The beginning of the 20th century brought an important change in the analysis of partial differential equations. It became clear that it was necessary to study their solutions in a more general sense than in the classical sense (i.e. pointwise). This led to the introduction of the concept of so-called weak solutions, which solve the corresponding integral equation in a suitable sense, and their subsequent analysis. A key tool for the analysis of weak solutions is the so-called Sobolev spaces and their properties. A very important role is played by the so-called Sobolev embeddings and their compactness, which is what Dr. Mihula from the Department of Mathematics will focus on in his habilitation lecture. First, however, he will explain the idea behind these notions, and then he will discuss compactness, which, among other things, is a common tool in the argument that a particular candidate for a weak solution is correct.
"The world around us is not black and white, and neither are seemingly abstract mathematical concepts such as compactness," says Dr. Mihula. Thus, the audience will have the opportunity to experience the shades of grey which surround the question of Sobolev embeddings compactness during his talk, as he will present recent results of a part of his research that deals specifically with the analysis of compactness and non-compactness of Sobolev embeddings.
In addition to research, Dr. Mihula is also involved in teaching. For the summer semester 2021/2022, he received the Dean's Award for Excellence in Teaching in the category of tutor and for the winter semester 2023/2024, the same award in the category of lecturer. He is very popular among students, they can meet him for example in the course Complex Analysis.
Dr. Mihula is also a member of the Czech Mathematical Society and the Union of Czech Mathematicians and Physicists, and he also represents young scientists in the EMS Young Academy, which was established by the European Mathematical Society to support young researchers in mathematics. In 2022, he was the chairman of the organizing committee of the conference Nonlinear Analysis, Function Spaces and Applications 12, which took place at the Department of Mathematics of the Faculty of Electrical Engineering of the Czech Technical University from 30th May to 3rd June of the same year and was attended by world mathematical experts.
