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Anonymous01/28/26, 17:34No.16900824
What are some of your favorite mathematical formulas?
I start with picrelated. It is a formula for the number of combinations a NxNxN Rubik's cube can have.
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Anon01/28/26, 17:55No.16900828
Working in a stable/triangulated category [math]D[/math] (motivic, derived, whatever) the “heart” is just the abelian slice cut out by a [math]t[/math]-structure: [math]\mathcal A := D^{\le 0}\cap D^{\ge 0}[/math], with [math]H^0(X):=\tau_{\ge 0}\tau_{\le 0}(X)\in\mathcal A[/math]. The point is you can define/use the heart as a certificate layer: if you have realizations [math]R_i:D\to\mathcal D_i[/math] into derived categories with known hearts [math]\mathcal D_i^{\heartsuit}[/math], then a natural candidate is [eqn]\mathcal A_R:={X\in D:\forall i,\ R_i(X)\in\mathcal D_i^{\heartsuit}}.[/eqn] Once a real heart exists, extensions become computable inside [math]D[/math] via [math]\mathrm{Ext}^n_{\mathcal A}(M,N)\cong\mathrm{Hom}_D(M,N[n])[/math] for [math]M,N\in\mathcal A[/math], and triangles give SES on [math]H^0[/math]. The actual hard seam is whether there is a canonical [math]t[/math]-structure on the motivic [math]D[/math] making the standard realizations [math]t[/math]-exact and the tensor compatible (so [math]\mathcal A[/math] is a tensor abelian category), not the formula itself.