Favorite mathematical object?
/sci/
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.
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Last: 11/9/2025, 10:49:57 PM
