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How I Learned to Love Algebraic Geometry
- cpone10 8y agoI liked the part about Alexander This one: One of these geniuses was Hartshorne’s thesis advisor, Alexander Grothendieck. From about 1960 to 1970, Grothendieck revolutionized algebraic geometry as part of an epic quest to prove some conjectures about number theory, the Weil Conjectures. He had the idea that these could be translated into questions about geometry and settled that way. But making this idea precise required a huge amount of work. To carry it out, he started a seminar. He gave talks almost every day, and enlisted the help of some of the best mathematicians in Paris.
- throwawaymath 8y agoIndeed. Grothendieck is one of the few names I associate with the word "genius" without qualification. Up there with people like von Neumann and Witten. There are a lot of really smart people in the world, but they stand out even among the best.
- umutisik 8y ago“Mathematicians study curves described by all sorts of equations – but sines, cosines and other fancy functions are only a distraction from the fundamental mysteries of the relation between geometry and algebra.” Is this statement backed up by theorems or is it being made because there are just more proven theorems in algebraic geometry and there is not much work on solution sets to more general equations?
- kevinventullo 8y agoI wasn't crazy about that line, but basically, sines and cosines make much less sense over say, the rational numbers, whereas polynomials work just fine. They're the most general "nice" function one can define over an arbitrary commutative ring. Going back to the 1800's, in the theory of Riemann surfaces (one-dimensional complex manifolds), the only meromorphic (complex differentiable) functions are ratios of polynomials! And all closed projective complex manifolds are algebraic varieties. See https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry https://en.wikipedia.org/wiki/Algebraic_geometry_and_analyti...
- wolfgke 8y ago> I wasn't crazy about that line, but basically, sines and cosines make much less sense over say, the rational numbers, whereas polynomials work just fine. They're the most general "nice" function one can define over an arbitrary commutative ring. This is rather a tautology since polynomials are exactly defined this way. I can imagine that if we built a (fictional) number system rather on properties of the Fourier transformation, sines and cosines would be very natural operations that would have a deep generelization when sufficiently abstracted.
- sdenton4 8y agoThe Fourier transform is just a change of basis, if you squint at it properly. So at least some of the time (eg, when you're not mixing frequency and time domains like a crazy person) you can convert to frequency space and the functions become much simpler, and usual polynomial operations are relevant again.
- joppy 8y agoSolution sets to more general equations are so difficult and pathological that a lot of modern mathematics just entirely rules them out as objects of study. For example, any time you hear the word “manifold”, it refers to a space which has none of these pathologies and is entirely smooth. So the entire theory of differentiable manifolds will never encourage anything which “pinches”, or drops down a dimension, etc. On the other hand, by restricting ourselves only to polynomials and their solution sets, it turns out that the singularities which arise are not too bad, and we can study them in detail. In other words, restricting to polynomials is the _only_ restriction we have to make, pretty much every solution set can be studied from there on.
- aaaaaaaaaaab 8y agoBut aren’t we missing out on something important by glossing over these “pathological” cases? Like how we kept ignoring nonlinear differential equations, because they aren’t that well-behaving as their linear counterparts... And then when we eventually looked into it, we found a completely new paradigm: chaos theory.
- joppy 8y agoFor sure we are missing something. But what hope do we have of understanding the more complicated case, before we figure out what we can know in simpler cases? The eventual goal is to move back to the more complicated cases.
- phkahler 8y agoWhat if the approach to the simpler cases is not applicable to the harder ones. Studying the simple may shape ones thoughts in a way that prevents seeing the way to harder problems. Maybe? Sometimes new things are discovered because someone didn't know any better and tried something they were not ready for. Innovation sometimes comes from the self taught.
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- notduncansmith 8y agoComplex numbers is one area where sine is very relevant. Fascinating article about the geometry of complex numbers: https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/
- kevinventullo 8y agoAlgebraic Geometry is a powerful tool of number theory because much of it works over any field. It allows one to translate geometric intuition (algebraic geometry over the complex numbers) into a more algebraic environment (finite, p-adic, or number fields). Going back further, algebraic geometry over the complex numbers was shown in the early 20th century to be in many ways equivalent to more classical analytic geometry: https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry https://en.wikipedia.org/wiki/Algebraic_geometry_and_analyti...
