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TOPOLOGY, GEOMETRY AND DEEP LEARNING

A Mathematics-to-AI Textbook

TOPOLOGY, GEOMETRY AND DEEP LEARNING
Anshuman Mishra

From Mathematical Foundations to the Future of AI

Most deep learning still lives in flat, Euclidean space. Real-world data does not.

This 15-chapter university textbook takes you from the basics of topology and geometry to the frontier of Geometric Deep Learning, Topological Data Analysis, equivariant networks, and optimization on manifolds.

Clear explanations. Worked examples. Research roadmaps.

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About the Book

Topology, Geometry and Deep Learning is a comprehensive, student-friendly textbook that bridges pure mathematics and modern artificial intelligence.

Spanning 15 carefully structured chapters, the book takes readers from the mathematical foundations of topology and geometry all the way to cutting-edge topics in Geometric Deep Learning, Topological Data Analysis, equivariant neural networks, and optimization on curved spaces.

Readers will learn:

  • How topology and geometry provide the right language for non-Euclidean data
  • Persistent homology, Betti numbers, and topological feature extraction
  • Graph Neural Networks, message passing, and geometric convolutions
  • Learning on manifolds, hyperbolic embeddings, and Riemannian optimization
  • Emerging research frontiers including simplicial neural networks, geometric foundation models, and topology-aware AI

Written in clear, accessible language with worked examples, conceptual reviews, self-assessment questions, and research roadmaps, this book is ideal for university students, researchers, and practitioners who want to move beyond standard Euclidean deep learning.

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MathematicsArtificial IntelligenceAI GovernanceDecision ModelEval EngineeringLarge language models

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Author

About the Author

Anshuman Mishra

Anshuman Kumar Mishra, M.Tech (Computer Science) Assistant Professor, Doranda College, Ranchi University

Prolific Author of 50+ Books on AI, Machine Learning & Computer Science | 20+ Years Experience

Anshuman Kumar Mishra is a dedicated educator, researcher, and highly prolific author with over 20 years of experience in Computer Science and Information Technology. Holding an M.Tech in Computer Science from BIT Mesra, he brings a rare combination of academic depth and practical teaching expertise.

Currently serving as Assistant Professor at Doranda College under Ranchi University, he has mentored thousands of students, helping them build strong foundations in programming, data science, and artificial intelligence. His student-centric teaching style emphasizes conceptual clarity, hands-on practice, and real-world application.

Anshuman is a prolific author with more than 50 books published across a wide spectrum of computer science and emerging technology domains. From foundational programming languages to advanced topics in Artificial Intelligence, Machine Learning, Reinforcement Learning, Decision Theory, and Computer Vision — his books are widely appreciated by students, educators, and professionals for their clear explanations, strong theoretical foundation, and practical approach.

His extensive body of work reflects his deep commitment to making complex subjects accessible and meaningful for learners at all levels. He is particularly recognized for creating well-structured learning paths that help readers progress from beginner to advanced levels with confidence.

Driven by the mission to democratize quality technical education, Anshuman continues to write and update books that bridge the gap between academic theory and industry practice.

When not teaching or writing, he actively follows and explores new developments in AI, Quantum Machine Learning, and Ethical Intelligence systems.

