We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders.

Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse ... more >>>

Let $H$ be an arbitrary family of hyper-planes in $d$-dimensions. We show that the point-location problem for $H$
can be solved by a linear decision tree that only uses a special type of queries called \emph{generalized comparison queries}. These queries correspond to hyperplanes that can be written as a linear ... more >>>

We give very short and simple proofs of the following statements: Given a $2$-colorable $4$-uniform hypergraph on $n$ vertices,

(1) It is NP-hard to color it with $\log^\delta n$ colors for some $\delta>0$.
(2) It is $quasi$-NP-hard to color it with $O\left({\log^{1-o(1)} n}\right)$ colors.

In terms of ... more >>>