Polyhedra, Moduli, and Quantum Geometry: Variational principles, spectral methods, and computation

How do the geometry of a polyhedron, the analysis of a surface, and a quantum state sum inform one another?

Igor Rivin develops these connections through geometric variational principles. Volume and length functions produce geometric equations; their second derivatives connect boundary response, numerical methods, and semiclassical amplitudes.

The monograph moves from hyperbolic and Euclidean polyhedra to Weil–Petersson geometry and moduli spaces, then to quantum tetrahedral amplitudes, Dehn filling, geodesic identities, and counting problems. Worked explanations introduce angular-momentum recoupling, Regge calculus, Turaev–Viro state sums, and unfolding techniques. A computational strand connects volume Hessians with finite elements, resistor networks, and sparse linear algebra.

Established inputs, deductions proved in the text, conditional statements, numerical evidence, and open questions are distinguished throughout. The book is intended for graduate students and researchers in geometry, topology, mathematical physics, and related computational subjects.

This paperback contains the complete monograph. A free online edition and downloadable computational companions accompany the book; access instructions are included in the printed edition.