Contents

Table of Contents

TABLE OF CONTENTS Chapter 1: Introduction to Topology, Geometry and Deep Learning 11 1.1 What Are Topology and Geometry? 11 1.2 Why Geometry Matters in Artificial Intelligence 13 1.3 From Euclidean Data to Non-Euclidean Data 15 1.4 What Is Topological Data Analysis? 17 1.5 What Is Geometric Deep Learning? 19 1.6 Relationship Between Topology, Geometry and Deep Learning 21 1.7 Examples of Geometric and Topological Data 22 1.8 Real-World Applications 23 1.9 Advantages and Limitations 24 1.10 Key Concepts and Summary 25 Chapter 1: Integrated Worked Examples 27 Chapter 1: Extended Conceptual Review 28 1.11 A Practical Decision Framework for Choosing a Mathematical 28 1.12 A Deeper Look at Structure-Preserving Learning 29 1.13 Visual Interpretation Guide 29 1.14 Glossary for Chapter 1 30 1.15 Chapter-End Self Assessment 31 1.16 Research Questions for Further Study 32 1.17 Chapter Roadmap: From This Chapter to Research 32 Chapter 2: Mathematical Foundations for Deep Learning 33 2.1 Sets, Functions and Relations 33 2.2 Vectors and Matrices 34 2.3 Vector Spaces 35 2.4 Distance and Similarity 36 2.5 Norms and Metrics 36 2.6 Inner Products and Angles 37 2.7 Linear Transformations 38 2.8 Eigenvalues and Eigenvectors 38 2.9 Graphs and Adjacency Matrices 39 2.10 A Gentle Introduction to Optimization 40 Topology, Geometry and Deep Learning 3 TABLE OF CONTENTS 2.11 Mathematical Notation Used in This Book 41 2.12 Worked Examples and Exercises 41 Chapter 3: Introduction to Topology 49 3.1 What Is Topology 49 3.2 Open and Closed Sets 50 3.3 Topological Spaces 51 3.4 Continuity and Homeomorphism 52 3.5 Connectedness 53 3.6 Compactness 54 3.7 Neighborhoods and Boundaries 55 3.8 Holes and Their Mathematical Meaning 56 3.9 Topological Invariance 57 3.10 Intuitive Examples of Topological Spaces 58 3.11 Why Topology Is Useful for Machine Learning 59 3.12 Chapter Summary and Exercises 60 Chapter 4: Geometry, Manifolds and Data Spaces 65 4.1 From Flat Spaces to Curved Spaces 65 4.2 What Is a Manifold? 66 4.3 Local and Global Geometry 67 4.4 Curves and Surfaces 67 4.5 Tangent Spaces 68 4.6 Riemannian Geometry 68 4.7 Geodesics and Geodesic Distance 69 4.8 Curvature 70 4.9 Data Manifolds 70 4.10 The Manifold Hypothesis in Machine Learning 71 4.11 Examples from Computer Vision and Science 72 4.12 Practical Exercises 72 4.13 Extended Conceptual Review 74 Chapter 5: Topological Data Analysis — From Data to Shape 79 5.1 What Is Topological Data Analysis? 79 Topology, Geometry and Deep Learning 4 TABLE OF CONTENTS 5.2 The Shape of Data 80 5.3 Point Clouds 81 5.4 Neighborhood-Based Representations 81 5.5 Simplicial Complexes 82 5.6 Čech Complexes 83 5.7 Vietoris–Rips Complexes 83 5.8 Filtrations 84 5.9 Betti Numbers 85 5.10 Connected Components and Holes 85 5.11 From High-Dimensional Data to Topological Structure 86 5.12 Applications of TDA 87 5.13 Limitations of Basic TDA 87 5.14 Extended Conceptual Review 89 Chapter 6: Persistent Homology and Topological Features 93 6.1 Motivation for Persistent Homology 93 6.2 Birth and Death of Topological Features 94 6.3 Persistence Diagrams 95 6.4 Persistence Barcodes 95 6.5 Persistence Intervals 96 6.6 Betti Curves 97 6.7 Stability of Persistent Features 97 6.8 Persistent Homology Pipeline 98 6.9 Extracting Machine Learning Features 99 6.10 Example with a Simple Point Cloud 99 6.11 Noise and Robustness 100 6.12 Applications in Classification and Clustering 101 6.13 Extended Conceptual Review 103 Chapter 7: Graphs, Networks and Graph Neural Networks 107 7.1 Why Graphs Are Important in AI 110 7.2 Nodes, Edges and Graph Attributes 111 7.3 Adjacency and Degree Matrices 111 Topology, Geometry and Deep Learning 5 TABLE OF CONTENTS 7.4 Graph Representations 112 7.5 Message Passing 112 7.6 Graph Convolutional Networks 112 7.7 GraphSAGE 113 7.8 Graph Attention Networks 114 7.9 Node, Edge and Graph-Level Prediction 114 7.10 Over-Smoothing and Over-Squashing 115 7.11 Graph Representation Learning 115 7.12 Applications of GNNs 116 7.13 Practical Example 116 7.14 Extended Conceptual Review 117 7.15 Worked Problems and Mini Projects 119 Chapter 8: Geometric Deep Learning 121 8.1 From CNNs to Geometric Deep Learning 122 8.2 What Is Non-Euclidean Data? 123 8.3 Grids, Graphs and Manifolds 124 8.4 Symmetry in Machine Learning 125 8.5 Invariance and Equivariance 126 8.6 Group Actions 128 8.7 Group-Equivariant Neural Networks 129 8.8 Geometric Convolution 130 8.9 Geometric Message Passing 131 8.10 Graphs, Meshes and Point Clouds 132 8.11 Geometric Deep Learning Architecture 134 8.12 Applications in Vision, Science and Robotics 135 8.13 Current Challenges 136 8.14 Extended Conceptual Review 137 8.15 Examination, Research and Practical Review 138 8.16 Worked Problems and Mini Projects 139 8.17 Chapter Summary, Key Terms and Self-Assessment 140 Chapter 9: Deep Learning on Manifolds and Geometric Spaces 141 Topology, Geometry and Deep Learning 6 TABLE OF CONTENTS 9.1 Why Standard Neural Networks May Be Insufficient 142 9.2 Learning on Riemannian Manifolds 143 9.3 