👉 Arch mathematics, named after Oswald Veblen, is a branch of mathematics that generalizes and extends the concepts of arithmetic, algebra, and analysis to non-Euclidean geometries, particularly hyperbolic geometry. It focuses on properties of numbers and structures that remain invariant under the hyperbolic metric, which is a model of hyperbolic space where angles and shapes behave differently than in Euclidean geometry. Key concepts include the Archimedean and non-Archimedean models, discrete groups, and the study of geometric series and sums in this setting. Arch math provides a framework for understanding various mathematical objects, such as modular forms, automorphic functions, and the arithmetic of hyperbolic manifolds, offering insights into deep connections between number theory, algebra, and geometry.
