FGA Explained. Articles by a bunch of people, most of them free online. You have Vistoli explaining what a Stack is, with Descent Theory, Nitsure constructing the Hilbert and Quot schemes, with interesting special cases examined by Fantechi and Goettsche, Illusie doing formal geometry and Kleiman talking about the Picard scheme.
For intersection theory, I second Fulton's book.
And for more on the Hilbert scheme (and Chow varieties, for that matter) I rather like the first chapter of Kollar's "Rational Curves on Algebraic Varieties", though he references a couple of theorems in Mumfords "Curves on Surfaces" to do the construction.
And on the "algebraic geometry sucks" part, I never hit it, but then I've been just grabbing things piecemeal for awhile and not worrying too much about getting a proper, thorough grounding in any bit of technical stuff until I really need it, and when I do anything, I always just fall back to focus on varieties over C to make sure I know what's going on.
EDIT: Forgot to mention, Gelfand, Kapranov, Zelevinsky "Discriminants, resultants and multidimensional determinants" covers a lot of ground, fairly concretely, including Chow varieties and some toric stuff, if I recall right (don't have it in front of me)
The analogy doesn't quite give a number theoretic version of the Poincare conjecture. See Sikora, "Analogies between group actions on 3-manifolds and number fields"
(arXiv:0107210): the author states the Poincare conjecture as "S3 is the only closed 3-manifold with no unbranched covers." The analogous statement in number theory is that Q is the only number field with no unramified extensions, and indeed he points out that there are a few known counterexamples, such as the imaginary quadratic fields with class number 1.
The paper also has a nice but short summary of the so-called "MKR dictionary" relating 3-manifolds to number fields in section 2. Morishita's expository article on the subject, arXiv:0904.3399, has more to say about what knot complements, meridians and longitudes, knot groups, etc. are, but I don't think there's an explanation of what knot surgery would be and so I'm not sure how Kirby calculus fits into the picture.
Edit: An article by B. Morin on Sikora's dictionary (and how it relates to Lichtenbaum's cohomology, p. 28): "he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora's formulas which leads us to clarify and to extend the dictionary of arithmetic topology."
Best Answer
One useful remark is that dimension reduction is a critical problem in data science for which there are a variety of useful approaches. It is important because a great many good machine learning algorithms have complexity which depends on the number of parameters used to describe the data (sometimes exponentially!), so reducing the dimension can turn an impractical algorithm into a practical one.
This has two implications for your question. First, if you invent a cool new algorithm then don't worry too much about the dimension of the data at the outset - practitioners already have a bag of tricks for dealing with it (e.g. Johnson-Lindenstrauss embeddings, principal component analysis, various sorts of regularization). Second, it seems to me that dimension reduction is itself an area where more sophisticated geometric techniques could be brought to bear - many of the existing algorithms already have a geometric flavor.
That said, there are a lot of barriers to entry for new machine learning algorithms at this point. One problem is that the marginal benefit of a great answer over a good answer is not very high (and sometimes even negative!) unless the data has been studied to death. The marginal gains are much higher for feature extraction, which is really more of a domain-specific problem than a math problem. Another problem is that many new algorithms which draw upon sophisticated mathematics wind up answering questions that nobody was really asking, often because they were developed by mathematicians who tend to focus more on techniques than applications. Topological data analysis is a typical example of this: it has generated a lot of excitement among mathematicians but most data scientists have never heard of it, and the reason is simply that higher order topological structures have not so far been found to be relevant to practical inference / classification problems.