(Roughly) Daily

Paulina Rowińska on how, in the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics…

Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.

The world is full of such shapes — ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.

This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces — leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis and physics.

Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”

So what are manifolds, and what kind of vocabulary do they provide?…

[Rowińska explains manifolds and the history of the development of our understanding of them, concentrating on the pivotal role of Riemann…]

… Manifolds are crucial to our understanding of the universe… In his general theory of relativity, Einstein described space-time as a four-dimensional manifold, and gravity as that manifold’s curvature. And the three-dimensional space we see around us is also a manifold — one that, as manifolds do, appears Euclidean to those of us living within it, even though we’re still trying to figure out its global shape.

Even in cases where manifolds don’t seem to be present, mathematicians and physicists try to rewrite their problems in the language of manifolds to make use of their helpful properties. “So much of physics comes down to understanding geometry,” said Jonathan Sorce, a theoretical physicist at Princeton University. “And often in surprising ways.”

Consider a double pendulum, which consists of one pendulum hanging from the end of another. Small changes in the double pendulum’s initial conditions lead it to carve out very different trajectories through space, making its behavior hard to predict and understand. But if you represent the configuration of the pendulum with just two angles (one describing the position of each of its arms), then the space of all possible configurations looks like a doughnut, or torus — a manifold. Each point on this torus represents one possible state of the pendulum; paths on the torus represent the trajectories the pendulum might follow through space. This allows researchers to translate their physical questions about the pendulum into geometric ones, making them more intuitive and easier to solve. This is also how they study the movements of fluids, robots, quantum particles and more.

Similarly, mathematicians often view the solutions to complicated algebraic equations as a manifold to better understand their properties. And they analyze high-dimensional datasets — such as those recording the activity of thousands of neurons in the brain — by looking at how those data points might sit on a lower-dimensional manifold.

Asking how scientists use manifolds is akin to asking how they use numbers, Sorce said. “They are at the foundation of everything.”…

“What Is a Manifold?” from @quantamagazine.bsky.social.

Apposite: Rowińska in conversation with Ira Flatow on Science Friday: “How Math Helps Us Map The World.”

* David S. Richeson, Euler’s Gem: The Polyhedron Formula and the Birth of Topology (Riemann’s work was an advance on the foundation that Euler laid in his 1736 paper on the Seven Bridges of Königsberg, which led to his polyhedron formula)

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As we get down with geometry, we might spare a thought for John Wallis; he died on this date in 1703. A clergyman and mathematician, he served as chief cryptographer for Parliament (decoding Royalist messages during the Civil War) and, later (as Savilian Chair of geometry at Oxford after the hostilities), for the the royal court. Wallis is credited with introducing the symbol ∞ to represent the concept of infinity, and used 1/∞ for an infinitesimal… which earned him (along with his contemporaries Isaac Newton and Gottfried Wilhelm Leibniz) a share of the credit for the development of infinitesimal calculus. He was a founding member of the Royal Society and one of its first Fellows.

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