A Map to Represent the Scope of Mathematical Research and Show Concrete Applications of the Discipline
How do we count protesters? What is the probability that two ministers share the same birthday? How can we encrypt and decrypt a message? Is it possible to make a sphere from a sheet of paper?
These are some of the questions that the “Universe of Mathematics” map aims to illustrate.
Created in 2023 by CNRS Mathematics (Insmi), this map visually represents the fields of mathematical research and demonstrates how mathematics addresses concrete questions in our daily lives.
Origins and Creation of the Universe of Mathematics Map
The idea for the Universe of Mathematics project emerged from the desire to compile and visually represent mathematical research, showcasing as many fields as possible.
" To create this map, the first step was to establish a relevant cartography of research areas in mathematics. We used several sources, such as the national synthesis and foresight on mathematics (2022) or the Mathematics Subject Classification (2020), which covers almost all research themes in mathematics (algebra, analysis, number theory, differential geometry, statistics, …), " explains Christophe Delaunay.
A selection of research areas was chosen to make the map as accessible as possible.
Why a Map Representing Space?
“In a terrestrial map, borders are marked, whereas in mathematics, the borders are very diffuse”: the universe is vast and allows passages between galaxies to avoid marking borders between research areas. A wormhole is also mentioned to represent these passages.
What Does the Universe of Mathematics Map Contain?
The map depicts 5 “major galaxies” of mathematics: topology and logic, arithmetic, algebra and geometry, probabilities and statistics, analysis.
In these galaxies, constellations and stars correspond to mathematical subjects: model theory, number theory, dynamical systems, differential geometry, probabilities, numerical analysis and scientific computing, decision theory, non-commutative geometry, discrete probability, etc.
In parallel, several thematic nods are found on the map, represented by drawings.
Additionally, thematic nods are found on the map, represented by drawings. For example, the constellation in the “Arithmetic” galaxy depicts an elliptical curve echoing the central field in number theory, the tightrope walker in the “Analysis” galaxy refers to control theory, and the musical score connects Langlands’ program to harmonic analysis. The map also features the towers of Hanoi, Möbius strip, Earth deforming space-time, Schrödinger’s cat, and more.
The map also highlights 11 concrete and mathematical questions. Among them: Are there more ways to shuffle a deck of 52 cards or atoms on Earth? Does the Barber of Seville shave himself? How to measure the complexity of a number? What happens during a collision between two vehicles?
This map is intended for mathematicians for their outreach activities, the broad ecosystem of mathematics, but also for the general public and all curious minds.