To my knowledge as a math-turned-ML guy, there are currently no useful geometric characterizations of deep net latent spaces that are both "deep" (in the sense of using advanced mathematics) and "useful" (in the sense of revealing properties of networks or their latent spaces that aren't understood otherwise). Of course if anyone knows better I'd love to hear about it.

Continuous geometric concepts don't play super well with the way we like to decompose model outputs into discrete entities (classes, words, visual properties). We can, e.g. find variables in celebrity face GAN latent spaces that seem related to face orientation, or hair color, sort of, over some variable range and under some input conditions, but that doesn't really translate cleanly into any typical mathematical characterizations, geometric or otherwise, where you'd be looking for some property to hold everywhere or at least have an atlas of connected local approximations to simple characterizations.

Instead, we get high-dimensional messes of spaces, and network gradients during training don't exhibit clean or easy to understand dynamics except in the simplest toy cases.

To paraphrase a more serious "math for ML" prof I've chatted with at times -- "doing math" classically involves being able to find a description with only a few free parameters for a complex phenomenon that may superficially appear to have many/infinite free parameters. It's possible that for large ML models trained on natural data, such a reduction just doesn't exist, you can't break the contributions of millions or billions of parameters down into a low-dimensional approximation. He was/is skeptical of us attaining deep mathematical insight into their operation, but he could always be wrong. I'd certainly love to see cool novel insights come out of mathematics that give clarity to what's been going on these past 15 years.