[HN Gopher] The Stacks Project, a new model for organizing and v...
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The Stacks Project, a new model for organizing and visualizing
mathematics
Author : mathgenius
Score : 57 points
Date : 2022-02-05 15:58 UTC (7 hours ago)
(HTM) web link (news.columbia.edu)
(TXT) w3m dump (news.columbia.edu)
| dang wrote:
| Related from 2013: https://mathbabe.org/2013/07/30/the-stacks-
| project-gets-ever...
|
| (via https://news.ycombinator.com/item?id=11054838, but nothing
| else there)
| mjfl wrote:
| How much of this math is deep work and how much of it is it the
| conjuring of obscure objects that haven't had much study and
| proving trivial things about them?
| ogogmad wrote:
| I think Deligne's Theorem is a poster child for the power of
| Modern Algebraic Geometry. Andrew Wiles's proof of Fermat's
| Last Theorem might also be relevant.
|
| Generally speaking, studying what the solution sets of
| polynomial equations is "like" is quite fundamental to a lot of
| mathematics. Doing this in a "deep" way can lead to a
| reimagining of much of modern mathematics:
| https://rawgit.com/iblech/internal-methods/master/notes.pdf
|
| [edit]
|
| For instance, lots of people use a straightforward
| generalisation of number systems called rings. But ring theory
| is quite abstract. Modern Algebraic Geometry shows that at
| least in the case of commutative rings, these are merely spaces
| of functions on a space called a ring's spectrum. You can
| visualise a ring's spectrum, unlike the ring itself. Many
| properties of a ring are just properties of its spectrum. This
| seems like a significant conceptual leap in the understanding
| of things that were studied since the 1800s without much
| geometric understanding.
|
| Oh yeah, and I'm not an algebraic geometer.
| foxes wrote:
| Algebraic geometry has many deep and powerful ideas. It started
| out by looking at the space described by the zeros of a
| polynomial, eg
|
| y^2 - x^3 - x = 0, x^2+y^2+1=0
|
| But actually you do not need to talk about the underlying space
| directly. If you want to talk about a space, all you actually
| need to think about are the possible functions on the space. If
| you want to talk about geometry, you only need the algebra of
| functions on that space, so in the example just the polynomials
| themselves, rather than having to say explicitly solve it for
| the points. You can use this big idea in a lot of other areas
| of mathematics and physics.
| hoten wrote:
| It's very difficult to navigate this on mobile.
| ruined wrote:
| wikipedia is the wiki encyclopedia hosted and managed by the
| wikimedia foundation.
|
| this article appears to be about a different and completely
| distinct wiki project.
| ogogmad wrote:
| It's a bit like a Wiki, except with more centralised
| editorialship. Confusing title indeed.
| gryn wrote:
| it's used in the sense of 'a wiki project which tries to act
| like an encyclopedia' unlike for example the wiki wiki web. a
| similar kind of wiki is scholarpedia.org.
|
| the emphasis is on the encyclopedia part.
| Someone wrote:
| I think HN's 'smart' code to remove clickbait fragments from
| titles didn't help here. The page's full title is
|
| _This Wikipedia of Algebraic Geometry Will Forever Be
| Incomplete. That's the Point._
|
| That _"This"_ , IMO, makes it clear that they use Wikipedia in
| a genericized way
| (https://en.wikipedia.org/wiki/Generic_trademark)
| ogogmad wrote:
| To what extent is this knowledge reducible to a form that someone
| can learn it quickly and get something done with it? Algebraic
| Geometry is just one field of mathematics, no?
|
| Related question: How do you use this resource?
|
| [edit] To make it clear: It's a wonderful thing that this exists.
| WoahNoun wrote:
| Modern Algebraic Geometry is a field that encompasses and
| somewhat unifies so many different areas of mathematics that it
| is extremely difficult to distill down quickly. So it really
| depends on what your currently mathematical background is. If
| you only know some Abstract Algebra (ring and field theory in
| particular) and Topology at the undergrad level, it will take a
| long-time for you to contribute to research mathematics, but
| you could probably get a feel for the subject after a couple
| years (shorter if our a graduate student focusing on it full-
| time). If you don't know anything about those two subjects,
| it's a very long slog. If you know differential geometry,
| category theory, complex geometry in addition to the above, you
| could probably pick it up relatively quickly.
|
| The definitions and machinery make sense if you have enough
| background, but the why and how we got here is often very
| unclear.
|
| If you don't care about schemes, stacks and current modern
| viewpoint of Algebraic Geometry, it's not hard to get a decent
| understanding of algebraic varieties which are the original
| motivation in the field. An undergrad book on the subject
| Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea
| does a great job of introducing the subject. And has a really
| cool project on calculating Groebner bases for polynomial
| equations in sin and cosine to define the configuration space
| of different types of robotic arms.
| ogogmad wrote:
| I'm trying to learn more modern algebra (including algebraic
| geometry). The modern stuff is of interest to me, but it
| feels overwhelming.
|
| I get that affine schemes are somehow the "geometric" dual of
| a commutative ring. A motivating example is Spec(R[X,Y]/(X^2
| + Y^2 - 1)), which is simple enough as a ring (if you remove
| the "Spec"), but as an affine scheme it is a circle. The
| slash is almost acting like a subset formation operation. I
| know enough category theory to see that the reason why the
| slash is acting that way is because equalisers are the dual
| construction to coequalisers; the slash (ring quotienting) is
| a coequaliser, and in the category of affine schemes it
| becomes an equaliser, and equalisers on "spaces" are supposed
| to form subsets somehow. Another example is the ring
| R[X]/(X^2), sometimes called the dual numbers, whose affine
| scheme (or Spec) is a lot like an infinitely small line
| segment. The fact that the affine scheme behaves like an
| infinitely small region of space is dual to the algebraic
| fact that the dual numbers are a _local_ ring.
|
| Finally, I have a vague understanding that a scheme is the
| result of gluing some affine schemes together. Sheaf stuff is
| involved.
|
| Anyway, the above summarises my understanding of schemes. I
| don't know any differential geometry as such. I rely a lot on
| naive, and sometimes not wholly rigorous intuition. I have no
| idea how you compute with this, especially given how
| elaborate the definitions are.
|
| [edit]
|
| I'm hoping this might present a shortcut for someone like me:
| https://www.ingo-blechschmidt.eu/research.html It's
| especially promising because the computations look more
| familiar to me.
| ABeeSea wrote:
| The dual numbers, in my opinion, are a lot easier to
| understand with some background in differential geometry.
| In differential geometry, if I_x is the ideal of smooth
| functions vanishing at the point x in the ring of smooth
| functions, then I_x / I_x^2 is a real vector space called
| the cotangent space (these elements are given the
| suggestive dx moniker) and dual of this vector space is the
| space of tangent vectors at this point. Another way to
| think about R[X,Y]/(X^2) dropping all the non-linear terms
| for X to create a flat (co)tangent space.
|
| Also the original motivation for sheaves was about creating
| a way to deal with multi-valued complex function. The
| complex log function is multi-valued so in intro complex
| analysis it's studied locally by choosing a branch of the
| range where it's singular valued. Thus it's impossible to
| "do differential geometry" by talking about a global ring
| of analytic functions. But you can talk about the "local
| ring of analytic functions" at a point and specific branch
| and glue these locally ringed spaces together to get global
| insight.
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