Topology and geometry in Artificial Intelligence: Manifolds high-dimensional learning and geometric deep models(Math and Artificial Intelligence)

Artificial Intelligence (AI) has rapidly transformed from a theoretical curiosity into one of the most powerful technological forces shaping the modern world. From self-driving cars to natural language processing, from healthcare diagnostics to advanced robotics, AI is driving the fourth industrial revolution. Yet, behind the success of deep learning and machine learning models lies a less visible but profoundly important mathematical foundation-geometry and topology. These fields, which traditionally belonged to pure mathematics, are now emerging as central to the future of AI. The last decade has shown that data is not merely a collection of numbers or vectors but often lies on complex geometric structures called manifolds. Understanding these manifolds, learning from them, and designing models that respect their structure has become a key challenge in modern AI. Similarly, topology provides tools to understand the shape of data, identify its intrinsic structure, and improve the interpretability of AI systems. This book, "Topology and Geometry in Artificial Intelligence: Manifolds, High-Dimensional Learning, and Geometric Deep Models," aims to serve as a comprehensive guide for students, researchers, and practitioners who wish to explore how topology and geometry fundamentally reshape our approach to AI. Unlike standard machine learning textbooks that emphasize algorithms and programming, this book focuses on the mathematical principles that enable AI to handle high-dimensional, complex, and structured data. Why This Book Is Important Bridging Mathematics and AI AI students often learn linear algebra, probability, and calculus, but few are introduced to topology and differential geometry in the context of machine learning. This book fills that gap by providing an accessible yet rigorous introduction to these advanced mathematical concepts and showing how they directly apply to real-world AI problems. Manifold Hypothesis and High-Dimensional Data One of the central assumptions in machine learning is the manifold hypothesis: that high-dimensional data (such as images, speech, or biological signals) actually lies on low-dimensional manifolds embedded in higher-dimensional space. This book explains this hypothesis in detail, demonstrating how it guides the design of algorithms for dimensionality reduction, visualization, and representation learning. Geometric Deep Learning (GDL) Traditional deep learning operates on Euclidean data, but much of the data in modern applications is non-Euclidean-such as graphs, social networks, molecules, or 3D surfaces. Geometric Deep Learning provides the framework to extend neural networks to these domains. This book explores GDL methods, from graph neural networks to hyperbolic embeddings, providing students with cutting-edge tools for research and applications. Topological Data Analysis (TDA) Beyond geometry, topology provides tools like persistent homology that help uncover the shape and connectivity of data. These methods are particularly valuable for understanding deep learning models, analyzing high-dimensional datasets, and ensuring robustness. The book presents TDA in an accessible way, with practical case studies that connect theory with application. Essential for Advanced AI Research As AI moves toward explainability, interpretability, and efficiency, researchers must go beyond "black-box" neural networks. Geometry and topology provide the language for describing the structure of models, data, and optimization landscapes. This makes the book invaluable not only for students but also for PhD researchers and practitioners exploring frontiers of AI research.