189-457B: Honors Algebra 4

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Professor: Henri Darmon

Marker: TBA.

Classes: MWF 8:35-9:25 AM, in BURN 1104.

Office Hours: M 2:00-3:00 and W 9:30-10:30 or by appointment, in Burnside Hall 1111.

Special review sessions:
Wednesday, April 23, 8:30-10:00 AM. In BH 1104.
Monday, April 28, 8:30-10:00 AM. In BH 1205. (But don't forget to vote!)
Tuesday, April 29, 8:30-10:00 AM. In BH 1104.

Practice final exam: Click here.

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.
Helena Heinonen will be working at the help desk from 2:30-4:30 on Thursdays and is particularly eager and qualified to assist Math 457 students.

Syllabus:
The course syllabus is divided into two parts:

1. The representation theory of finite groups. Definition and examples. Semisimplicity and Maschke's theorem. Duals, homs and tensor products. Characters, and their orthogonality. Induced representations. Character tables. Frobenius reciprocity. Fourier analysis on finite groups.

2. Galois Theory. Basic field theory. Constructibility by ruler and compass, and solvability by radicals. Splitting fields. The fundamental theorem of Galois Theory. The Galois correspondence. Computation of Galois groups. Applications.

The different parts of abstract algebra interact with each other in a rich variety of ways. Representation theory (or at least the tiny part of that vast subject that will be covered in the class, concerned with representations of finite groups over the field of complex numbers) is a mature, complete theory which marries ideas from group theory and linear algebra. Galois theory likewise establishes a fundamental bridge between the theory of fields and of groups.
In particular, Algebra 4 will rely very much on the basic notions around groups, fields and vector spaces, and rings and modules, acquired in the Algebra 3. This will also give you a chance to solidify and deepen your grasp of these fundamental mathematical objects.

Main texts: The following textbooks are good references for the material we will cover in the course:

Part 1. Representation theory

The book by Jean-Pierre Serre is a classic and something of a gold standard for presenting subject.
The lecture notes of Benjamin Steinberg are a well written accessible introduction to the material, aimed at advanced undergraduates.

There are also a number of excellent videos available on line about representatin theory, notably the on-line course by Richard Borcherds and a lecture series at a CIMPA summer school by René Schoof and Laura Geatti.

Part 2. Galois Theory

The book by Emil Artin is somewhat the counterpart of Serre's book for Galois theory, presenting what has become by now the standard treatment of the foundational parts of the subject.
The book by James Milne is an excellent and slightly more modern treatment of the same subject.
Finally, this pedagogically ambitious book by Tom Leinster aims for a well-motivated account of the subject.

Just like for representation theory, there is a wealth (maybe even too much) of on-line ressources devoted to Galois theory.
For instance, another lovely on-line course by Richard Borcherds.

Optional Textbooks:
As in last semester, 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.

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, February 26, at the usual time (8:35-9:25).
Here is a practice midterm to assist you in your studying.

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.