Μαθηματικά και Λογοτεχνία

If you were asked to picture how electrons move, you could be forgiven for imagining a stream of particles sluicing down a wire like water rushing through a pipe. After all, we often describe electrons as “flowing” in an “electric current.”

In reality, water and electricity flow in completely different ways. Whereas water molecules move together to form a swirly, coherent substance, electrons tend to fly past one another. “Water is seeing nothing but other water,” said Cory Dean, a physicist at Columbia University, “but in an electronic system, in a wire, that’s manifestly not the case.” Water molecules unite to flow, but each electron acts on its own.

This every-particle-for-itself movement serves as the foundation for all of electronic theory. It explains why a warm wire resists more than a cold wire, and why a round wire conducts as well as a square wire. But since the 1960s, theorists have suspected that electrons can be coaxed to act more like their watery counterparts, and to form an electron fluid.

In recent years, a string of experiments has confirmed that prediction. Making electrons behave like water might someday lead to the development of new kinds of electronic devices. And extending the familiar theory of water to electrons could spawn a new way of thinking about quantum materials.

🫗 Read the full story: https://www.quantamagazine.org/physicists-make-electrons-flow-like-water-20260211/

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Tensors are instrumental in physics, machine learning and even biology. Einstein once begged a friend to help him understand them, fearing he was going mad. Joseph Howlett explains how they work: https://www.quantamagazine.org/the-geometric-tool-that-solved-einsteins-relativity-problem-20240812/

When it comes to hard problems, computer scientists seem to be stuck. Consider, for example, the notorious problem of finding the shortest round-trip route that passes through every city on a map exactly once. All known methods for solving this “traveling salesperson problem” are painfully slow on maps with many cities, and researchers suspect that there’s no way to do better. But nobody knows how to prove it.

For over 50 years, researchers in the field of computational complexity theory have sought to turn intuitive statements like “the traveling salesperson problem is hard” into ironclad mathematical theorems, without much success. Increasingly, they’re also seeking rigorous answers to a related and more nebulous question: Why haven’t their proofs succeeded?

This work, which treats the process of mathematical proof as an object of mathematical analysis, is part of a famously intimidating field called metamathematics. Metamathematicians often scrutinize the basic assumptions, or axioms, that serve as the starting points for all proofs. They change the axioms they start with, then explore how the changes affect which theorems they can prove. When researchers use metamathematics to study complexity theory, they try to map out what different sets of axioms can and can’t prove about computational difficulty. Doing so, they hope, will help them understand why they’ve come up short in their efforts to prove that problems are hard.

In a paper published last year, three researchers took a new approach to this challenge. They inverted the formula that mathematicians have used for millennia: Instead of starting with a standard set of axioms and proving a theorem, they swapped in a theorem for one of the axioms and then proved that axiom. They used this approach, called reverse mathematics, to prove that many distinct theorems in complexity theory are actually exactly equivalent.

“I was surprised that they were able to get this much done,” said Marco Carmosino, a complexity theorist at IBM. “People are going to look at this and they’re going to say, ‘This is what got me into metamathematics.’”

Keep reading:
https://www.quantamagazine.org/reverse-mathematics-illuminates-why-hard-problems-are-hard-20251201/

In mathematics, ubiquitous objects called groups display nearly magical powers. Though they’re defined by just a few rules, groups help illuminate an astonishing range of mysteries. They can tell you which polynomial equations are solvable, for instance, or how atoms are arranged in a crystal.

And yet, among all the different kinds of groups, one type stands out. Identified in the early 1870s, Lie groups are crucial to some of the most fundamental theories in physics, and they’ve made lasting contributions to number theory and chemistry. The key to their success is the way they blend group theory, geometry and linear algebra.

What makes a Lie group special is that it can be visualized as a smooth, continuous shape called a manifold. Lie groups might look like a circle, or the surface of a doughnut, or a high-dimensional sphere. Some look even stranger: The group of all rotations of a ball in space, known to mathematicians as SO(3), is a six-dimensional tangle of spheres and circles.

The manifold nature of Lie groups has been an enormous boon to mathematicians and physicists.

🌐 Keep reading: https://www.quantamagazine.org/what-are-lie-groups-20251203/

Mathematical Thinking Isn’t What You Think It Is | Quanta Magazine The mathematician David Bessis claims that everyone is capable of, and can benefit greatly from, mathematical thinking.