A COMPREHENSIVE REVIEW OF TOPOLOGICAL METHODS IN MODERN MATHEMATICS
Keywords:
Topology, algebraic topology, differential topology, computational topology, topological data analysis (TDA), homotopy theory, homology, category theory, dynamical systems, applied mathematics, artificial intelligence, complex systems, geometric structures, modern mathematical methods.Abstract
This article provides a comprehensive review of topological methods and their role in modern mathematics. The paper analyzes the theoretical foundations of topology, explores its evolution from classical to contemporary frameworks, and highlights its applications in various fields such as geometry, analysis, algebra, data science, and mathematical physics. Special attention is given to the use of topological techniques in the study of complex systems, computational topology, and topological data analysis (TDA), which have recently gained significance in applied mathematics and artificial intelligence. The review also examines the intersection of topology with other disciplines, including graph theory, category theory, and dynamical systems, emphasizing how topological perspectives contribute to problem-solving in both pure and applied contexts. By synthesizing recent research trends, the article underlines the importance of topological thinking as a unifying language in mathematics and its potential for future developments.
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