Interval Computations and their Categorification
Nikolaj M. Glazunov
TL;DR
The paper develops a category-theoretic framework for interval computations applied to Minkowski's conjecture on the critical determinant of the region $|x|^p + |y|^p < 1$ for $p>1$. It introduces interval manifolds, presheaves, and functors to capture interval computations, and extends this with interval operads and interval programs viewed as functors, aiming for a compositional, structural approach to the underlying lattice-theoretic optimization. A nondifferential interval method is interpreted as a contracting map on the moduli space of admissible lattices, enabling a formal proof outline for the Minkowski conjecture in broad parameter ranges. The framework unifies interval analysis, lattice theory, and category theory to provide a systematic semantic basis for interval-based proofs and computations in geometric number theory, albeit with proofs omitted in the presentation. Overall, it presents a foundational, categorical lens for interval computations in mathematical problems involving moduli spaces and lattices.
Abstract
By the example of the proof of Minkowski's conjecture on critical determinant we give a category theory framework for interval computation.