Explore the method of Gauge transform in this lecture by Jeffrey Galkowski, part of the SMS 2023 series on Periodic and Ergodic Spectral Problems. Delve into the construction of the Gauge transform, examining crucial aspects such as cutoff, top order, unitary invariance, and Fourier multipliers. Investigate resonance zones, finite dimensional decomposition, and unitary transformation. Analyze the monotonicity, unique solutions, and sum of eigenvalues. Gain insights into analyticity, determinants, and derivatives as they relate to the Gauge transform method.