Lukas Schimmer

University of Copenhagen QMATH

University of Copenhagen:

Summer 2018: Advanced Mathematical Physics

Course description: The question of why the negative electron in the hydrogen atom does not fall into the positive nucleus cannot be explained by Classical Mechanics. In this course a self-contained proof of the stability of atoms and molecules within the model of non-relativistic Quantum Mechanics was presented. The first part contained an introduction to one-body Quantum Mechanics which included the proof of the stability of Hydrogen, as well an investigation of excited states. This culminated in the proof of the famous Lieb–Thirring inequality. The second part introduced many-body Quantum Mechanics and the main result of this course, the stability of the second kind of atoms and molecules. To prove this, several important results about non-interacting particles and electrostatic inequalities had to be established.

Course notes: I have written lecture notes in LaTeX which are available here:

Lecture notes on the Stability of Matter

The content is largely based on The stability of matter in Quantum Mechanics by E. H. Lieb and R. Seiringer as well as Analysis by E. H. Lieb and M. Loss.

California Institute of Technology:

Spring 2017: Ma 191c-sec1 Introduction to Random Matrices

Course description: This course provided an introduction to random matrix theory. It introduced relevant concepts, such as the concentration inequalities and the moment method. The matrix norm of a random matrix was investigated and several proofs of Wigner's semi-circle law were given. The result was then given wider context in the theory of free probability. The course also presented more specific results concerning Gaussian ensembles and the edge of the spectrum.

Winter 2017: Ma 108b Classical Analysis

Course description: This course dealt with the fundamental question of how one should think of the size of a set of real numbers. It provided a ground-up introduction to the theory of Lebesgue measure, which included the discussion of Cantor sets and non-measurable sets. Using the method of Lebesgue integration, several powerful techniques of Analysis, such as the dominated convergence theorem, Fubini'stheorem and the Lebesgue differentiation theorem, were established.

Winter 2017: Ma 98b Advanced Functional Analysis

Course description: This reading course covered several chapters of Functional Analysis by P. Lax adapted to the student's interests.

Fall 2016: Ma 111a Topics in Analysis

Course description: The spectral theorem provides conditions under which a bounded operator can be diagonalised. Using the theory of Banach algebras this result was proved and several equivalent formulations were established. The result was generalised to unbounded self-adjoint and normal operators. The course included various applications of the spectral theorem. In the last part of the course the theory of semi-groups of operators was introduced and the famous Hille–Yosida theorem was proved.