Nonlinear acoustics has been a topic of research for more than 250 years. Driven by a wide range and a large number of highly relevant industrial and medical applications, this area has expanded enormously in the last few decades. Here, we would like to give a glimpse of the mathematical modeling techniques that are commonly employed to tackle problems in this area of research, with a selection of references for the interested reader to further their knowledge into this mathematically interesting field.

We introduce the idea of curvature, including how it developed historically, and focus on the scalar curvature of a manifold. A major current research topic involves understanding positive scalar curvature. We discuss why this is interesting and how it relates to general relativity.

Also available in Spanish.

Algorithms are mathematical procedures developed to solve a problem. When encoded on a computer, algorithms must be “translated” to a series of simple steps, each of which the computer knows how to do. This task is relatively easy to do on a classical computer and we witness the benefits of this success in our everyday life. Quantum mechanics, the physical theory of the very small, promises to enable completely novel architectures of our machines, which will provide specific tasks with higher comput- ing power. Translating and implementing algorithms on quantum computers is hard.

Waves of diverse types surround us. Sound, light and other waves, such as microwaves, are crucial for speech, mobile phones, and other communication technologies. Elastic waves propagating through the Earth bounce through the Earth’s crust and enable us to “see” thousands of kilometres in depth. These propagating waves are highly oscillatory in time and space, and may scatter off obstacles or get “trapped” in cavities. Simulating these phenomena on computers is extremely important. However, the achievable speeds for accurate numerical modelling are low even on large modern computers.

In this snapshot we present the concept of topological recursion – a new, surprisingly powerful formalism at the border of mathematics and physics, which has been actively developed within the last decade. After introducing necessary ingredients – expectation values, random matrices, quantum theories, recursion relations, and topology – we explain how they get combined together in one unifying picture.