Introduction to Homology Theory and Algebraic Topology

Understanding the shape of data and geometric spaces is a fundamental challenge in modern mathematics and computer science. This course offers a clear, structured path into homology theory, translating complex topological concepts into concrete algebraic structures. By studying these concepts, you will transition from intuitive geometric ideas to rigorous algebraic formulations, enabling you to compute homology groups and analyze the global structure of spaces. You will also explore how these classical mathematical tools are applied to modern data analysis. What you'll learn: - Understand the foundational concepts of topological spaces, simplicial complexes, and triangulation. - Define and construct chain complexes, boundary operators, and cycle groups. - Compute simplicial homology groups for various geometric shapes and surfaces. - Apply the Euler characteristic and understand its topological invariance. - Explore the basics of persistent homology and its applications in topological data analysis. - Practice algebraic computations through step-by-step written proofs and exercises. The course begins with essential topological definitions and geometric intuition before moving systematically through simplicial homology, algebraic chain complexes, and modern computational applications. Designed for undergraduate students in mathematics, computer science, or data science, this course requires only a basic background in linear algebra to get started. Start reading today to unlock the power of algebraic topology and see geometry through an algebraic lens.