Vertex operator algebras exhibit a feature much like Lie algebras in that they admit too many modules for the category of all their modules to exhibit nice structure. However, good choices of module category can lead to categories with very rich structure. For example the categories of admissible modules over rational vertex operator algebras are modular tensor categories, as proved by Huang. I will present some recent work on making the study of vertex operator algebra module categories more tractable by replacing them by Hopf algebras, an arguably simpler algebraic structure. The guiding example will be the free boson.
There is always more that can be learnt from the free boson
Valued CSP is an optimization version of the Constraint Satisfaction Problem. Instead of finding any homomorphism from A to B, our goal in VCSP is to find a homomorphism that has a low cost. The complexity of VCSP is well understood with a dichotomy proved by Kolmogorov, Krokhin and RolĂnek in 2015.
In this talk we will look at how to define the category of valued structures to get nice products in this cateogory. In particular, we will want weighted polymorphisms of a structure A to correspond to mappings from powers of A to A. This will require generalizing the notion of valued relational structure.