1. Introduction  
2. UMAP  
3. Justifying UMAP’s Cost Function by Algebraic Topology and Category Theory  
4. Neighbor Embedding: Comparison with t-SNE and LargeVis  
5. Discussion on Repulsive Forces and Negative Sampling in the UMAP’s Cost Function  
6. DensMAP for Density-Preserving Embedding  
7. Parametric UMAP for Embedding by Deep Learning  
8. Progressive UMAP for Streaming and Out-of-sample Data  
9. Conclusion