Common questions about Topology

Short answers, pulled from the story.

What is topology and how does it define the relationship between a coffee mug and a doughnut?

Topology is a branch of mathematics that cares nothing for the exact measurements of an object but everything about how it is connected. A pliable torus can be deformed into a coffee mug by creating a dimple and enlarging it while shrinking the central hole into the handle without tearing or gluing. This ability to distinguish between objects based on their connectivity rather than their size or shape defines the entire field.

When did Leonhard Euler publish his paper on the Seven Bridges of Königsberg and what was the result?

The paper on the Seven Bridges of Königsberg was published in 1736 and is regarded as one of the first practical applications of topology and the birth of graph theory. Euler proved that a route crossing each of the seven bridges exactly once without retracing any steps was impossible because of the connectivity properties of the landmasses. This result did not depend on the exact shape of the bridges but only on which bridges connected to which islands.

Who coined the term topology and when did the English form appear in print?

Johann Benedict Listing coined the term Topologie in 1847 in his work Vorstudien zur Topologie. The English form topology appeared in Listing's obituary in the journal Nature in 1883. Listing had been using the word in correspondence for ten years before its first appearance in print, distinguishing qualitative geometry from the ordinary geometry that treated quantitative relations.

What is the hairy ball theorem and what does it state about combing hair on a sphere?

The hairy ball theorem of algebraic topology states that one cannot comb the hair flat on a hairy ball without creating a cowlick. This fact is immediately convincing to most people even though they might not recognize the more formal statement that there is no nonvanishing continuous tangent vector field on the sphere. The result applies to any kind of smooth blob as long as it has no holes proving that the shape of the sphere is irrelevant to the outcome.

Who won the 2016 Nobel Prize in Physics for their work on topological orders and what was their contribution?

David Thouless, Duncan Haldane, and Michael Kosterlitz were awarded the 2016 Nobel Prize in Physics for their work on Topological orders. Their research involved the possibility of obtaining a one-way current which is a current protected from backscattering. This was first discovered in electronics with the famous quantum Hall effect and then generalized in other areas of physics such as photonics.

When was the 2022 Abel Prize awarded to Dennis Sullivan and for what contributions?

The 2022 Abel Prize was awarded to Dennis Sullivan for his groundbreaking contributions to topology in its broadest sense. His work focused particularly on the algebraic, geometric and dynamical aspects of the field. Modern topology depends strongly on the ideas of set theory developed by Georg Cantor in the later part of the 19th century.

Read the full story about Topology →