The original proposal suggested a systematic development of Discrete Harmonic Analysis (Objective 0), focusing on three domains: the Grassmann scheme (Objective 1), high-dimensional expanders (Objective 2), and the quantum hypercube (Objective 3).
As part of Objective 0, a systematic theory of complexity measures of Boolean functions on various domains has been developed, by the PI, his collaborators, and several students (part of it remains unpublished).
As an application, we were able to determine the structure of maximum size 2-intersecting families of permutations, a classical problem in extremal combinatorics which has so far resisted attacks.
Our systematic theory applies to domains such as the slice, the symmetric group, and the perfect matching scheme, but does not apply to domains such as the Grassmann scheme.
Regarding Objective 1, the period just prior to the initiation of the project saw the advent of a new general tool, global hypercontractivity, by Noam Lifshitz.
This tool serves as a unifying principle for Discrete Harmonic Analysis on many domains, and in particular, gives the "right" way to consider the Grassmann scheme.
Due to this, we directed our efforts to the complexity measures direction, so far unsuccessfully.
As part of Objective 2, we developed Discrete Harmonic Analysis on high-dimensional expanders, by viewing them as "approximately sequentially differential".
As an application, we construct explicit hard instances for Sum-of-Squares. Our work was later improved by a group which went beyond simplicial complexes, but otherwise closely followed our footsteps.
Regarding Objective 3, unfortunately we were unable to find proper collaborators, and due to this, we decided to direct our attention at several other related topics: Discrete Harmonic Analysis on the symmetric group and the slice, and Approximate Polymorphisms.
The symmetric group is perhaps the most natural object of study which goes beyond Boolean Function Analysis.
The PI was part of the initial effort of developing Discrete Harmonic Analysis on the symmetric group via the route of global hypercontractivity.
Following this, we developed a notion of "Fourier expansion" for functions on the symmetric group, as well as on the perfect matching scheme.
Additionally, we proved a tight structure theorem for Boolean function close to degree 1, improving on several earlier works by the PI.
This is an analog of the classical Friedgut-Kalai-Naor theorem for the symmetric group; the proof is much more involved.
The slice is an even simpler domain than the symmetric group, closely related to the biased Boolean cube.
When the slice is relatively balanced, it indeed behaves very similarly to the Boolean cube with the matching bias.
This is no longer the case when the slice is heavily skewed.
We determined the exact point at which this happens, by identifying the "junta threshold" for constant degree functions.
Related to the preceding two topics is our work on "sparse juntas" which describes the structure of Boolean functions close to degree d on the biased Boolean cube (or on the corresponding slice).
Our highly tight structure theorem enabled us to determine several "critical exponents".
The other area which we focused our attention on was Approximate Polymorphisms, an area motivated by Universal Algebra (and its applications to the study of CSPs in computer science), Property Testing, and PCPs.
We were able to show that in the case of functional Boolean predicates, approximate polymorphisms are close to exact polymorphisms; in upcoming work, we generalize this to arbitrary Boolean polymorphisms.
Additionally, we were able to determine the "pseudorandomness threshold" for the functional AND predicate.
