Summer Term 2024

The lectures are Mondays and Wednesdays at 2:00pm at großer Hörsaal, Philosophenweg 12

Motivation:

Nonlinear dynamics is an interdisciplinary part of mathematical physics, with applications in such diverse fields as mechanics, optics, chemistry, biology, ecology, to name but a few. Equations with nonlinearities show a much more diverse behavior than their linear counterparts, for instance self-sustained oscillations, nonlinear competition (as linear superposition does not hold anymore), chaotic dynamics and pattern formation. Pattern formation, in turn, is one of the most fascinating and intriguing phenomena in nature: it takes place in a wide variety of physical, chemical and biological systems and on very different spatial and temporal scales: examples are convection phenomena in geosciences and meteorology, but also patterns occurring in chemical reactions and bacterial colonies. In some circumstances, pattern formation is undesired, for instance the formation of spiral waves leading to cardiac arrhythmias in the heart muscle. In other contexts, pattern formation is even essential for the functioning of a system as in cell division and embryo development.

Contents:

The lecture will start with an introduction to nonlinear dynamics on the level of ordinary differential equations (ODEs), introducing concepts like phase space analysis, attractors, (in)stability of solutions and bifurcations, as well as nonlinear oscillations.

We will then proceed to study spatio-temporal behavior, i.e. partial differential equations (PDEs) and discuss the main questions in pattern formation: when will a homogeneous state become structured, i.e. unstable towards a pattern? What are the generic scenarios/types of patterns? When are patterns stable and are they unique? What determines the wavelength / period in time / amplitude of a pattern? Importantly, a universal description of pattern dynamics exists, that is independent of the system-specific pattern formation mechanism. The method to obtain this description is called multiple-scale reduction, resulting in an amplitude equation (also called center manifold), which is nothing but the famous Ginzburg-Landau equation (Nobel Prize in Physics 2003, originally derived for superconductivity).

Finally, nonlinear waves and solitons (localized waves) will be discussed. They again occur in many systems, from coupled nonlinear springs to hydrodynamic surface waves and nonlinear optics. In addition, solitons have intriguing mathematical properties that will also be discussed.

Prerequisites:

The course is designed for physics students in advanced bachelor and beginning master semesters (students from other disciplines are also welcome). It will be given in English. A basic understanding of physics and differential equations is sufficient to attend. Exercises will be discussed in the tutorials (please register).

Literature:
- SH Strogatz, Nonlinear dynamics and chaos, Westview 1994
- Cross M C and Hohenberg P C, Rev. Mod. Phys. 1993.
- Cross M C and Greenside H, Pattern formation and dynamics in nonequilibrium systems (Cambridge, Cambridge Univ. Press, 2009).

The lecture course provides an introduction to the strongly-correlated physics of QCD and Quantum Gravity. The related physics problems are treated within the Functional Renormalisation Group (fRG), and a survey of alternative approaches is provided.

    Outline

    I The Functional RG
        Euclidean QFT
        Functional Renormalisation Group
        Critical Phenomena & Fixed Points

   II  QCD
        Introduction
        Non-Abelian gauge theories & confinement
        Chiral symmetry breaking in QCD
        QCD at finite T
        A glimpse at the QCD phase diagram

  III Quantum Gravity
        Introduction
        RG approach to quantum gravity
        Gravity and matter
        Cosmological applications*