Richard Brauer (1901–1977) was a German–American mathematician who is regarded as the founder of a highly active mathematical area known as modular representation theory. This area grew from group theory, which can be thought of as the mathematical study of symmetries. In this snapshot, we hope to impress on the reader the legacy left by Brauer and celebrate the 60th anniversary of “Brauer’s problems”, a list of 43 conjectures and objectives suggested by Brauer in 1963. These problems inspired an entire branch within character theory, studying “local-global conjectures”.

This snapshot compares two techniques of shuffling a deck of cards, asking how long it will take to shuffle the cards until a “well-mixed deck” is obtained. Surprisingly, the number of shuffles can be very different for very similar looking shuffling techniques. 

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 effects into account. Surprisingly, despite the presence of randomness, our probabilistic prediction of future states is stable with respect to changes in the surrounding environment, even if the original prediction was unstable. This snapshot will unveil the core mathematical mechanism underlying this “regularisation by noise” phenomenon.

Everyone loves a good game, but when the players can access the counterintuitive world of quantum mechanics, watch out!

“God does not play dice with the universe” – Albert Einstein

“Not only does God play dice but… he sometimes throws them where they cannot be seen.” – Stephen Hawking

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 physics, chemistry, and biology. In mathematics, and algebraic geometry in particular, there are many instances of similar  periodic tables”, describing fundamental classification results. We will go on a tour of some of these.