Advances in algebraic topology: Theoretical and practical perspectives

Abstract

Algebraic topology has undergone transformative advancements between 2010 and 2024, significantly enriching its theoretical foundations and expanding its interdisciplinary applications. This review explores key developments in homotopy theory, persistent homology, and directed algebraic topology. Computational innovations, such as the use of spectral sequences, have enhanced the calculation of homotopy groups, resolving longstanding challenges in stable homotopy theory. Persistent homology has emerged as a cornerstone of topological data analysis, demonstrating robustness through stability results and computational tools, enabling its application in biology, image processing, and material science. Directed algebraic topology has found applications in robotics, optimizing motion planning and modeling dynamic systems with inherent directionality. These developments highlight the increasing relevance of algebraic topology in addressing real-world challenges. This article synthesizes contributions from these advancements, evaluates their implications for mathematics and applied sciences, identifies existing research gaps, and proposes future directions for continued innovation and broader interdisciplinary integration.