Trends in mathematics

What's new in mathematics? This question is not that easy to answer, given the ubiquity and rapid development of this science in countless areas.

Blackboard with mathematical formulas
© Jürgen Haacks, Kiel University

Biomathematics is the most recent form of the discipline. As its name suggests, it is a mixture of mathematics and biology and its objective is to describe and analyse biological problems with the aid of mathematical methods. Currently, this is used in modelling COVID-19 infections. Irrespective of this, the traditional tools of mathematics are involved here: analysis and algebra, probability theory, statistics and much more besides.

Generally, according to Professor Detlef Müller, the trends in mathematics relate both to the increasing number of references to application-based fields and to the rapid and equally important developments in many traditional areas of pure mathematics. For example, new ideas and methods of harmonic analysis have led to significant advances in the theory of differential equations. Professor Müller is also considering these aspects within a new project funded by the German Research Foundation (DFG).

Important applications include new findings on the theory of wave propagations and what are known as Schrödinger equations in quantum physics. Because these equations have since become accessible even to numerical handling via high-performance computers, they can be used in chemistry, for example, to improve understanding of and, above all, predict reaction processes.

Physical string theory and mathematical differential geometry have also been significantly mutually beneficial. In string theory, elementary particles are modelled as strings of one-dimensional threads with the aim of reconciling the theory of relativity and quantum theory. Also, the so-called wavelet functions, whose basic theory was developed by leading harmonic analysts and were then taken up intensively in numerics, have only been a subject since around the 1980s. In contrast to breaking them down into sine and cosine functions as taught at school, here mathematicians work with oscillating functions that are only additionally expanded to a limited degree, as is the case with many signals.

In application, the wavelets play an important role in data compression, for example, when shrinking images to JPEG format. These dynamic functions are also used in geophysics or computer tomography, among other fields.

Author: Martin Geist