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Charlene Kalle

Mathematical Institute, Leiden University

Qualitative Differential Equations · Nov 11 - Nov 25, 2023

From November 12 to November 25, 2023, I visited the LUMS Mathematics Department in Lahore to teach in the course Qualitative Differential Equations that is part of the International Mathematics Master (IMM). In these two weeks I taught ten sessions in total, one session each day. The topic of the first week was measure theory, including an introduction to invariant measures for dynamical systems. In the second week we focused on ergodic theory and the statement and proof of the Birkhoff Ergodic Theorem. This famous result on dynamical systems is a mathematical formalisation of Boltzmann's hypothesis from statistical mechanics which roughly states that for large systems of interacting particles in equilibrium time averages equal the space average. This material builds on the theory of discrete time dynamical systems that was introduced earlier in the course by Prof. Stefano Luzzatto (ICTP) and complements the part on continuous time dynamical systems that was taught by Prof. Adnan Khan (LUMS).

There are 15 students in the course, 11 of which were regularly present during the classes. Most of them are from Pakistan, two are from Algeria. Four of the students are women. The students had various backgrounds. Besides the IMM students, there were bachelor students from Mathematics, a master student from Physics and several PhD students in Mathematics. On average the level of the students was high. They are all highly motivated and participate very well in the classroom. They ask very good questions. A noticeable difference between the students is that several of them had already seen measure theory before, while for some it was completely new (especially for the students coming from a physics background). This made the first week a bit challenging, but with hard work everyone eventually managed.

For the discrete time dynamics part of the course, we deliberately chose active learning methods. Rather than lecturing, I set up the classes to have group discussions about exercises and about the interpretation of the mathematical results.

For these discussions I took short student presentations as a starting point. During the two weeks of my visit the students in class presented the solutions to several exercises to their fellow students. I emphasised that these presentations were not for grading purposes, but for all of the class to try and understand the material better together. This led to all students feeling comfortable (sometimes after some encouragement) to step forward even if they did not have a finished solution to the exercise ready. For me, these presentations showed that all students have developed a basic understanding of the material and some students have an excellent grasp of the theory. Currently, the students are working on joint projects that have a deadline at the end of the semester. We divided and discussed the projects before I left. I have the feeling that the students learned a lot from this crash course in ergodic theory, both in terms of the new material and in working style.

Besides the teaching activities, I also gave two talks to a general audience during my visit. On Wednesday, November 22 I had the opportunity to visit the Forman Christian College University and gave a talk on normal numbers. The visit was very well arranged and it was very interesting to see another campus outside LUMS and interact with the mathematicians there. On Friday, November 24 I gave a talk in the John Conway spirited lecture series at LUMS. Even though the day was a public holiday due to smog, the live audience was large, and several more people joined online. I think that the John Conway spirited lecture series is very valuable in that it gives a large and general audience the possibility to join in on accessible mathematics talks and I hope I did my part to honour this nice initiative.

I was lucky that during my visit I could witness one of the other important outreach activities of the LUMS Mathematics Department, namely the math circles. It was very interesting to see how the staff was able to engage children from many different backgrounds and ages in a fun afternoon of themed mathematics activity. I will take this experience home and will try to incorporate at least part of it in my home country. I have already found people who are interested in the idea. Another special experience during my stay was the invitation of Syed Babar Ali to meet him in his factory on Friday, November 24.

The highly adequate support offered by the staff members at the LUMS Mathematics Department, the warm welcome by every single person I met in Pakistan, the professional activities and facilities at the LUMS Mathematics Department (especially the women in science wall), and the facilities on the LUMS campus including the REDC guest house all thoroughly impressed me. In my opinion, the IMM is a very valuable initiative and I think that LUMS provides an excellent environment for its embedding. The current set-up facilitates the exchange and interaction between the LUMS Mathematics Department, the larger mathematics community in Lahore, and the international mathematics community very well.


Applicants are divided into two groups: "International Applicants" who have foreign citizenship and "Pakistani Applicants" who either hold Pakistani citizenship, have dual citizenship, or are Pakistanis residing outside the country.

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