Have you ever wondered how your computer knows it can trust certain downloads but not others? This snapshot describes some security concerns and one algebraic way of dealing with them. We'll see an interesting procedure ...

A bijection transforms one type of mathematical object into another. Such transformations provide new perspectives on these objects, revealing surprising properties and uncovering new mysteries. We discuss bijections from ...

Ptolemy's theorem is a classical result from ancient Greek mathematics, concerning the lengths of sides and diagonals of a polygon drawn in a circle. In this snapshot, I will explain why this theorem is still important ...

Nonlocal models consider interactions over a range of distances, not just at a single point. In this snapshot, we give a short introduction to nonlocal modeling, explain how it differs from its local counterpart, and present ...

The famous 4-Color Theorem from graph theory states that the vertices of any planar graph can be colored with four colors, so that no neighboring vertices have the same color. The 4-Sample Theorem from algebraic statistics ...

Nowadays, there are different medical imaging techniques to collect data about the progression of Alzheimer's disease in a patient's brain. These data describe different phenomena which are still not understood from a ...

The geometry of lines is a fundamental part of mathematics and the way we interact with the physical world. Core concepts such as distance and angle, and the accompanying theory of trigonometry, have been studied since ...

In this snapshot, we introduce the study of slices of polytopes – geometric shapes with flat sides – and examine the area of these slices. This is connected to combinatorics and polynomials and is surprisingly complex, ...

Statistical modelling is the art of connecting abstract models with scientific or practical questions, so that the models' probabilities throw light on these questions. Standard statistical theory summarizes the results ...

Networks can be found all around us. From the friendships we form, both in person and online, to the websites we visit on the Internet, and the telephones we use to send messages and make phone calls. Understanding the ...