and normal operators.

The different parts of abstract algebra interact with each other in a myriad ways and it is never clear which comes first, the chicken or the egg. Indeed, this is one of the charms of the subject. In particular, your prior exposure to linear algebra will certainly be taxed in this course, even before vector spaces are introduced in the above syllabus.

Assignments:
Assignments are to be turned in on Wednesdays and will be returned, graded, the following Monday. There will be around 6 assignments in all during the semester, which will be assigned every two weeks.

Grading Scheme : There will be two possible schemes, and I will take the maximum of those.

1. 20% Weekly assignments, 30% Midterm, 50% Final.
2. 20% Weekly assignments, 80% Final.

Midterm Exam:. The midterm exam will be held, in class, on Wednesday, October 23 at the usual time (8:35-9:25).

To help you with your studying and give you a sense of what the midterm might look like, here is a sample midterm from last year. This year's midterm will follow the same format, but contain different questions, and , may also cover some of the topics from ring theory that we covered in the last weeks. (Last year's midterm was given earlier when we had only covered group theory.)

Final Exam:. The final exam will be held in the Gym, Fieldhouse Auditorium, Rows 40-41 on Wednesday, December 18 from 2:00 to 5:00 PM.

The customary statements

McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures.

In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.

In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change.

Shortlist of Resources: The Department of Mathematics and Statistics, as well as the University at large, have many resources to help you succeed in this course and throughout your degree. A more extensive list of resources can be found in the MyCourses Webpage for this Course (under ``Content''), as well as the Department's OSW webpage. A shortlist of the most commonly used resources is available here:

1. The Wellness Hub is a centralized website for student physical and mental health resources.

2. The Math Help Desk is staffed by knowledgeable math students who can help answer your questions related to your courses. They have tutors on M-F from noon-5 PM in Burnside Hall room 911.

3. Statistics Online for Students is a resource which offers online help on statistics courses in the department. Click on the link above and you will be added to a Microsoft Teams group where the schedule will be announced.

4. SUMS is the Society of Undergraduate Mathematics Students. Join their Facebook group to get on their listserv, connect with other students in the department, and participate in some of their activities/social events.

5. GSAMS is the Graduate Student Association for Mathematics and Statistics (GSAMS). All graduate students are represented by GSAMS, and they hold regular events throughout the semester to promote graduate student community.

6. Advising is an important resource to guide you throughout the duration of your degree, and advisors can help with answering questions related to your degree program. Check out the Department's Advising Website to find out how to get in touch with a Departmental Advisor, should you need to do so.

7. The Office for Mediation and Reporting is a McGill centralized office used to file a formal report of discrimination, harassment, or sexual violence; learn about policies and processes; or be connected to additional supports.

8. The Office for Sexual Violence, Response, Support and Education provides support for all members of the McGill community who have been impacted by sexual violence (whether it be sexual harassment or assault, gender-based or intimate partner violence, or cyberviolence) and works to foster a culture of consent on campus and beyond.

9. The Office of the Dean of Students (Case Management) is a collaborative process between a student, a Case Manager, and other concerned parties with the intention of improving the student's academic and personal outcomes. Case managers are trained in confidentiality, disclosures of sexual violence, and mental health first aid.