ꭓle Ormsby 𓆙

I just analyzed 1.9M recipes and can now report that the yummy simplicial complex is arguably *not* flag. The following flavor trios have far fewer recipes containing all three ingredients than the recipes with pairs of ingredients would predict.

Here's a slice of Young's lattice — integer partitions with at most m pieces each of size at most n — as a little Sunday treat: e-infinity.space/young-lattice/ It's built as the quotient of the Boolean lattice Bₘₙ by the action of 𝔖ₘ≀𝔖ₙ ≤ 𝔖ₘₙ. 🧮

The Mushroom at the End of the World in Your Salt. jacobsensalt.com/products/mat...

New paper day! 🥳 🧮 Pontryagin-Thom, π₀ of the motivic sphere spectrum, and principal quadratic forms arxiv.org/abs/2610.29065 wherein I provide a novel proof of Morel's theorem identifying π₀(𝕊ₖ) with the Grothendieck-Witt ring GW(k) of quadratic forms.

See the paper arxiv.org/abs/2610.29065 or my slides e-infinity.space/files/taters... for more info. I'll speak about this paper at the AMS Western Sectional meeting at ASU on Nov 7.

New paper day! 🥳 🧮 Pontryagin-Thom, π₀ of the motivic sphere spectrum, and principal quadratic forms arxiv.org/abs/2610.29065 wherein I provide a novel proof of Morel's theorem identifying π₀(𝕊ₖ) with the Grothendieck-Witt ring GW(k) of quadratic forms.

... if we can determine π₀(𝕊ₖ) without recourse to graded endomorphisms and Milnor-Witt K-theory. This paper demonstrates that we can, and I hope that it opens the door to theorems where Morel's pattern can't be followed, e.g. in equivariant stable motivic homotopy theory.

New paper day! 🥳 🧮 Pontryagin-Thom, π₀ of the motivic sphere spectrum, and principal quadratic forms arxiv.org/abs/2610.29065 wherein I provide a novel proof of Morel's theorem identifying π₀(𝕊ₖ) with the Grothendieck-Witt ring GW(k) of quadratic forms.

Beyond this (section 6 of the paper), I demonstrate that the kernel of PT is generated by what I call 'principal quadratic divisors', with the proof passing through an elaboration of Scharlau's quadratic reciprocity law on ℙ¹ — the quadratic version of principal divisors on ℙ¹ having degree 0.

New paper day! 🥳 🧮 Pontryagin-Thom, π₀ of the motivic sphere spectrum, and principal quadratic forms arxiv.org/abs/2610.29065 wherein I provide a novel proof of Morel's theorem identifying π₀(𝕊ₖ) with the Grothendieck-Witt ring GW(k) of quadratic forms.

The main contributions here are to 1. define Aᵘ(Γₖ) and recognize that it's the correct domain for PT, 2. prove this via a Position Lemma (an elementary "re-topping" method) that allows us to avoid non-reduced points,

New paper day! 🥳 🧮 Pontryagin-Thom, π₀ of the motivic sphere spectrum, and principal quadratic forms arxiv.org/abs/2610.29065 wherein I provide a novel proof of Morel's theorem identifying π₀(𝕊ₖ) with the Grothendieck-Witt ring GW(k) of quadratic forms.

The paper introduces a motivic Pontryagin-Thom construction PT taking a pair (L,u) — L a finite étale k-algebra, u a unit in L — to a framed correspondence e(L,u). Think of e(L,u) as the points Spec(L) in 𝔸¹ with framing (tangent data) u.