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Algebraic Geometry as studied by Mathematicians often deals with very abstract and general issues that I admit that I don’t completely understand myself. Nice
by mindcrime 4y ago
Algebraic Geometry as studied by Mathematicians often deals with very abstract and general issues that I admit that I don’t completely understand myself.
Nice to see this kind of honesty and/or modesty from an instructor!
Reminds me of the time I took Discrete Math at a local community college. We all showed up at the room for the scheduled first session, and found no instructor. We waited around about 20 minutes, and were about to leave, when a grey-bearded gentleman who looked to be of the instructorly persuasion rocked up and started unlocking the door. So we follow him in, and he gets in front of the class and says something approximately like:
"Hi, so-and-so, who was scheduled to teach this class, had to back out at the last minute and so the school called me. I'll be your instructor. I was a Physics major 60 years ago, and can do this math, but I've never taught this before and don't really remember anything about it. So we'll all be learning it together as we go."
It was an, ahem, interesting experience, to say the least. Luckily for me I'd had Discrete before and was only taking it due to a technicality that kept my previous credit from transferring.
- Sharlin 4y agoTo be fair, it’s Jim Blinn, one of the pioneers of computer graphics, so his admission could be read as a polite way of saying that AG as studied by mathematicians is needlessly abstract and lacking any real-world applications :)
- lupire 4y agoI don't think Jim Blinn would be so crass toward his colleagues.
- SideQuark 4y agoAG as studied by mathematicians has produced an incredible amount of applications that would not exist without that level of abstract tooling. And it's highly nontrivial to learn those tools (as someone that has worked through Hartshorne... And then put 20 years into applications).
- auggierose 4y agoWhat are some of those applications?
- SideQuark 4y agoBesides the other list, places results from AG have uses in tech include statistics, control theory, robotics, error-correcting codes, phylogenetics, geometric modelling, string theory, game theory, graph matchings, solitons , integer programming, computer vision, computer aided design..... Some of these are likely why Blinn decided to make an entire course on using it just in computer graphics alone.
- auggierose 4y agoThanks, I am aware of the general uses, I was just wondering what you in particular have used it for :-)
- SideQuark 4y agoI've used it professionally for error correcting code design, robotic control, crypto (ecc and multivariate polynomial), lots of computer vision uses, grobner bases to simplify algorithms, and geometric modeling. Perhaps more if I look over lists of past projects...
- matt-noonan 4y agoHere are a few off the top of my head, as a mathematician-turned-programmer who never has been an algebraic geometer. - Elliptic curve cryptography (https://en.wikipedia.org/wiki/Elliptic-curve_cryptography https://en.wikipedia.org/wiki/Elliptic-curve_cryptography) - Grobner bases, with many applications. Example domains: coding theory, robotics, signal processing... (https://math.stackexchange.com/questions/32421/applications-of-gröbner-bases https://math.stackexchange.com/questions/32421/applications-...) - Physics [solitons] (https://kasmana.people.cofc.edu/SOLITONPICS/ https://kasmana.people.cofc.edu/SOLITONPICS/) - Physics [string theory] (https://royalsociety.org/~/media/people/new-fellows-2014/Presentation-slides/Bridgeland.pdf https://royalsociety.org/~/media/people/new-fellows-2014/Pre...) - Automata theory, via "tropical" algebraic geometry (https://link.springer.com/article/10.1007/s00233-019-09999-8 https://link.springer.com/article/10.1007/s00233-019-09999-8) This is not even considering applications of AG to other areas of pure mathematics, which are extensive.
- dhosek 4y agoYeah, I was looking over the materials and thinking, where’s the Algebraic Geometry? I don’t think that Algebraic Geometry (as opposed to algebraic geometry in the sense that the author here is using it) has any applications to computer graphics, but definitely can be applied to other aspects of computer science (I can imagine it being potentially useful in cryptographic applications), although my own Algebraic Geometry knowledge is more limited than I would like it to be.
- kweingar 4y agoAlgebraic geometry is also used in robotics
- kevinventullo 4y agoProjective space, syzygies, Bezier curves, cubic curves aren’t algebraic geometry?
- kcexn 4y agoI'd say they're more algebraic representations of regular geometries. Algebraic Geometry as a mathematical field is interested in solving highly non-trivial geometric problems (think, from differential geometry, functional analysis, etc.), using tools from Abstract Algebra (think Galois Fields, Lie Groups etc.) It saddles a bridge between traditional geometry, and abstract algebra, and allows insights from one field of mathematics to be applied to the other. As such, it allows practitioners skilled in these tools to make many useful inferences about incredibly complicated systems. It's also incredibly dense. In part because many of the tools of algebra are incredibly involved. But also, in part because to define an algebraic object in a way that is equivalent to a geometric object, sometimes requires a fairly complicated definition.
- edflsafoiewq 4y agoAlgebraic geometry existed before Grothendieck lol. It's still algebraic geometry even it looks boring next to etale cohomology of infinity stacks or whatever.