Infinite-dimensional methods in commutative algebra

Workshop on Algebra and Representation Theory, Held on Oregonian Grounds

June 26-30, 2022
University of Oregon
Eugene, OR

Expected background and suggested reading

For this WARTHOG workshop to be effective, we decided that students should be familiar with certain basic topics in commutative algebra and representation theory. Do not worry if you do not have this prerequisite yet, there should be time to read up before the workshop begins (though WARTHOG is fast approaching, so if multiple topics are new you should start learning ASAP). Below we give suggested readings. For these background topics, the workshop will recall the needed facts and give examples — but we cannot expect anyone who's never seen the representation theory of finite groups or the definition of an algebraic variety to be able to follow most of the lectures.

Most of these background topics can be found within two well-known textbooks:

  • [E] Eisenbud, Commutative Algebra.
  • [FH] Fulton and Harris, Representation theory.

    The expected background for attendees of this year's WARTHOG is:

  • Finite groups and their representation theory over the complex numbers. [FH] Chapters 1-2.
  • Algebraic facts about polynomial rings and their modules. Hilbert's basis theorem, and Hilbert's syzygy theorem. The concept of a free resolution. [E] Chapters 1-2.
  • The beginnings of algebraic geometry. Algebraic sets and their coordinate rings, morphisms between algebraic sets/varieties, the relationship to algebras and algebra morphisms. Projective varieties. [E] Chapter 1.

    (For most of these background topics, it is most important to know the definitions and statements of theorems, and to get a feeling for examples, rather than being very familiar with the proofs.)

    The following topics are advantageous to know, but not necessary:

  • Algebraic representations of GL_n(C), e.g. highest weight theory. [FH] Chapters 11, 12, 13, 15.
  • Representation theory of symmetric groups in characteristic zero, e.g. Specht modules, the Pieri rule. [FH] Chapter 4.
  • Schur functors and Schur-Weyl duality. [FH] Chapter 6.
  • Regular sequences and Koszul complexes. [E] Chapter 17.

    We recommend that you pick something on this list that you don't know (priority being the basics) and try to read up before the workshop. 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.

    Additional references

    Below we list some good references for learning some of the specific material discussed in the workshop. It is not expected that you will have read them.

    This is a survey paper on Stillman's conjecture and some of its proofs. We'll cover a lot of the topics in this paper.

  • [ESS] Daniel Erman, Steven V. Sam, Andrew Snowden. Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials. Bull. Amer. Math. Soc. 56 (2019), 87-114. arXiv.

    A good introduction to infinite-dimensional algebra with symmetries:

  • [D] Jan Draisma. Noetherianity up to symmetry. In Combinatorial Algebraic Geometry (pp. 33-61). Springer, Cham. arXiv.

    This expository paper gives an introduction to GL-equivariant commutative algebra. It also contains background on symmetric group representations, GL representations, and Schur--Weyl duality:

  • [SS] Steven V Sam, Andrew Snowden. Introduction to twisted commutative algebras. arXiv.