In how many ways can you go for a walk along a lattice grid in such a way that you never meet your own trail? In this snapshot, we describe some combinatorial and statistical aspects of these so-called self-avoiding ...

100 people leave their hats at the door at a party and pick up a completely random hat when they leave. How likely is it that at least one of them will get back their own hat? If the hats carry name tags, how difficult ...

Graphs are mathematical objects composed of a collection of “dots” called vertices, some of which are joined by lines called edges. Graphs are ideal for visually representing relations between things, and mathematical ...

Big data and deep learning are modern buzz words which presently infiltrate all fields of science and technology. These new concepts are impressive in terms of the stunning results they achieve for a large variety of ...

Formation control is one of the fundamental coordination tasks for teams of autonomous vehicles. Autonomous formations are used in applications ranging from search-and-rescue operations to deep space exploration, with ...

We describe a connection between quantum mechanics and nonlinear wave equations and highlight a few problems at the forefront of modern research in the intersection of these areas.

When the rigorous foundation of calculus was developed, it marked an epochal change in the approach of mathematicians to geometry. Tools from geometry had been one of the foundations of mathematics until the 17th century ...

We explore mathematical models for physical problems in which it is necessary to simultaneously consider equations in different dimensions; these are called mixed-dimensional models. We first give several examples, and ...

Throughout the history of dynamical systems, instruments have been used to calculate and visualize (approximate) solutions of differential equations. Here we describe the approach of a group of physicists and engineers ...

In this snapshot we will outline the mathematical notion of curvature by means of comparison geometry. We will then try to address questions as the ways in which curvature might influence the topology of a space, and ...