First Fundamental Course of RTG: Mathematical Analysis

Text by Florian Noethen

Organizers: Jens Rademacher and Florian Noethen

From May 4-5, 2021 the first fundamental course of the Research Training Group (RTG) took place via Zoom. Over 20 of our new PhD students attended the 2-half-day course on mathematical analysis and learned about important concepts of mathematics that underlie the different areas of the TRR 181.

Fundamental courses and planning:

The TRR 181 is an interdisciplinary project that contains researchers from very diverse backgrounds. To bridge the gaps between knowledge and languages of different disciplines, we established our own graduate school, the RTG. It provides various benefits to our PhD students and postdocs, such as a mentoring program and different types of courses. Mandatory for all our PhD students are the fundamental courses. They create a basis of knowledge for all main disciplines of our project: mathematics, oceanography, and meteorology. The course on mathematical analysis was the first in the series of fundamental courses.

The organization of the course was not without obstacles. Due to corona, the start of the second phase of the project and the RTG were postponed by half a year. Hence, also the fundamental courses had to be postponed. We decided to wait until the summer semester to have most of our new PhD students on board, but not wait much longer, so that our PhD students can benefit from the fundamental courses early on. The first discipline treated in the fundamental courses was mathematics, as it is needed for the other disciplines as well. Mathematics was split into two parts: analysis and numerics. We began preparing for the courses on mathematics by asking all project leaders about the most important concepts of mathematics needed to understand their subprojects. With these suggestions in mind, a team of project leaders and postdocs organized the course on mathematical analysis. Contentwise, we decided to give an introduction to seven different topics divided among the lecturers, while for the format, we opted for short lecture videos (2h total) and a practical part during a 2-half-day Zoom meeting. The students were tasked to watch the videos before the meeting.

Day 1:

The first day of the course started with some opening words and organizational aspects by Lea Diederichsen and me. Then, Ivan Ovsyannikov took over with the first topic: ‘ODE basics’. The students were encouraged to ask questions on his video in which he discussed the existence and uniqueness of solutions to Ordinary Differential Equations (ODEs). After answering the questions, Ivan went through several exercises on solution techniques for ODEs with the students.

Next up was my slot on ‘Center manifold’. In the video, center manifolds were introduced as a reduction technique to study local dynamics near stationary states. After a brief recap of the video and questions, the students were divided into small groups and sent into breakout rooms. Each group had the choice between two task in which the knowledge from the lecture video had to be applied.

The last slot of the first day was on ‘PDE basics’ by Camilla Nobili. She decided to focus on one of the most important Partial Differential Equations (PDEs) for our TRR 181, the Navier-Stokes equations. In her video, she singled out different types of PDEs contained within these equations and discussed their properties. Again, after a recap and questions, the students were sent into breakout rooms in which they recomputed some of the formulas from the video under slightly different assumptions.

Day 2:

With the most fundamental topics out of the way, the second day moved closer to the heart of the TRR 181. Marcel Oliver started with the slot on ‘Model reduction’. His video showed the derivation of the 1-layer quasigeostrophic equations from the shallow water equations as a limit of rapid rotation. He demonstrated techniques such as non-dimensionalization, a separation into slow and fast modes in the Fourier representation, and epsilon-expansions to arrive at the first order balance condition. During his slot, the students had to derive the second order balance condition by comparing higher order terms of the epsilon-expansions from his lecture.

The next slot contained two topics that were merged into one: ‘Linear waves’ by Anton Kutsenko and ‘Nonlinear waves’ Bing-Ying Lu. Anton Kutsenko highlighted that linear waves can be obtained analytically if one knows the eigenfunctions of the Laplace operator on the specific domain. Moreover, he showed us properties of linear waves, such as reflection and transmission, analytically and in terms of a simulation. Bing-Ying Lu pointed out the differences between linear and nonlinear waves. While linear waves satisfy a superposition principle, nonlinear waves come from nonlinear PDEs for which the sum of two solutions is in general not a solution anymore. Nevertheless, there are special nonlinear waves, so-called soliton waves, that interact in a similar fashion to linear waves. Furthermore, there are so-called shock waves in nonlinear systems that destroy the classical concept of solutions. During the meeting, both Anton Kutsenko and Bing-Ying Lu gave brief recaps of their videos and showed additional simulations or examples. Various exercises on linear waves were discussed and an example of a nonlinear wave was computed in small groups.

The last slot of the course was by Jeffrey Carpenter on ‘Hydrodynamic Instabilities’. For the preparation, he referred to an old, but still great video from the National Committee for Fluid Mechanics Films. The video showed several types of flow instabilities achieved in tank experiments. While the video focused on experiments, Jeffrey Carpenter provided the theoretical background in a lecture during his slot. For the Rayleigh equation, he presented criteria for critical parameters that determine whether the flow is stable or unstable. Central in his talk were Rayleigh’s inflection point theorem and Fjortoft’s extension theorem, which the students had to apply to example cases.

Overall the lecturers did a great job at introducing the individual topics. However, a 2-half-day course cannot possibly cover all the important details. Thus, for further reading we uploaded a list of references to our internal website. Moreover, for latecomers or RTG members who want to revisit the course, the lecture videos, the lecture notes, and the additional material from the course are also available on our internal website.