189-457B: Honors Algebra 3

-------------- Course blog --------- Assignments ----------------

Professor: Henri Darmon

Marker: Shousen Lu, Fengyi Sun, and Sindbad Walter.

Classes: MWF 10:35-11:25 AM, in ENGTR 1080 (Trottier Building).

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

Tutorials: There will be no tutorials for this course. However, there is 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.

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 25 at the usual time (10:35-11:25).

The Pathy Foundation Fellowship is an intensive 12-month experiential learning opportunity for graduating students to lead a self-directed project in a community they have a meaningful connection with. Fellows are provided with comprehensive training, personal and professional development coaching, up to $40,000 in funding, and wrap-around supports to design, plan, and implement an innovative project in their community. This year-long Fellowship is designed for students who show a demonstrated capacity and potential to develop and evolve as leaders and change makers and have a meaningful connection with a community of their choosing. Applications for Cohort 9 of the Pathy Foundation Fellowship are open until January 5th, 2024! Visit https://linktr.ee/pathyfellowship to learn more and start your application today.

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.