Tensorgami

The mathematical library

Expositions

This library follows familiar calculations into the structures they retain. Choose a current study, a connected reading route, or a subject.

Current expositions

Root covers, graph complexes, and derivatives provide starting points for following familiar calculations into geometry, arithmetic, and reconstruction.

Choose an entry point and follow the connections between the three studies

Galois theory · Fourier duality

Fourier, Symmetry, and Reconstruction

Start with polynomial roots (§1)

The labelled-root calculations in §§3–5 use linear algebra, polynomial rings, finite group actions and basic Galois theory. Stop after the cubic; return to §2 for the cover construction.

Cardano, Ferrari, and categorical Fourier duality

How root covers and their algebras produce Cardano's and Ferrari's substitutions, then lead to Fourier–Mukai duality, motives, and higher Galois reconstruction.

Concrete applications include reducing genus-two integrals to elliptic integrals, reconstructing vector bundles, counting points over finite fields, and bounding exponential sums.

Read the expositionPDF · 47 pagesTeX source

Background and reading routes

Graph theory · Homological algebra · Motives

What a Boundary Map Can Prove

Start with the tree criterion (§1)

Start with §1, then the worked counts and resistance in §3 using graphs and linear algebra. Take the matrix-tree formula as given on a first pass; §2 develops the determinant-line mechanism used in its proof.

Derived gluing, torsion, Hodge inequalities, and motives

Determinant lines govern tree counts, Hodge theory constrains chromatic coefficients, and an elliptic attachment reveals motivic Galois symmetries invisible in the graph.

Worked examples show how torsion weighting removes a positive correlation, and how two nodal curves with the same graph can have different motivic Galois groups.

Read the expositionPDF · 46 pagesTeX source

Background and reading routes

Differentials · Derived geometry · Homotopy theory

What a Derivative Retains

Start with the rectangle (§1)

Begin with calculus. Polynomial rings and modules enter in §§4–5; section 5 is a useful first-order stopping point.

From side strips and oriented boundaries to infinitesimal geometry

The rectangle product rule and oriented boundaries lead to Kähler differentials, cotangent complexes, higher brackets, spectral derivatives, and Goodwillie calculus, through explicit calculations.

Worked examples distinguish an invisible Frobenius direction, a node’s lifting obstruction, derived self-intersection, and the K-theory and rational data retained by spectral derivatives.

Read the expositionPDF · 39 pagesMarkdown manuscript

Background and reading routes

Other mathematical routes

Follow a pair of polynomial proofs, choose an arithmetic construction, or begin with an analysis argument. The guide connects the notes within each route.

Local control in analysis

Begin with a product oscillation estimate and Darboux sums, then read about compactness in Lebesgue's criterion. The weighting argument extends the theme to exceptional sets and convergence in measure.

Full catalogue

Reading paths

Some notes offer a full web edition; others introduce a PDF argument. Each entry states its background and links to the available reading edition and downloads.

Algebra & geometry

Number theory

Analysis & topology

Problems