Given that PCA is heavily antiquated these days, I'd say that asking your candidates to know algebraic topology (the basis behind many much more effective non linear DR algorithms like UMAP) is far better... But in spite of the field having long ago advanced beyond PCA, you're still using it to gatekeep.
The initialization strategy for UMAP is important enough that asking about that in practice is probably more important than anything out of Ghrist's book as an interview question
UMAP (and t-SNE) aren't the same as PCA. UMAP is pretty close to t-SNE and I think expanding PCA (Principle Component Analysis) and t-SNE (teacher Stochastic Neighbor Embedding) explain the difference. Neighbor embedding is a visualization technique and not the same as determining principle components. PCA preserves global properties while t-SNE and UMAP don't. They are good techniques for _visual_ dimensional reduction, but they aren't going to tell you the dominant eigenvectors of the data, or _dimensional reduction_. This is a bit of a pet peeve of mine.
