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Thursday, March 28, 2024

Math, Matter, and the 2016 Nobel Prize in Physics

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THE ancient Pythagoreans, so hypnotized by the power of mathematics to describe the world, formed mystical beliefs about numbers. While other ancient philosophers believed everything was made of water, or fire, or air, the Pythagoreans believed that everything was made of numbers. 

The ideas of the Pythagoreans greatly influenced Plato. In his latter work, Plato argued that the concepts of mathematics were more real than the world we can detect with our senses. No matter how many times you draw a circle, Plato reasoned, those drawings would be nothing but shadows of Real Circles. Real Circles, Plato said, exist in the “World of Forms”— the Real World. 

Modern scientists and mathematicians are seldom as mystical in their views, at least in public (and when sober). Still, the success of abstract mathematical ideas in describing the observable world is something that keeps many of them awake in the wee hours of the morning. 

In the case of David J. Thouless, F. Duncan M. Haldane, and J. Michael Kosterlitz, this success became theirs. Thouless, Haldane, and Kosterlitz were awarded the 2016 Nobel Prize in physics for their use of mathematical concepts to probe how matter can be made to behave in strange ways.

Using tools from a field of math called topology, this year’s Nobel Prize winners investigated the behavior of matter at very cold temperatures. 

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What is topology? And what is the relevance of its study to your life? Topology is very closely related to geometry. Both branches of mathematics are concerned with shapes. 

If you were a mathematician, you would describe topology and geometry as the study of spaces. To a mathematician, a space is a set of points with certain properties. Lines and shapes, which are sets of points, are examples of spaces. There are other kinds of spaces, but for our purposes lines and shapes are good enough.

If both are concerned with shapes, what distinguishes geometry from topology? Geometry is more concerned with the local properties of the shape, such as how far apart different points in the shape are. Topology is more concerned with the global properties of the shape. That is, topology tries to answer questions like, “what properties of this shape is shared by others?”

One interesting example of a shape would be a sphere. Think of a globe. The points on the surface of a globe can be described by their latitude and longitude. Since you need exactly two numbers to describe the points on a sphere, we say a sphere is a two-dimensional shape (or, if you were a mathematician, a two-dimensional space).

If you were studying the geometry of a globe, you would be asking questions like, “What is the shortest path from this point on the globe to that other point?” If you were studying its topology, you would be asking questions like, “What makes a globe similar to a cylinder but different from a donut or a bagel?”

The answer to that last question is the number of holes. A globe and a cylinder both have no holes. A donut has one hole while a bagel has two. Since a coffee cup with a handle has one hole, it shares that property with a donut. In fact, if you had a piece of clay shaped like a donut (what mathematicians call a torus), you can deform it into something that looks more like a coffee cup without messing with the hole in the middle.

The number of holes in a shape is an example of a topological property. A topological property is a property of a shape that does not change when you deform it. You can get a different topological property only when you tear apart or glue together different parts of the shape. Imagine you had a piece of clay shaped like a sphere. The piece of clay has zero holes. To turn it into something with one hole, you need to either punch a hole in the middle or join together parts that were originally far from each other.

This year’s Nobel Prize winners did not need to deform clay or break a lot of coffee mugs to do their research. Instead, they dealt with shapes in the abstract. Using the tools of topology, they dealt with them as spaces. In other words, the globes, donuts, coffee cups, and bagels they studied were abstract sets of points existing only in the World of Forms.

Which is what makes the next step in this story quite surprising. 

What Thouless, Haldane, and Kosterlitz did was apply the concept of topological properties, which are properties of abstract shapes, to describe how very cold atoms and electrons behave. And the very cold atoms and electrons don’t need to be related to any donuts or coffee cups. For example, if you get a very thin sheet that can conduct electricity and cool it down to almost absolute zero, some of its electrical properties can be mathematically described as topological properties. 

In other words, they were able to connect questions about the number of holes in abstract shapes to the amount of electricity a collection of actual atoms can conduct. For example, they showed that since the number of holes in a shape come in whole numbers (0 for a sphere, 1 for a donut or coffee mug, 2 for a bagel), certain properties of a material should also come in multiples of whole numbers. They also showed how these properties could change in a way that reflects changes in the number of holes in abstract shapes.

What’s it all for? 

First, the odd behavior of semiconductors was initially just a curiosity. Today, we use those properties to make computers. Who knows what engineers might cook up using the strange properties of matter Thouless, Haldane, and Kosterlitz have studied? Some of them are already thinking about applications like quantum computers. The quantum leap such applications can result in are likely beyond our imagination.

Second, they remind us how important the study of abstract, seemingly esoteric mathematics is. They remind us that the imagination of mathematicians, who are, after all, humans, is just as important in studying the world as our senses and all the telescopes and microscopes we use to improve on these senses.

Plato and the Pythagoreans will be pleased with the choice of this year’s winners.

 

Pecier Decierdo is a science communicator for The Mind Museum.

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