For this WARTHOG workshop to be effective, we decided that students should be familiar with algebraic geometry and coherent sheaves before the workshop begins. Similarly, we hope students are familiar with homological algebra. Do not worry if you do not have this prerequisite yet, it should not be difficult to read up before the workshop begins, but you should definitely expect to put in some time to get practice with this material. One should expect that the workshop will review the important facts and give examples — no one is expected to be an expert — but we cannot expect anyone who's never seen a coherent sheaf or an interesting short exact sequence of coherent sheaves to be able to follow most of the lectures.
The main topics in our background list, sorted in order of priority, are:
For each of these we'll give a list of concepts, like coherent sheaf or adjoint functor, and we indicate whether we want participants to be well-practiced with the concept (internalized the definition, worked examples and exercises) or just familiar with the concept (thought about the definition, seen examples, have a feeling for what goes into it). Work your way down the priority list; if you already know the essential material, use your prep time to read ahead on the optional material, or to become more well-versed with the concepts you're only familiar with.
We will be setting up a Googlegroup for this workshop, and we hope this will be a useful forum to ask questions or find online study groups. We give some references and guidance for reading below. Note also that Wikipedia does a good job with many mathematical definitions.
Algebraic Geometry and Coherent Sheaves. Essential. Everything here should be well-practiced.
References: the standard reference [Ha] is a good one, chapters II.1-5. Another good reference with exercises is [Va], Chapters 13 and 16.1-3. For a gentle first introduction, see [Sh] Chapter 1.
Homological Algebra. Very important. There is a huge amount of machinery under the hood (e.g. abstract triangulated categories, construction of the derived category) which makes the subject more technical to learn, but isn't really used much in computing examples. We want to focus on applications in algebraic geometry.
Well-practiced:
References: The goal is perhaps to understand [Hu] Chapter 3, while returning to chapters 1 and 2 or to other references as needed, while not overemphasizing their machinery. Another good reference is chapters 1 and 2 of [Or]. The notes [Ca] give a loose exposition of this material, with examples and applications in algebraic geometry. A solid and comprehensive textbook is [We], but with no applications to algebraic geometry.
Additional algebraic geometry. Advantageous, but optional. Begin with familiarity, and gain practice if time.
References: [Ha], [Va] Part V, [Ca]. Perhaps more references will be added here later.
Representation theory. Advantageous, but optional. Again, this is a huge field with a massive amount of machinery going into the proofs, but we will more be interested in using the results and playing with examples than knowing how they are proven. Begin with familiarity, and gain practice if time. Focus on GLn.
References: An efficient route through this material can be found in [Ta], Chapters 1-3. The classic reference is [FH], Chapters 11-13 and 15.
Even though one of the most-studied objects in this workshop will be the affine Grassmannian, there is not an immense amount to be gained by reading up on it beforehand. It is hard to find an appropriate introduction with exercises, and there are many distracting technicalities. The techniques for working with the affine Grassmannian will be introduced during the workshop. Participants who are still interested in gaining familiarity with the affine Grassmannian could try the short survey [El].
Of course, the expert participant is encouraged to read the papers on which this workshop will be based (see below), though this is not expected of most participants.
The content of the lectures will be aimed at understanding the following papers (but you will not need to read them):
More topic-adjacent references to come.