In this project, we attempt to reformulate various notions from classical commutative algebra (such as flatness, regularity, smoothness, etc.) in an entirely categorical manner, so as to be able to later write down their analogues in derived algebraic geometry without having to develop extra theory. We will also be presenting certain application…

Sato-Tate Groups and Distributions of y^{\ell} = x(x^{\ell}-1)

Definition of Serre's twisting sheaf (work in progress)

The Bombieri-Lang Conjecture via rational-point sparsity persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

The Beilinson conjectures via regulator-L-value persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

Motivic Architecture via realization-filtration persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

The Grothendieck-Serre Conjecture via principal-bundle triviality persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

The Zilber-Pink Conjecture via unlikely-intersection persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

Anabelian geometry via fundamental-group reconstruction persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

Motivic Realizations via regulator-filtration persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

Anabelian Reconstruction Program via multi-fundamental-group persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

The Bloch-Kato conjecture via Selmer-L-value persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

Unlikely Intersections Ecology via atypical-family persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

The Fontaine-Mazur conjecture via geometric-Galois persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

The Section Conjecture via section-packet persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

The Andre-Oort conjecture via special-subvariety persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

The Tate conjecture via cycle-class persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

Abelian unramified geometric class field theory

Unlikely Intersections via atypical-locus persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.

The abc Conjecture via height-defect persistence on the manifold-constrained canonical lane. Reproducible local-to-global theorem package.