Issue Details

HOMOLOGY UNVEILED: EXPLORING ALGEBRAIC AND GEOMETRIC APPROACHES

ABSTRACT

Homology theory serves as a fundamental tool in modern mathematics, with applications spanning algebraic topology, algebraic geometry, and various other fields. This comprehensive study explores the intricate interplay between algebraic and geometric approaches to homology, shedding light on the underlying structures and connections that enrich our understanding of topological spaces. In this examination, we delve into the foundational concepts of homology, starting with singular and simplicial homology theories, emphasizing their algebraic aspects. We then transition to a geometric perspective, exploring the geometric interpretation of homology through techniques such as homotopy theory and the theory of spectra. This interdisciplinary approach reveals the profound linkages between algebraic and geometric viewpoints, providing new insights into the topological invariants that homology theory seeks to capture. Moreover, we investigate advanced topics, including persistent homology and applications in data analysis, demonstrating how these concepts bridge the gap between pure mathematics and practical problem-solving. By synthesizing these diverse aspects of homology theory, this study aims to foster a deeper appreciation for its elegance and utility, highlighting its role as a unifying force in the mathematical landscape. Through a synthesis of theory and applications, this comprehensive study offers a valuable resource for mathematicians, researchers, and students interested in the profound and intricate world of algebraic and geometric homology.

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