Constant degree functions on various domains

The classical structure theory for functions on the Boolean cube shows that a Boolean function has constant degree iff it can be computed by a constant depth decision tree. The same structure persists for functions on other domains, such as the symmetric group and balanced slices. The plot thickens for highly unbalanced slices, where there is a "junta threshold" which is known exactly.

The situation for functions on the Grassmann scheme was, until recently, only known for Boolean degree 1 functions. Recently we were able to extend the theory to any constant degree, using completely different techniques.

Joint work with Ferdinand Ihringer (SUSTech) and ChatGPT (OpenAI).