On Monday the 13th, I went to the UMN Topology seminar where I saw a talk from Dr. Mike Hill of UCLA. Admittedly I’m completely lost when it comes to most homotopy theory, but I saw “combinatorics” in the abstract and so I went to the talk hoping that I could get something out of it. A fellow UMN grad student Sasha Pevzner introduced me to the “three things” exercise for getting something out of a talk. So here are my three things from this talk:

  1. The “transfer systems” of the cyclic \(p\)-group \(C_{p^n}\) form a lattice which is ismorphic to the Tamari lattice. Lately I have been studying for my upcoming oral preliminary exam, and have been studying posets, in particular the Fundamental Theorem of Finite Distributive latices, so as a note-to-self, I’ll say that the Tamari lattice is not distributive. Here’s how I checked: 
     sage: L = posets.TamariLattice(4)
 sage: L.is_distributive()
 False

    Thanks, Sage!

  2. A question - “what is a \(G\)-spectrum?” From the context of the talk it seems like the idea is that it should generalize a \(G\)-set, but the (nLab page)[https://ncatlab.org/nlab/show/G-spectrum] on it is a bit tough for me to understand.

  3. Another question - “what is known about transfer systems for other groups?” I ended up asking Dr. Hill this question after the talk, and it turns out that not much is known for other groups, partly because their subgroup structure is complex, and the starting point for much of the work was looking at the subgroup lattice.

Dr. Anna Weigandt’s talk (see yesterday’s post) underscored the idea of turning a commutative algebra question into combinatorial question for me. Another main main takeaway for me from this talk is the idea of turning a topological question into combinatorial question.