https://sguzman.github.io/marginalia/tags/algebraic-geometry/Recent content in Algebraic-Geometry on MarginaliaHugoen-usThu, 12 Feb 2026 00:00:00 +0000https://sguzman.github.io/marginalia/posts/evolution-and-frontiers-of-algebra/Thu, 12 Feb 2026 00:00:00 +0000https://sguzman.github.io/marginalia/posts/evolution-and-frontiers-of-algebra/A graduate-level survey of algebra’s evolution from ancient, rhetorical problem-solving traditions to modern abstract and structural formulations. It traces key historical milestones (e.g., the rise of symbolic notation, the solution of higher-degree equations, and the 19th-century emergence of group and Galois theory), maps major contemporary subfields (groups, rings, fields, modules, representation theory, Lie/Hopf algebras, homological algebra, category-adjacent viewpoints), and highlights interdisciplinary applications in science and technology. The report also examines philosophical and pedagogical debates around abstraction and “structuralism,” and sketches forward-looking frontiers such as higher algebra, quantum/categorical methods, and computer/AI-assisted discovery.https://sguzman.github.io/marginalia/posts/geometry-semantics/Thu, 12 Feb 2026 00:00:00 +0000https://sguzman.github.io/marginalia/posts/geometry-semantics/This report traces the evolving meanings of “geometry” in European and American mathematics from the era of Leonhard Euler (18th century) to the present day. Over nearly three centuries, “geometry” has expanded from denoting the classical study of shapes and Euclidean space to a sprawling family of subfields and methodologies. We examine what mathematicians of each era understood “geometry” to mean, how new theories and external pressures reshaped those meanings, and how “geometry” functioned both as a subject area and a style of reasoning. Key transitions include the rise of analytic and coordinate methods, the introduction of projective and non-Euclidean geometries, the 19th-century unification of geometry via transformation groups (Klein’s Erlangen Program), the axiomatization of geometry by Hilbert, the 20th-century branching into differential, topological, algebraic, and computational geometries, and the influence of physics and computing. We also explore enduring tensions—synthetic vs. analytic methods, intuition vs. rigor, local vs. global perspectives, continuous vs. discrete structures, algebraic vs. geometric mindsets—and how the term “geometry” at times unified mathematicians and at other times fragmented into specialized “geometries.” Through historical narrative, case studies, and quotations from major figures, we show how “geometry” evolved from the study of tangible figures in Euclidean space to a unifying language of mathematical structure, and why it remains a plurality of approaches today.