For me, it's [math]\widehat{\infty}[/math]. It denotes the formal neighborhood (formal disc) of the point [math]\infty\in\mathbb{P}^1[/math].More conretely: if [math]X[/math] is a smooth curve over some field [math]k[/math], write [math]\hat{\bar{\mathcal{O} } }_{X,x}[/math] for the completed local ring at [math]x\in X[/math], let [math] \mathbb{D}_{X,x} := \mathrm{Spf}(\hat{\bar{\mathcal{O} } }_{X,x})[/math] be the formal disc at [math]x[/math], and let [math] \mathbb{D}_{X,x}^\times := \mathrm{Spec}(\textrm{Frac}(\hat{\bar{\mathcal{O} } }_{X,x}))[/math] be the punctured formal disc. Then [math]\widehat{\infty} := \mathbb{D}_{\mathbb{P}^1,\infty}^\times[/math].This is quite the interesting object in algebraic geometry. For [math]X=\mathbb{P}^1[/math] with affine ccordinates [math]t[/math] on [math]\mathbb{A}^1 = \mathbb{P}^1\setminus \{\infty\}[/math], put [math]s=1/t[/math] as a uniformizer at infinity. We then get:[eqn] \hat{\bar{\mathcal{O} } }_{\mathbb{P}^1,\infty}\cong k[\! [x]\!] = k[\![t^{-1}]\!][/eqn][eqn] K_\infty := \textrm{Frac}( \hat{\bar{\mathcal{O} } }_{\mathbb{P}^1,\infty})\cong k(\!(s)\!)=k(\!(t^{-1})\!).[/eqn]Thus:[eqn]\widehat{\infty} = \mathrm{Spf}\, k[\![s]\!][/eqn][eqn]\widehat{\infty}^\times = \mathrm{Spec}\, k(\!(s)\!).[/eqn]It basically marks a local "port" where you impose a level structure and couple global geometry on [math]\mathbb{P}^1[/math] to local loop-group/Kac-Moody representation theory at the place infinity.So when algebraic geometers write, say, [math]\mathrm{Bun}_G(\mathbb{P}^1,\hat{0},\widehat{\infty})[/math], they mean G-bundles on P^1 together with chosen trivializations over the formal discs at 0 and infinity.
>neighborhood of infinto the trash it goes
>>16838292you lost infinitranny
>>16838336>>16838642>retards can't into projective geometryI'm neither OP nor versed in algebraic geometry, but even I can understand these basics.
>>16838292Universal turing machine
>>16838292Hypercomputation
>>16838292Infinity Owl!
>>16838994This is basically what happens when you spend too much time shitposting on 4chan: You legitimately start to believe that every concept a schizo stole from an established field and lobotomized to the point of having no mathematical value is itself schizobabble. Schizos usually aren't that creative to come up with these things on their own.Just because Tooker once read a paper on (formal) algebraic geometry which had infty-hats as formal neighborhoods of projective infinity, badly aped the concept, and spammed this board with it doesn't mean there is no such concept.
>>16838292Explain as if I'm not algebraic geometrist.What's a curve over a field?Then you take a point on that curve, take it's close neighborhood and what?
>>16840443a smooth curve over a field k is just a 1-dimensional (pure dimension) smooth scheme that is separated and of finite type over k. Smoothness of schemes is basically analogous to smoothness as defined in commutative algebra if you've ever seen it: https://stacks.math.columbia.edu/tag/01V4a formal neighborhood (like infty hat in the OP) is not a smaller open set around a point, it is the infinitesimal thickening of that point capturing all jets, Spf here means "formal spectrum," a generalization of the prime spectrum of a ring https://ncatlab.org/nlab/show/formal+spectrumPut simply, topological neighborhoods keep many nearby points but forget infinitesimal data (i.e. nilpotents in the algebraic sense), whereas formal neighborhoods keep only the point (topologically) but remember infinite‑order Taylor data. You can think of it as analogous to complex analysis if you've ever done it: take k=C, the analytic germ at x of some smooth curve X over C has ring C{t} of convergent power series and we get a canonical map C{t}C[[t]]. The formal disc is like the analytic disc with "infinite radius", where only formal taylor coefficients matter, convergence is irrelevant.so considering hat infty gives one the loop group/Kac-Moody action at infty needed to define the localization functor that lands in D-modules on the so-called "thick affine flag". "thick" here again to emphasize full formal power‑series structure
>>16840384justice for tooker
>>16840503he could've at least chosen a different symbol. Gaitsgory was talking about hat infty way back in 2009
>>16840512This notation is pure cancer. At this point you might as well draw it as a commutative diagram.
A solid of constant width, those are pretty fun and interesting mathematical creatures. The shape which has the minimum volume among all 3D shapes which have the same constant width is apparently an unsolved problem according to wikipedia which makes this subject extra interesting.
>>16840547How can it be unsolved lmao constant width just means a maximum radial length making the shape an exaggerated axle. Keep radius of disk, make it as thin as material properties support, then stick a line through the center again of minimum size supported by material. Due to autistic definition of width you run into shape definitions so merely make the spoke have the same length as the width of the disk. Easy, give me my fucking fields medal.
>>16840554Constant width means that if you squeeze the shape between two parallel planes, the planes will always be the same distance apart regardless of how the object is oriented or rotated between the planes.
>>16840563>that descriptionThis website is 18+.
>>16840582Nta but width of a set (in R^d) is a common notion in measure theory. It's basically sup_v(sup_x v•x-inf_x v•x), where v is some normal vector out of S^(d-1) and x an arbitrary element of the set whose width you want to determine. This is the same as what the guy said, just with hyperplanes instead of planes in R^d
>>16838292How do I become smart like you mathfags?
>>16841210You said it in the grown up way.
>>16841265Read all of this https://stacks.math.columbia.edu/browse
>>16841400Thanks
>>16841400It may be a bit much but it's genuinely an excellent resource for algebraic geometry and it's preliminaries. I'd recommend it's chapter on commutative algebra over any book unironically
