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Earlier quoted context omitted.
And they have a RSS feed, although it's a bit tricky to figure out, since the relevant header tag for that is set up incorrectly, pointing to a useless empty "comments" feed even from their main page. The actual feed for articles is https://www.quantamagazine.org/feed/
Nice find, thank you. Your sleuthing is appreciated.
I always found interesting that the English mathematical terminology has two different names for "stuff that locally looks like R^n" (manifold) and "stuff that is the zero locus of a polynomial" (variety). Other languages use the same word for both, adding maybe an adjective to specify which one is meant if not clear from the context. In Italian for example they're both "varietà"
> The term “manifold” comes from Riemann’s Mannigfaltigkeit, which is German for “variety” or “multiplicity.”
https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold
I'm not going to pretend to understand it all but they do make pretty pictures!
Man, I wish that the modern internet -- and great stuff like this -- had been around when I took GR way back when. My math chops were never good enough to /really/ get it and there were so many concepts (like this one) that were just symbols to me.
This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition. It’s actually much more well written than the majority or articles we usually come across.
This reminds me of how physicists will define a tensor. So a second rank tensor is the object that transforms according as second rank tensor when the basis (or coordinates) changes. You might find it circular reasoning but it is not, This transformation property is what distinguishes tensors (of any rank) from mere arrays of numbers. Looking at things from abstract view does allow us not to worry about how we visual…
This is a tendency among physicists that I find a bit painful when reading their explanations: focusing on how things transform between coordinate systems rather than on the coordinate-independent things that are described by those coordinates. I get that these transformation properties are important for doing actual calculations, but I think they tend to obfuscate explanations. In special relativity, for example, a…
Something's gone badly wrong here. "Without learning Cyrillic" is the normal way to learn Russian. Pick a slightly less prominent language and 100% of learners will do it without learning anything about the writing system.
I first learned about manifolds through Introduction to Smooth Manifolds by John M. Lee. The book is dense but beautifully structured, guiding you from basic topology to smooth maps and tangent spaces with clear logic. It demands focus, yet every definition builds toward a deeper picture of how geometry works beneath the surface. Highly recommended.