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Spring 26 Special Session: Homological Algebra Mini-Course

Starting from April 4, we will hold a one-month mini-course on homological algebra.

We will meet 4:00-6:00 PM EST. The meeting location is Smith Hall 204. If more people are interested in this special session, we can arrange to meet on Zoom.

Here is a tentative syllabus:

  • Spectral sequences
  • Sheaf cohomology
  • Derived functors and derived categories
  • Triangulated categories and stable infinity-categories
  • Cohomology operations and Steenrod squares
  • (Optional Application:) Lie algebra (co)homology and Hochschild-Serre spectral sequences
  • (Optional Application:) Floer homology

Currently we have four speakers, and if you are interested in contributing a talk we are happy to make arrangements. We will have dinner after each talk every week. A more detailed syllabus, including references (mainly research papers) will be provided shortly after, along with the title and abstract of each talk and (non-guaranteed) typed lecture notes.

April 4, Yikai Teng:

Filtered complex and spectral sequences

Abstract: In this talk, we define the category of spectral sequences. Additionally, we introduce one of the most common ways to construct a spectral sequence, i.e., from a filtered chain complex. As an application, we sketch the proof of Serre spectral sequence for fiber bundles.

April 13 (This talk is on Monday!), Jose Alfredo: Homological algebra, what is it, and why?

Most of us have seen homological algebra in topology using of Z-modules, but it can be developed in many other settings, such as algebraic geometry, number theory, and Lie algebras. All these theories can be studied simultaneously by introducing abelian categories. This first talk will introduce the definitions and many examples. I will explain why it is computationally interesting to still consider abelian categories in general, and state the embedding theorem as an important result that allows us to transfer many results from the category of modules to abstract abelian categories.

April 18, Jose Alfredo: Derived Functors

One of the reasons homological algebra is computationally efficient is the naturality of the long exact sequence, and we will see how this can be further exploited. In this talk I will introduce universal delta functors and state the main ingredients that allow us to prove that every derived functor is a universal one, giving us a good foundation for homological algebra in abstract abelian category we are working in. This is fundamental in areas such as algebraic geometry and group cohomology. This second talk will follow chapter 2 of Weibel.