The Kansas Extension Seminar


INTRODUCTION

Welcome to the website for the Kansas Extension Seminar, an online reading and research course for graduate students (roughly speaking) anywhere who are interested in commutative algebra or closely related areas.

There are several motivations for this seminar. During my mathematical travels, I noticed that many graduate students have acquired good technical know-how, but do not always realize there are interesting problems that they can apply those techniques to (perhaps with a little extra learning). Secondly, I always enjoy talking math with different people, and technology has made online discussion/collaboration easier than ever before. Thirdly, I spent quite a bit of time already online on Mathoverflow, but has rarely interacted with students who are actually in commutative algebra. Many people told me that they enjoyed reading MO but are too shy to participate. The last impetus came when I learned about the Kan Extension Seminar ran by Professor Emily Riehl, and read her excellent article about it on the Notices. In fact, you may notice that the design for this website, up to its name, is influenced a lot by her. I thank Emily for some helpful and encouraging correspondence about the project in early 2019.

SEMINAR STRUCTURE

During Summer 2019 (roughly June-August), the participants will discuss with me about their chosen topics and the particular papers/notes they will read. Then they will start reading the relevant materials, with my periodic guidance if necessary. As soon as possible, we will also discuss some open questions related to the topic and possible approaches. During September/October 2019, the participants will take turn to present an online seminar (one hour) on their particular topic. It will be expository, giving the basic background, fundamental results/techniques and some open questions. Online discussions will be continued during Fall 2019. In November/December 2019, there will be a second round of seminar talks, with either more in-depth results or hopefully some new ones by the participants themselves! Hopefully, at some point not long after the semester the participants can produce useful summaries or research preprints on their topics.

Please feel free to contact me with any questions regarding the course or suggested topics.

LIST of TOPICS and SAMPLE PAPERS

Below is a list of broad potential topics (each participant's eventual topic will probably be much more specific) and some sample papers/surveys. They largely reflect my current interests in commutative algebra and related areas, but people are encouraged to suggest their own, see item I. The papers are just samples, and do not necessarily describe the full pictures or the potential ideas that I have in mind. If you are interested in the topic but do not find the papers listed interesting, contact me or explain in your application.

  1. Homological and K-theoretic features of complete intersections. Avramov-Buchweitz's cohomological support. Serre's intersection multiplicity, Hochster's invariant and generalizations. Vanishing of Ext, Tor, and Chow groups. Sample papers: 1,2,3,4,5.
  2. Higher multiplicities: generalized versions of Hilbert-Samuel or Hilbert-Kunz multiplicities and connections to algebraic geometry. Sample papers: 6,7,8,9,10.
  3. Higher nerves of simplicial complexes and connections to combinatorial topology, Helly-type theorems, and Lyubeznik complex in local cohomology. Sample papers: 11,12.
  4. Categories associated to commutative rings. The category of reflexive modules, maximal Cohen-Macaulay modules, totally reflexive modules, finite length and finite projective dimension modules and their connections to singularities that arise in birational geometry. Sample papers: 13,14,15.
  5. Local cohomology: associated primes, cohomological dimensions, cohomologically full rings. Sample papers: 16,17,18,19.
  6. Various closures of ideals and connections to special modules and algebras. Sample papers: 20,21,22.
  7. Non-commutative resolutions of singularities. Sample papers: 23,24,25,26.
  8. Monomial ideals: bounds on regularity and projective dimension, symbolic powers, higher chordality. Sample papers: 27,28,29,30.
  9. You are more than welcome to suggest your own topic. Indeed, coming up with a good one by yourself is a plus. Obviously, for the process to work well the topic should be reasonably close to what I am able to handle. Some ideas can be found from my list of papers (with comments), Mathoverflow questions and answers or the list of accessible papers in commutative algebra.
TO APPLY

The seminar is open to anyone who does not have a PhD before May 2018. To benefit the most from it, the participants should probably have: some previous research experience, a solid background in commutative algebra, self-motivation, access to reasonable internet connection and be able to spend a fair amount of time each week (say, at leas a couple of focused hours) working toward the goals. Due to the time commitment, I expect to be able to accommodate only up to 6 participants in the 2019 version. I will try my best to have a diverse group of participants, so even if you think your preparation is a bit short, but are really interested, please apply. Most importantly, your research interests do not have to match too closely with the topics listed, one of the goals of the project is to find new connections.

To apply, please email me at hdao AT ku.edu a single email containing the following:

Please put some thoughts into the application, especially on the links between what you already knew and what you would like to learn more about. Even if there is no space to admit you in this round, I would be happy to offer some advices or point you to possibly useful reading sources. Application deadline: May 20th, 2019. Participants will be notified by the end of May.

PARTICIPANTS AND RESULTS

To be announced.


CONTACT INFO

My email is hdao AT ku.edu and here is my personal website .


REFERENCES

  1. Support varieties and cohomology over completeintersections
  2. Some Homological Properties of Modules over a Complete Intersection, with Applications
  3. On the vanishing of Hochster's theta invariant
  4. Cohomological support and the geometric join
  5. Hochster's theta invariant and the Hodge–Riemann bilinear relations
  6. Limits in commutative algebra and algebraic geometry
  7. An observation on generalized Hilbert-Kunz functions
  8. On generalized Hilbert-Kunz function and multiplicity
  9. Some computations of the generalized Hilbert-Kunz function and multiplicity
  10. Length of local cohomology of powers of ideals
  11. Higher Nerves of Simplicial Complexes
  12. Connectivity of hyperplane sections of domains
  13. The dimension of a subcategory of modules
  14. The radius of a subcategory of modules
  15. Dimension of the singularity category of a variety with rational singularities
  16. On the associated primes of local cohomology
  17. On the relationship between depth and cohomological dimension
  18. Cohomologically full rings
  19. Deformations of log canonical and F-pure singularities
  20. A guide to closure operations in commutative algebra
  21. Closure operations that induce big Cohen–Macaulay algebras
  22. Grothendieck topologies and ideal closure operations
  23. Non-commutative resolutions and Grothendieck groups
  24. Lectures on Noncommutative Resolutions
  25. Noncommutative resolutions using syzygies
  26. Flops and Clusters in the Homological Minimal Model Program
  27. Bounding the Projective Dimension of a Square-Free Monomial Ideal via Domination in Clutters
  28. The Type Defect of a Simplicial Complex
  29. Exposed circuits, linear quotients, and chordal clutters
  30. Higher chordality: From graphs to complexes