I'll discuss the following applications of the finite Fourier transform to combinatorial topology.

1. We study the homology of certain arithmetically constructed spaces called Sum Complexes. In particular, it is shown that sum complexes on a prime number of vertices are hypertrees, i.e. have vanishing rational homology. One ingredient in the proof is a remarkable theorem of Chebotarev concerning submatrices of the Fourier matrix. (joint work with N. Linial and M. Rosenthal)

2. Uncertainty principles roughly assert that a nonzero function and its Fourier transform cannot both be concentrated on small sets. We will point out a connection between discrete uncertainty inqualities and the topology of sum complexes.

3. We give a Fourier characterization of the top homology of balanced complexes. This leads to an extension and a short proof of a recent result of Reiner and Musiker on a topological interpretation of the coefficients of the cyclotomic polynomial.