Leavitt path algebras, which are algebras associated to directed graphs, were first introduced about 20 years ago. They have strong connections to such topics as symbolic dynamics, operator algebras, non-commutative geometry, representation theory, and even chip firing. In this talk we sneak a peek at these fascinating algebras and their interplay with several seemingly disparate parts of mathematics.

Abstract:
The Crane-Yetter state-sum model is a cornerstone example of a 4-dimensional topological quantum field theory (TQFT) associated to a modular tensor category. While often said to be fully extended, the precise sense in which this is true depends delicately on the target higher category and the SO(4)-fixed point data required by the cobordism hypothesis. In this talk, I will revisit the extension problem for Crane-Yetter using tools from stable homotopy theory. Specifically, I’ll review how invertible TQFTs are classified by maps of spectra, and how this perspective sheds light on the possible ways to extend the Crane-Yetter partition function down to points. The main tool is an instance of the homotopy hypothesis, namely the relation between Picard groupoids and connective spectra, of which I will show explicit examples in low dimensions. This framework leads to concrete answers: for example, I will show that there are exactly six ways to extend Crane-Yetter to a fully extended TQFT with target BrFus, all assigning the same modular category to a point.