Faculty of Science and Professor of Medical Statistics, University of Stavanger
Dennis Sullivan works with the math that sees no difference between a donut and a cup of coffee. This year he received the Abel Prize.
This is history. Opinions expressed in the text are at the author’s expense.
A triangle is not a square, and neither are an approximation. We learn that in primary school. But this is not always true. It depends on which eyes you are looking at.
If someone asks you the way, and you answer “Turn right at the first intersection, then left when the road splits,” you are obscuring the details. She has not yet told how far away the first intersection is, or if there was a turn before the road split. You’ve reduced your answer to the engineering gist of the problem.
Geometry is about shape. There are lengths, heights, and angles. Details matter. But not everything is always the same. Topology is a qualitative approach to geometry, in which unnecessary details are removed away. The key word is rubber band.
From the same rubber band, you can make a circle and an oval. Triangle and square as well. Thus the circle, ellipse, triangle, and square can be seen as symmetrical. That is why topology is often called rubber engineering. The approach has many everyday parallels.
Beware the pit Beware the pit
Tube passes in London billion people annually. Then artistic cartoonist Harry Beck advanced The iconic tube map in 1933, inspired by electronic circuits and gave the illusion of geographic accuracy. What is important is determining the next stop – not specific distances or whether there will be a turn or a turn.
The center of the tube map is the empty space between lines. Holes in the map. Topology loves holes.
Donut tag hole. Also has a cup of coffee: the handle. In topology, there is no difference between a donut and a cup of coffee. A plate and a vase are the same too: there are no holes.
My desk has zero holes, so it’s a plate. While I have one hole across and I’m a donut.
Classifying objects based on basic geometric properties is useful in solving complex problems. It’s about looking behind the plaster wall and wallpaper and revealing the structure itself. Then you want to find the similarities where you see the differences.
Every birthday, we hang a string of lights with stars on it in the living room window. A wire hangs from above, and from below hang rows of glowing stars. My job is to suspend the light chain. It’s a stupid job.
Each year I carefully pack the candle decorations in a bag. And every Christmas, I catch again the clutter of wires and stars entangled in each other. It is a struggle to reveal everything.
but me Veterinarian Dr Yes it works. The light string is several straight strands connected together. Like a tube map. Or plate: zero holes. Simply finish until the tangle is untangled and the link can be cut.
With unknown problems it is even worse.
You can try to solve the tangle of wires with your own hands. It is not possible with a large set of data with numbers that you want to know how to connect. It could be the molecular structure of complex chemical compounds or the friendly structures on social media. The complex surfaces of information are intertwined.
Would it be a knot if we tried to solve it, or would it be a simple and obvious system?
On the surface, all problems seem unique. But most problems are similar to others, and by ignoring unnecessary details, the topology can reveal seemingly different problems that are essentially the same.
By ignoring unnecessary details, the topology can reveal seemingly different problems which are essentially the same
We collect huge amounts of data. Digital pictures, price lists, endless rows of numbers. And the more data we have, the more difficult it is to visualize it in a meaningful way.
One of the tools we have is one of the most important elements in topology: diversity. Variations closely resemble regular flat floors, but may look different from a distance. Globe and donut are diverse, but they are not a cube or pointed mountaintop.
If we can trace the numbers in the data set to a variety, we can separate them into different parts, which can then be visualized using patterns and color codes.
Diversity rating is important. It is easy in one and two dimensions, but difficult in three dimensions. In four dimensions, the mission is full of mystery. From five dimensions and up, it gets easier again. Then it gets a little bit more space – more arm strokes – and we can use topological surgery to create new geometric objects that are easier to work with.
In the search for how things connect, it is easy to get lost. That is why many subjects are fond of mathematics. Mathematics is concerned with basic properties and structures in itself – It is not intended to be used. Therefore, mathematical results have general validity and can be used by all subjects that use mathematics. A lot.
American Dennis Sullivan was awarded the 2022 Abel Prize “for his pioneering contribution to topology in the broad sense”. His early work was on diversity taxonomy and topological surgery, and he has contributed to a complete picture of diversity in five dimensions and above.
It may seem unrealistic. But then it wasn’t the case anyway. Topology has proven to be an excellent way of thinking with important applications in physics, chemistry, economics, and computer science.
There are many who enjoy looking under the surface.
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