Helmut Eschrig Knihy

A concise but self-contained introduction of the central concepts of modern topology and differential geometry on a mathematical level is given specifically with applications in physics in mind. All basic concepts are systematically provided including sketches of the proofs of most statements. Smooth finite-dimensional manifolds, tensor and exterior calculus operating on them, homotopy, (co)homology theory including Morse theory of critical points, as well as the theory of fiber bundles and Riemannian geometry, are treated. Examples from physics comprise topological charges, the topology of periodic boundary conditions for solids, gauge fields, geometric phases in quantum physics and gravitation.

Density functional methods are central to a vibrant and evolving field within computational atomic, molecular, solid state, and nuclear physics. Many computational physicists utilize these methods as a straightforward recipe, often without delving into their underlying logic. Despite their significant success, there remains some skepticism among more theoretically inclined researchers. In the 1970s and 1980s, density functional concepts gained traction in mathematical physics, notably with Thomas Fermi theory and Hohenberg-Kohn theory. The 1994 celebration of the thirtieth anniversary of Hohenberg-Kohn-Sham theory prompted lectures on density functional theory for senior students and postgraduates, emphasizing its logical foundation. During the preparation of these lectures, it became evident that while numerous monographs and reviews exist on density functional theory, they predominantly focus on practical applications and improved approximations. The theoretical underpinnings are often scattered and inadequately presented in the literature. This realization inspired the creation of a printed version of the lecture notes, culminating in the current text, which aims to provide a clearer understanding of the logical basis of density functional theory.