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In the main part of this PhD thesis, we consider a periodically realizedmicroscopic superconductor described by BCS theory, which is subject toexternal electromagnetic fields. We show that the superconductor is properlydescribed by Ginzburg--Landau theory in the macroscopic and weak magnetic fieldlimit. The main novelty of our results is to allow for a non-vanishing magneticflux through the unit cell of the lattice of periodicity. These main resultsare supplemented by various unpublished notes in the field of BCS theory.Furthermore, we preface the presentation of these results with a comprehensiveintroduction suitable for master's or PhD students. Thereby, we hope tocontribute to filling the gap of missing introductory literature in the field.  The thesis comprises a second topic, in which we provide ideas for setting upquantum lattice systems in order to prove exponential estimates for theadiabatic theorem. These notes are the result of studies in this field, whichhave been conducted during a research stay at the University of BritishColumbia (UBC) in Vancouver, Canada.

BCS Theory in the Weak Magnetic Field Regime for Systems with Nonzero  Flux and Exponential Estimates on the Adiabatic Theorem in Extended Quantum  Lattice Systems