Attendance:

No attendance policy will be enforced, but consistent attendance -- in person, except as dictated by the Covid situation -- is wise.

Course objectives and learning outcomes:

In Math 701 and 702, successful students will:

• Master algebraic topics concerning groups, rings, fields, and other algebraic structures.

• More importantly, develop their sense of algebraic thinking, so that they can more easily absorb additional algebraic material in the future.

• Lay the foundation for research in topics including algebraic number theory, algebraic combinatorics, representation theory, algebraic geometry, commutative algebra, algebraic topology, etc., etc., etc.

• Prepare for the Qualifying Exam in algebra.

Office Hours:

• Mondays and Tuesdays, 4:00-5:00, in COL 1022D (eventually) or via Teams (all semester). At the beginning of the semester office hours will be held via Teams only, or in-person on the Horseshoe by request.
• Tuesdays, 9:00-10:00, outdoors on the Horseshoe. (Via Teams in case of inclement weather.)

Books:

There are a number of other excellent books on algebra as well. Perhaps the most interesting is that by Lang. In my opinion Lang's book is not very good for the beginner, but is excellent for someone who has seen the material before and wants to review or see a different perspective. In particular, it is great to read when you are studying for quals.

Microsoft Teams:

Please either check Teams daily, or have it forward new announcements to your email. You can ask me questions directly via Teams. Although private questions are okay, please ask your question publicly if you are comfortable doing so -- maybe others will be interested in the answer as well!

Course Requirements:

• (70%) Written homework assignments will be assigned at most weekly.

• (10%) Midterm

• (10%) Final Exam

• (10%) Seminar Reports

The exams will be take-home and pledged, closed book/notes, 4 hours for both the midterm and the final, and similar to what will appear on the qualifying exam. The midterm will be passed out on a date to be announced, and you will have a week to do it. The final exam will be distributed on a date to be announced, and due the last day of finals period.

For the seminar reports you are asked to attend at least four seminars, conference talks, or colloquia and write with a brief report of what you learned, what you found interesting, or what questions you have. The format is up to you; Ravi Vakil's three things is one good choice. You may write privately, but posting publicly to the Teams chat is encouraged.

Seminars may be in-person or online. USC-sponsored seminars include Algebra, Geometry, and Number Theory and the department colloquium. There are also many external online seminars, for example the Number Theory Web Seminar.

You should periodically go to these if a topic catches your interest. Even if you expect the seminar to go over your head, it's worthwhile to go anyway: in part, we learn mathematics the same way babies learn language. Here is Ravi Vakil's advice on how to go to a talk.

Special Accommodations:

If you have a disability, have particular Covid-related concerns, or otherwise need special accommodations, please be in touch with me and I will do my best to accommodate your needs.

Grading Scale:

• A (75+): You have a strong chance of passing the qualifying exam if you prepare reasonably. This also represents a strong foundation for learning further related topics.
• B+ (60+): With additional effort, you should be able to pass the qualifying exam if you prepare. This represents a partial foundation for learning further related topics.
• B (50+): You have demonstrated some mastery of the subject material, but should put in a lot of additional effort if you want to pass the qualifying exam or take followup courses in algebra.
• C (40-49), D (30-39), F (0-29): This represents a serious problem.

Homework Assignments:

To be posted, either here or on Microsoft Teams.

Rough schedule of topics:

This is for both 701 and 702, and is very much subject to change.

The course will include a short unit on linear algebra at the very beginning of the term. This will showcase the themes of the course in a setting that will be very familiar, and yet possibly unfamiliar at the same time. (Matrices will be mentioned only briefly if at all, and they will certainly not be row reduced at the blackboard.)

For a more highbrow perspective, I recommend Lang's book. For a still more highbrow perspective, read this or this if you dare.

• Linear Algebra (2-3 weeks): Dummit-Foote, Ch. 11-12; Axler, Linear Algebra Done Right.

• Group Theory (~6 weeks): Dummit-Foote, Ch. 1-5.

• Ring Theory (~4 weeks): Dummit-Foote, Ch. 7-9. For the most part we will deal with commutative rings.

• Module Theory and Tensor Products (~3 weeks): Dummit-Foote, Ch. 10, 11.5; Atiyah-Macdonald, Introduction to Commutative Algebra.

• Field and Galois Theory (~6 weeks): Dummit-Foote, Ch. 13-14.

• (if possible) Introduction to Commutative Rings and Algebraic Geometry (~3 weeks): Dummit-Foote, Ch. 15; Atiyah-Macdonald.

• (if possible) Introduction to Algebraic Number Theory (2-3 weeks): Dummit-Foote, Ch. 15.3, 16.2-16.3; Neukirch, Algebraic Number Theory.