Math 507: Advanced commutative algebra

Fall quarter 2022

Lectures: Thursday 9:30 - 12:20 in PDL C-401
Instructor: Jarod Alper (
Office hours: Wednesday 3-5 pm in PDL C-549

Textbooks: There is no required textbook for the class. We recommend the following texts:
Syllabus: This course is an example-based inquiry into a selection of advanced themes in commutative algebra. We will also discuss what makes an effective math lecture, and sstudents will improve their ability to communicate algebraic ideas by practicing giving short expositions. The topics covered will depend on everyone's interests.

Possible lecture topics:

Presentation schedule:

Expectations: This is not a class to sit back and nod your head. Class participation is required. You are expected to choose topics of your interest, learn these topics extremely well, and do your best to present the material effectively to your classmates. The classroom will be a welcome and informal environment where we learn from our mistakes. The expectation is that you give either a 1 hour lecture or two 30 minute lectures, but we will be flexible.

You will also be required to submit weekly "Reflections on three things." The idea is that during the previous week's lectures, you record three things (e.g definitions, theorems, themes, examples, questions) that you don't know, are confused about, or simply would like to know more about. You then look up these three things in whatever sources you find, you read about them, and then play around with them until you've improved your understanding. You should summarize what you've learned in a 1/2-1 page document. This should be submitted in class. For more background on "Three things," see Ravi's Vakil's description.

Advice on choosing topics: Anything goes!

You can choose a subtopic from the above list, e.g. Hilbert-Samuel polynomials and its relation to dimension. (You won't be able to cover everything listed in a given bullet point.) You can choose a topic that you maybe have learned in an earlier algebra course but would like to learn in more depth. You can try to summarize a research article related to commutative algebra, or you can try to give a big picture view of say applications of commutative algebra to representation theory (or algebraic topology, number theory, ...). You can also pair up with other students if you want to coordinate your lectures.

It can be challenging to develop a big picture of what commutative algebra is all about, and what type of research problems the commutative algebra community is interested in. My advice is to attend seminars and talk to others. You should also read survey articles, introductions to books, introductions to research papers, mathoverflow posts, and whatever else you find. One great resource is which contains wonderful lists of survey articles and accessible papers.

Sources for improving your communication skills

For giving effective math talks, see Improving your writing skills will also improve your speaking skills: