To understand our world, we classify things. A famous example is the periodic table of elements, which describes the properties of all known chemical elements and gives us a classification of the building blocks we can use ...

In the '70s, physicists introduced a new type of symmetry – supersymmetry – to address some unresolved issues in particle physics models. Its mathematical foundations involve the representation theory of the associated ...

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 ...

We describe some of the beautiful mathematical structures that arise from the study of the associativity equation. Our journey takes us from combinatorics to abstract algebra, with brief excursions through geometry and ...

This snapshot introduces the theory of trisections of smooth 4-manifolds, an area of exploration in low-dimensional topology aiming to make four-dimensional spaces more understandable. Along the way, we discuss the concepts ...

We introduce Tadao Oda's famous question on lattice polytopes which was originally posed at Oberwolfach in 1997 and, although simple to state, has remained unanswered. The question is motivated by a discussion of the ...

Differential equations make predictions on the future state of a system given the present. In order to get a sensible prediction, sometimes it is necessary to include randomness in differential equations, taking microscopic ...

Is it possible to approximate arbitrary points in space by vectors with rational coordinates, with which we, and computers, feel much more comfortable? If yes, can we approximate those points arbitrarily close? In this ...

We describe how simple observations related to vectors of length 1 recently led to the proof of an important mathematical fact: the sharp Stein–Tomas inequality from Fourier restriction theory, a pillar of modern harmonic ...

The solutions to a surprising number of mathematical questions can be classified by the ADE Coxeter–Dynkin diagrams. This snapshot will show you a selection of these questions and how they correspond to the ADE Coxeter–Dynkin ...