Polyhedra, Moduli,
and Quantum Geometry

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Polyhedra, Moduli,
and Quantum Geometry

Variational principles, spectral methods, and computation

Igor RivinDepartment of Mathematics, Temple University

Read the monograph from polyhedral volume and moduli measures through quantum geometry, trace formulas, mixing and sharp counting.

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Contents

Introduction: the connections and the results

Part I

Polyhedra and moduli measures

Polyhedral realization supplies coordinates; the measure comparison turns them into volume formulas. Fixed-deficit Euclidean shapes then reveal a period geometry at the flat limit.

  1. 1 The volume problem and its normalization
  2. 2 Ideal polyhedra, angle cells, and a partition function
  3. 3 The ideal tetrahedron: an explicit volume integral
  4. 4 Hyperideal polyhedra: an actual local integrand
  5. 5 Boundary lengths, cone angles, and continuation
  6. 6 Equal parameters and the torus–sphere correspondence
  7. 7 Euclidean shapes and Deligne–Mostow periods
  8. 8 From ideal polyhedra to the Euclidean limit of WP geometry

Part II

Variational geometry and computation

The volume Hessian becomes a discrete Dirichlet operator. Boundary elimination, Gaussian determinants, and barrier methods connect the geometry to computation.

  1. 9 Volume Hessians, finite elements, and computational complexity

Part III

Quantum geometry, degeneration, and surgery

An introduction to the quantum constructions leads to exact period comparisons, a controlled tetrahedral degeneration, and the classical and Gaussian data of Dehn filling.

  1. 10 A geometric introduction to the quantum constructions
  2. 11 Polyhedral degeneration and quantum volume asymptotics
  3. 12 An exact quantum comparison and the boundary measure
  4. 13 The four-point residue: fusion, area, and an explicit density
  5. 14 The normalized quantum-group comparison and the compact limit
  6. 15 A uniform quantum Airy transition for the regular family
  7. 16 Dehn filling, variational response, and quantum surgery

Part IV

Curves, trace formulas, and quotients

Cutting and gluing geodesics produces identities and twist integrals. Twisted trace formulas retain the multicurve coefficients and their lengths.

  1. 17 Unfolding and the problem of finding constant series
  2. 18 Figure-eight curves and simple lifts
  3. 19 Cutting into pants and varying twists
  4. 20 An exact torus twist calculation and a constant orbit series
  5. 21 Matching across cuffs: positions and directions
  6. 22 Cusps, McShane gaps, and the orthospectrum
  7. 23 Simple multicurve coefficients of a twisted trace formula
  8. 24 Quantitative counting and coefficient cancellation
  9. 25 Global coefficient filters and a length-valued Euler product

Part V

Mapping-class dynamics and mixing

Weighted endpoint counts encode correlations. The general converse specifies the required estimates, while the marked-torus model supplies an explicit calculation on one genus-two stratum.

  1. 26 When counting errors imply mixing
  2. 27 Filling-pair coefficients and a weighted mixing converse
  3. 28 Genus two: compact polyhedra and a marked-torus model
  4. 29 From simple-curve coefficients to controlled observables

Part VI

Sharp counting and arithmetic

The local shape of a norm ball governs geometric discrepancy. Primitive counting adds a separate arithmetic cancellation problem, and exact diagnostics keep the two inputs visible.

  1. 30 Diagnosing the sharp remainder without fitting its main term
  2. 31 Arithmetic cancellation and the geometric remainder

Part VII

Open problems and conjectural directions

The comparisons suggest concrete further estimates, integrations, and gluing problems. Their hypotheses identify what the next results would need to establish.

  1. 32 Questions suggested by the calculations

Appendices and references

About this edition

All chapters, proofs, appendices, bibliography, tables and diagrams are converted from the canonical TeX source. The computational evidence index records the distinct scope of each maintained companion. The earlier Section 9 edition remains a separate frozen checkpoint, including its older source archive.

Manuscript date: 30 October 2026. Research checkout commit 4681abb777c813808f2ea4758461c4e89d510ff7. This web working edition includes the complete manuscript and computational companions.