Manifold-Valued Data 145 9.4 Riemannian Metrics 145 9.5 Exponential and Logarithmic Maps 146 9.6 Tangent-Space Representations 148 9.7 Geodesic-Based Learning 149 9.8 Hyperbolic Geometry and Deep Learning 151 9.9 Hyperbolic Embeddings 152 9.10 Spherical and Other Non-Euclidean Spaces 153 9.11 Applications to Hierarchical Data 154 9.12 Challenges in Manifold-Based Learning 155 9.13 Extended Conceptual Review 157 Chapter 10: Topology Meets Neural Networks 162 10.1 Why Combine Topology with Deep Learning 162 10.2 Topological Features as Neural Network Inputs 165 10.3 Persistent Homology for Feature Extraction 167 10.4 Topological Regularization 170 10.5 Topology-Aware Loss Functions 172 10.6 Persistent Diagrams and Neural Networks 173 10.7 Topological Layers 174 10.8 Topological Representation Learning 176 10.9 Topological Analysis of Neural Networks 177 10.10 Deep Topological Analytics 179 10.11 Robustness and Generalization 181 10.12 Applications 182 10.13 Practical Workflow 184 10.14 Chapter at a Glance 186 10.15 Core Conceptual Questions 186 10.16 Comparison of Topology Integration Strategies 187 10.17 Common Misconceptions and Corrections 188 Topology, Geometry and Deep Learning 7 TABLE OF CONTENTS 10.18 Examination-Focused Review 188 10.19 Higher-Order Thinking Questions 188 10.20 Research-Oriented Questions 189 10.21 Research Insight 189 10.22 Practical Research Checklist 189 10.23 Mini Projects 190 10.24 Worked Problems 190 10.25 Key Terms 191 10.26 Chapter Summary 191 10.27 Self-Assessment Checklist 191 Chapter 11: Geometric Neural Networks and Equivariant Learning 192 11.1 The Role of Symmetry 192 11.2 Invariant vs. Equivariant Functions 195 11.3 Group-Equivariant Neural Networks 197 11.4 E(n)-Equivariant Networks 198 11.5 SE(3)-Equivariant Learning 199 11.6 Equivariant Message Passing 201 11.7 Geometric Attention 203 11.8 3D Point-Cloud Learning 205 11.9 Molecular and Physical Systems 207 11.10 Scientific Machine Learning Applications 209 11.11 Computational Challenges 211 11.12 Research Directions 213 11.13 Chapter at a Glance 214 11.14 Core Conceptual Questions 214 11.15 Comparison: Invariant, Equivariant, and Ordinary Learning 215 11.16 Common Misconceptions and Corrections 216 11.17 Examination-Focused Review 216 11.18 Higher-Order Thinking Questions 216 11.19 Research-Oriented Questions 217 11.20 Research Insight 217 Topology, Geometry and Deep Learning 8 TABLE OF CONTENTS 11.21 Practical Research Checklist 217 11.22 Mini Projects 218 11.23 Worked Problems 218 11.24 Key Terms 219 11.25 Chapter Summary 219 11.26 Self-Assessment Checklist 219 Chapter 12: Geometric Optimization and Learning on Curved Spaces 221 12.1 Why Euclidean Optimization Is Not Always Enough 222 12.2 Optimization on Manifolds 223 12.3 Riemannian Gradients 224 12.4 Riemannian Optimization 225 12.5 Retractions and Exponential Maps 225 12.6 Manifold-Aware Neural Network Training 226 12.7 Hyperbolic Optimization 227 12.8 Optimization Challenges 228 12.9 Stability and Convergence 229 12.10 Applications in Deep Learning 230 12.11 Practical Examples 230 12.12 Open Research Questions 232 Chapter 13: Advanced Topological Deep Learning 237 13.1 Simplicial Neural Networks 238 13.2 Simplicial Complexes in Machine Learning 239 13.3 Cellular Complexes 239 13.4 Cell Complex Neural Networks 240 13.5 Higher-Order Interactions 241 13.6 Beyond Pairwise Graph Relationships 242 13.7 Topological Message Passing 242 13.8 Persistent-Learning Architectures 243 13.9 Topological Attention 243 13.10 Topological Regularization 244 13.11 Topological Generative Models 245 Topology, Geometry and Deep Learning 9 TABLE OF CONTENTS 13.12 Applications in Complex Systems 246 13.13 Current Research Challenges 247 Chapter 14: Research Frontiers in Topology and Geometry for AI 252 14.1 Geometric Graph Neural Networks 253 14.2 Graph Structure Learning 254 14.3 Graph Transformers and Geometric Transformers 255 14.4 Topological Representation Learning 256 14.5 Geometric Foundation Models 257 14.6 3D and 4D Geometric Learning 257 14.7 Molecular and Protein Structure Learning 258 14.8 Physics-Informed Geometric Deep Learning 259 14.9 Robotics and Spatial Intelligence 260 14.10 Medical and Biological Applications 260 14.11 Robust and Trustworthy Geometric AI 261 14.12 Scalability and Computational Efficiency 263 14.13 Open Problems for Researchers 264 Chapter 15: Future of Topological and Geometric Deep Learning 270 15.1 From Deep Learning to Geometric Intelligence 271 15.2 Topology-Aware Foundation Models 272 15.3 AI for Scientific Discovery 273 15.4 Learning Physical Laws Through Geometry 274 15.5 Neural Networks on Dynamic Manifolds 274 15.6 Topological and Geometric Generative AI 275 15.7 Multimodal Geometric Learning 276 15.8 Geometric AI for Robotics 276 15.9 AI for Molecules, Proteins and Materials 277 15.10 Explainable Topological and Geometric AI 278 15.11 Open Problems and Research Opportunities 279 15.12 How Students Can Start Research in This Field 280 15.13 Suggested Research Projects

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