The Italians solved this problem a long time ago.

Isn't this what happened with IMRN?

It would be helpful to remind the rest of us what is Hilbert's 11th problem. Those who aren't actively studying a particular Hilbert problem have trouble remembering which is which.

@Uncool: Start with the one numbered 1.

@Uncool: The paper you mention is an overview, if I remember right. There is a series of numbered papers that have all the details.

This is not true. It fails quite generally. See papers by Li Sheng Tseng with Tsai and Yau and others.

An algebra is $d$-generated if $d$ is the smallest integer such that there exist $d$ elements whose iterated products, however associated, span the algebra. The only Lie algebra that is $1$-generated in this sense is the trivial one-dimensional Lie algebra. For example, $sl(2)$ is $2$-generated in this sense although it is $3$-dimensional. M. Kuranishi showed that over a field of characteristic $0$ any semisimple Lie algebra is $2$-generated (doi.org/10.1017/S0027763000010059).

@GerryMyerson: "repository"

I'm not sure how one talks about the winding number without integers, and I'm not sure how one does topology without the winding number.

Much of what is called "screw theory" in the engineering literature amounts to special cases of general computations in Clifford algebras. I agree that the tendency to present this material by describing how it works in every different example (e.g. in McCarthy's book on linkages) makes it hard to get clear what are the basic ideas. The book "Geometric Robotics" by Selig is the textbook that I know that does the best job of treating this material in a way that is accessible to a mathematician.

If one broadens the context to include ethics in mathematical sciences so that it includes statistics, then there is a lot written. For example, Andrew Gelman's blog statmodeling.stat.columbia.edu is partly about the ethics of use and practice in statistics.

The reference given by Jim Humphreys is now freely available here: doi.org/10.2969/aspm/01110147

The gap between explaining what is a conformal map and what is conformal field theory is gigantic.

@alvarezpaiva: one confounding issue, relevant to the symplectic world, is where does volume fit? In the two-dimensional world area is part of geometry, whatever one means by geometry, In higher dimensions this is less clear. Perhaps one should speak also of "volumetric geometry". I think (maybe wrongly) of the "symplectic topology" terminology as due to Arnold. Note that Arnold/Khesin called their book "topological methods in hydrodynamics" not "geometric methods in hydrodynamics" and only write there of "geometry" when working with a metric on the diffeomorphism group.

@alvarezpaiva: I would put it this way - the debate about how to use "geometry" may or not be a worthwhile enterprise - I'm inclined to be cautious about assigning the word any tight definition - but then the use of "geometry" in "symplectic geometry" and "riemannian geometry" is very different in some senses. That's fine, the use of "geometry" in "algebraic geometry" is probably closer to the symplectic sense than the riemannian sense.

@alvarezpaiva: in projective differential geometry there is an underlying connection on a principal bundle with finite-dimensional structural group. This is not so for symplectic manifolds without further structure. In this sense symplectic manifolds are akin to differentiable manifolds and the study of their global properties is akin to differential topology, so could be called "symplectic topology". Whether it is worthwhile to make clean terminological distinctions between geometry and topology is another matter (where would combinatorics fit?).