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A unitary linear operator which resolves a function on $\mathbb{R}^N$ into a linear superposition of "plane wave functions". Most often used in physics for calculating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a quantum state in position co-ordinates into one in momentum co-ordinates and contrawise. There is also a discrete, fast Fourier transform for discretised data.

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Which property lead to the conclusion that BZ is a torus?

In calculating the Chern number in 2D BZ, we assume that BZ is a torus. However, under periodic gauge, we have $\psi_k(r) = \psi_{k+G}(r)$, which means that $u_{k}(r) = u_{k+G}(r) e^{iG\cdot r}$. Sinc …
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