- heinrichf 8y agoMore than over any field, over any ring. For example, that allows you to look at families of curves or talk about reduction modulo p (if you do number theory) very nicely/naturally. A mathoverflow thread with cool uses of schemes: https://mathoverflow.net/questions/59071/what-elementary-problems-can-you-solve-with-schemes?answertab=votes#tab-top https://mathoverflow.net/questions/59071/what-elementary-pro...
- selimthegrim 8y agoThe book by Gelfand, Kapranov and Zelevinsky is a magical mystery tour of precisely this.
- foxes 8y agoIt would be nice if they went in to a bit more detail about what they see as the connection between these two subjects. As they say, a variety is the zero set of some polynomial equations. In particular a Riemann surface can be represented as a polynomial equation P(x,y)=0. This can quantised as you might expect x->x, y-> h d/dx, and you can study the equation P(x,h d/dx) \psi(x) = 0. The solution \psi can be written as a formal power series. For a nice choice of P(x,y), these power series can be a generating function for "something", and can contain a lot of interesting algebraic information.
- reacharavindh 8y agoI like that you can use latex in the comments section..
- graycat 8y agoI really liked his first paragraphs! I also liked physics and was a math grad student to learn the math for physics! But I tend to agree with his first statements about polynomials: They look too restrictive to be highly promising for physics. It appears that he drifted off a central focus on physics and got interested in some math that may, long shot, have something to do with some detailed aspects of string theory. It looks like he is more interested in the math than the physics. Okay, but to me it's a loss for physics.
- maaaats 8y agoWhat are some good resources for grasping or learning more about algebraic geometry?
- ginnungagap 8y agoYou'll need a good background in commutative algebra to learn algebraic geometry, if you have that and want to see the modern approach, schemes and everything else that Grothendieck did, I suggest Vakil's notes, they're freely available from his homepage. (Disclaimer: I only studied the first 15 chapters and the one on Kähler differentials which should be the 21st, but I suppose the second half is as good as the first, I'll find out for sure next term)
- MaysonL 8y agoVakil's notes have an interesting epigram from Grothendieck: I can illustrate the … approach with the … image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!… A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration … the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet finally it surrounds the resistant substance. — A. Grothendieck
- ginnungagap 8y agoVakil seems very fond of that quote, the full title of his notes is "The Rising Sea - Foundations of Algebraic Geometry"!
- edflsafoiewq 8y agoTry some projective geometry to get your feet wet.
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- melling 8y agoJoan Baez’s dad, his uncle, gave him the physics book that he wrote. Hopefully, someday when we see Joan Baez represented in movies, we’ll see her physicist father, perhaps giving a physics book to his 8 year old nephew.
- chmaynard 8y agoIn 1968, I enjoyed taking a physics course at Harvard Summer School that was taught by Dr. Albert Baez. He had a home in Cambridge and graciously invited the entire class over for a party after the course ended. The highlight of the evening was an impromptu performance by his daughter Joan.
- dhimes 8y agoHe was also the crew-cut guy running demos in film loops I saw in school when I was a kid in the '60s and '70s.
- johncarlosbaez 8y agoCool! It sounds fun. Back then she used to sing at the so-called Nameless Coffeehouse in Harvard. Unfortunately I was 7 years old in 1968, so I missed all this.
- agumonkey 8y agooh I never realized they were siblings .. https://ncatlab.org/johnbaez/show/HomePage https://ncatlab.org/johnbaez/show/HomePage
- eggy 8y agoCan geometric algebra be used in algebraic geometry? I understand geometric algebra is useful in physics involving differential geometry and calculus, and that algebraic geometry is heavy on the algebra of polynomials and a complete field of study in mathematics, whereas geometric algebra is more like an object (Clifford algebra, etc.).
- danharaj 8y agoGeometric algebra is about the algebra of a widget called a grassmanian, which figures very prominently in algebraic geometry.
- eggy 8y agoI have been playing with John Browne's Grassmann Algebra in Mathetmatica for my geometric algebra studies, but this is the first I've encountered the Grassmannian. Thanks, I'll look into it.
- rsp1984 8y agoAlgebraic Geometry, not to be confused with Geometric Algebra [1]! I guess naming things was never the strong side of mathematicians... [1] https://news.ycombinator.com/item?id=13239632 https://news.ycombinator.com/item?id=13239632