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Section 1.1 Applied mathematics

Concerned with the mathematical formulation of technical systems, physical problems, and natural phenomena, the field of applied mathematics has undergone drastic changes over the last decades. Traditional applied mathematics was often motivated by theoretical thought experiments. A. Einstein is famous for using thought experiments to formulate plausible theoretical concepts about nature, which can then be formulated as mathematical models and be analysed. Deep mathematical methods were developed for the analysis of these models to gain insight into the engineering, physical, or natural system at hand. A beautiful theory of Applied Functional Analysis developed, which we present here.

Traditionally, data did not play a big role in this process, and data analysis would be left in the capable hands of Statisticians. If quantitative mathematical predictions were needed, applied mathematicians would focus on numerical methods. In fact, numerical analysis and scientific computing became a driving force of applied mathematics for many years. Experimental data would be included, somehow, but a full statistical analysis was not at the centre of interest.

This has changed. The fast development of data collection methods in all areas of sciences in recent years has modified the demands on applied mathematics. Some data sets are so vast that they can be seen as continua of data. Modern applied mathematicians engage in direct collaboration with the sciences, and science requires that their data are included in the modelling from the very beginning. Science expects us to provide an explanation of the available data and possibly make testable predictions. The traditional theoretical approaches are still possible, but applied mathematics has expanded to include inference, statistical learning, data analysis, and AI as applied math tools.

In my view, an applied mathematician in the 21st century needs to gain skills in:
  1. development of new mathematical models (modelling)
  2. theoretical analysis of mathematical models,
  3. numerical solutions of these models,
  4. data inferences and statistical learning.

This textbooks focuses on the theoretical aspects of applied mathematics (i.e. item (2)). This does not mean that the material is old. On the contrary, some parts of this book include quite modern approaches, for example the rainbow of function spaces, and the chapter of semigroup theory. Other chapters cover very traditional material, such as the chapters on operators and function spaces, and the chapter on variational methods.