Plática dada por Santiago López de Medrano (Instituto de Matemáticas de la UNAM) en el Segundo Congreso Nacional de Geometría Algebraica el viernes 2 de marzo del 2018 Casa Matemática Oaxaca, en Oaxaca de Juárez, México
In the study of a certain type of intersections of real quadrics with dihedral symmetry a relation was stablished between their smoothness and the minors of the Vandermonde matrix with the n-th roots of unity as entries, also known as the Discrete Fourier Transform (DFT) Matrix.
In this talk we consider the analogous situation of complex projective varieties with cyclic or dihedral symmetry, for which their smoothness is precisely equivalent to the fact that certain minors of the DFT matrix do not vanish. This question turned out to be very difficult to answer in general and we have obtained only some partial results.
From numerical evidence we arrived to some conjectures and results about those minors. It turned out that our conjectures were true, one of them by and old theorem by Chebotaryov stating that in the case n is prime all the minors are non-zero. The other one we proved ourselves. Recently we have obtained some results for the case where n is a prime power.
This question is of interest in the field of Signal Recognition, starting from work by Terry Tao where he rediscovered and applied Chebotaryov's theorem. This was followed by papers in Electrical Engineering journals among which some recent ones contain part of our results.
An important part of this research is joint work with Matthias Franz.