Holomorphic Methods in Physics [NAT3062] WiSe 2025/26

The module Holomorphic Methods in Physics aims to provide an overview of advanced aspects of function theory and its applications in mathematical physics.

This makes the lecture particularly suitable for physics students with an interest in mathematics from the fifth semester onwards.

This is not just a “typical analysis lecture” with all the usual inequalities. Instead, we will combine techniques from various fields of mathematics. For instance topology, algebra, geometry and functional analysis.

1. Recap of basics of complex analysis

2. Biholomorphic maps

3. The Mittag-Leffler Theorem and the Weierstrass Product Theorem

4. Spaces of holomorphic functions

5. Quantization schemes – simple case

6.  Toeplitz operators and anti Wick-ordering

7. Quantization schemes for compact Lie groups
   
   

Further possible topics could be: Sheaves and Sheaf cohomology, Riemann surfaces, …

Analysis and Linear Algebra, any basic course in complex analysis (e.g. Analysis 3 for physics), familiarity with basic notions of measure theory and Hilbert space theory; acquaintance with group theory is not required but we will study complex analysis on groups. We will introduce the definitions from algebra that we need.

Knowledge in theoretical physics (quantum mechanics/quantum field theory) is not necessary but can be helpful.

The lecture is based on the following reference works, among others:

• Holomorphic methods in analysis and mathematical physics von Brian C. Hall (arXiv)

• Funktionentheorie 2 von Reinhold Remmert und Georg Schumacher (Springer)

• Complex Analysis von Serge Lang (Springer)

References to further reading are also provided in the Moodle course where appropriate.