Lecture 1: The many-electron Schroedinger equation for atoms and molecules.

Stationary states. Born-Oppenheimer approximation. Brief overview of interesting many-body observables: multiplicities, spin, angular momentum, density, pair density, occupation numbers. Brief overview of further approximations: basis sets, molecular orbital theory, Hartree-Fock theory, multiconfiguration method, density functional theory.

Lecture 2: The one- and two-body density matrices of an N-electron system.

Definitions, basic properties, re-interpretation of nonlinear models (Hartree-Fock; multiconfiguration) as closure assumptions on higher-order density matrices. Rudiments of second quantization. Proof of ''naive Pauli principle'' that occupation number of any single-electron state is at most one. Examples of electron pair states which can be multiply occupied. Connection with BCS theory of superconductivity.

Lecture 3: Bond-breaking as a bifurcation problem.

As shown in Lecture 1 it is common in the physics and quantum chemistry literature to write down nonlinear approximations in fewer variables to the Schroedinger equation. But many of the mathematical consequences of the nonlinearity (e.g. bifurcations; multiplets of appearing and disappearing equilibria) are rarely if ever investigated in the literature. I will only treat the simplest possible example (Hartree-Fock equations for minimal-basis-H_2; these are a coupled system of two nonlinear algebraic equations in four complex variables), but will show that the behaviour is already extremely rich and interesting. In particular, as the interatomic distance is increased, a ''bond-breaking'' bifurcation occurs in which the topology of the ground state changes from a trivial U(1) to a three-dimensional projective space. This behaviour will be contrasted with that of the linear Schroedinger equation. I will close with an outlook on ''nonlinear effects'' in quantum many-body theory.

In this lecture I will present a geometrical framework for the reduction of the classical and quantum n-body problem with respect to translations and rotations, and explain how it is related to the more traditional coordinate-based approaches. Topics may include Jacobi coordinates, rovibrational and Watson Hamiltonians, fiber bundles and gauge theories, covariant derivatives, Coriolis and Yang-Mills gauge fields, the Iwai monopole, and Kaluza-Klein identities. The presentation and balance of topics will depend on the audience.

Lecture 2. Internal spaces, monopoles and string singularities in the n-body problem

This lecture will concern the structure of the internal spaces that occur in the n-body problem, including interesting subsets such as monopoles (singular sets that act as sources of the Coriolis gauge field) and singularities (strings) of the gauge potential. I will concentrate on the 3- and 4-body problem, and address the question of singularities of the Schroedinger wave function.

Lecture 3. Gauge theory of small vibrations in the n-body problem

This lecture will concern the problem of small vibrations in a rotating system. This is an old problem in molecular dynamics, one that traditionally involves the Eckart coordinates and Eckart frame. I will explain the geometrical meaning of Eckart's conventions, why they are important for understanding small vibrations, and generalizations.

2) The uncertainty principle in quantum mechanics is a commutator relation between conjugate operators, e.g., position and momentum. Likewise, the uncertainty principle for energy and time is invoked which is NOT based on a commutator relation since time is not an operator. This asymmetry in the uncertainty relations has provoked a number of (unsuccessful) attempts to construct a time operator. We have followed a different route: accepting the asymmetry, we have tried to justify it. This can be done by formulating time as a concept which can be derived from space (without relativity, just logically!). Semiclassics plays an important role in the derivation which will be presented. (Third lecture.)