The event Modern Mathematics is full on and yesterday was Topology Day. So I have tried to write interesting facts I found about topology. I have to admit that this topic was not completely new to me and even if I wasn’t extremely good at it at university, I enjoyed its concepts a lot. Also, thank you for all of you that have sent me suggestions for this day.
Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for “geometry of place”) and analysis situs (Greek-Latin for “picking apart of place”). Leonhard Euler’s Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field’s first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.
Topology can be formally defined as “the study of qualitative properties of certain objects (called topological spaces) that are invariant under a certain kind of transformation (called a continuous map), especially those properties that are invariant under a certain kind of transformation (called homeomorphism).”
The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside. To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism. Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from “squishing” some larger object.
Interesting applications of topology:
1. In physics, topology is used in several areas such as quantum field theory and cosmology. In cosmology, topology can be used to describe the overall shape of the universe. This area is known as spacetime topology.
2. Knot theory, a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis. Topology is also used in evolutionary biology to represent the relationship between phenotype and genotype. Phenotypic forms that appear quite different can be separated by only a few mutations depending on how genetic changes map to phenotypic changes during development.
Do you know of any other applications of topology? Let me know in the comment box bellow. Moreover, I would advice you to read the article “In space, do all roads lead to home?” to find out more about other interesting facts.
Also, check for a funny way of explaining some topological concepts:
And because there should be no day without some book recommendations, here are some of the ones I find most useful and interesting:
Main Source: Wikipedia;
There are many other sources for learning about topology and I advice you to search and give it a try. At least a small glance over it and some of its wonderful concepts will make you see the world with different eyes.
Don’t forget that maths is everywhere! Enjoy! ~LThMath~