Weeks 1-2: All things regular: regular sequences, regular local rings, Koszul cohomology, depth,
projective dimension,
Auslander-Buchsbaum-Serre's theorem that a noetherian local ring is regular if and only if it has finite
projective dimension.
Syllabus: We will cover advanced topics in commutative algebra and algebraic
geometry that are not often seen in a first-year course. The specific topics that we cover will depend on student
interest. The focus of this course may deviate from
other traditional courses in the following two ways:
examples, examples, examples! We wil explore each topic through an example-based inquiry. We may cover less
material overall, but for each topic we do cover, we will get a chance to enrich our understanding by working
playing around with explicit mathematical objects.
exposition by students: each student will be responsible for choosing
a topic of their choice and present the material to their peers in a 30 minute talk. We will have discussions on
what makes a good math talk versus a bad math talk. These short talks give students the opportunity their
communication and speaking skills.
Possible topics:
Flatness: Definitions, examples, Tor, going-down theorem for flatness,equational criterion for
flatness, local criterion for flatness, fibral flatness, graded modules and flatness, ...
Algebraic groups : Group schemes, algebraic groups, affine algebraic groups, actions of algebraic
groups, representations, diagonalizable groups, reductive groups.
Geometric Invariant Theory (GIT) : Quotients of algebraic varieties by group actions, Hilbert-Mumford
criterion, good vs. geometric quotients, HKKN stratification of the unstable locus.
Toric varieties : cones and affine toric varieties, fans and toric varieties, properties of toric
varieties in terms of the combinatorics of the fan.
Deformation theory: embedded deformations of closed subschemes, deformations of schemes, deformations of
coherent sheaves, infinitisemal deformations, obstructions, Grothendieck's Existence Theorem, Artin
Algebraization, Artin's Axioms.
Birational geometry: ampleness, bigness, nefness, base point freeness, semiampleness, ampleness
criterion.
Formalization of commutative algebra and algebraic geometry: How do you define and work with rings and
varieties in Lean?
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.
Expectation 1: You present at least one 30 minute lecture on a topic of your interest.
Expectation 1: You submit weekly reflections on three things. These should be submitted on Mondays by
9:30 am on Canvas. It should be somewhere between 1/2 and 2 pages in length.
Two research things: The idea is that from either the lectures of this course or from seminars that you
attended over the last week, you record two algebro-geometric things (e.g definitions, theorems, examples,
questions) that you don't know, are confused about, or simply would like to know more about. You then look up
these two things in whatever sources you find, you read about them, and then play around with them until you've
improved your understanding.
One teaching thing: From the previous week's lectures (or from other courses or seminars), think of one
speaking technique that you found particularly effective. Please record the speaking technique, why you find it
effective, and how you plan to incorporate it into your own talks.
Advice on choosing presentationtopics: Anything goes!
You can choose a subtopic from the above list, a topic you would like to learn about, ora topic that you've already
been exposed to but would like to learn in more depth. You can try to summarize a research article (either a classic
or a new paper), follow a textbook exposition of an article, present a big picture overview connecting disparate
themes, 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 and algebraic geometry is all about, and
what type of research problems the 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.