Talk Notes

These are the slides for a lecture series on lattices and thick subcategories I held at ICRA21 in Shanghai. 

Lecture 1 discusses lattices and where they may arise, eventually focusing on lattices of thick subcategories.  An example illustrates how representation theoretic examples in general differ from the tensor triangulated situation.

Lecture 2 discusses properties of lattices in more detail. Which properties are exhibited by lattices of thick subcategories? We find out that most possibilities occur, with one major exception: Lattices of thick subcategories are always algebraic, hence  a distributive lattice of thick subcategories is automatically a spatial frame.

Lecture 3 discusses the idea of approximating thick subcategories by spaces. As we have seen by now, lattices of thick subcategories of triangulated categories coming from representation theory are usually not, on the nose, controlled by a space. Can we still find a space that approximates it in a meaningful way? We discover that the answer is "Yes".

These are the notes for a lecture series on cluster categories of infinite rank I held at a masterclass in Copenhagen. 

Part 1 discusses a purely combinatorial construction of a class categories, which we will justify should be considered as infinite rank versions of a cluster category of type A. In Parts 2 and 3 we discover alternative ways to describe certain cases of these categories coming from different perspectives. In Part 4 we address our by now burning question: How can we address the fact that clusters are not related by finite steps of mutations? Is there even a cluster algebra type object which recognizes our clusters? (The answer is yes). Finally, in Part 5 we illustrate how natural structures on our categories translate to the combinatorial models.