Just confirming that I, another mathematician (and a pretty rusty one; I’ve been working in industry for decades) would also look at that and think “that sure looks like word salad to me” in a way that I wouldn’t for “an excellent Cohen-Macaulay domain is normal”.
Unpacking the thought process a little, though I still expect this to be incomprehensible to people who aren’t mathematicians: I don’t know what the “Lagrangian graph category” might be, but I couldn’t swear there isn’t one; but if there is, it doesn’t seem like a thing that would have things called “manifold rings” in it. “Manifold ring” is not a phrase I have ever heard but I could imagine someone using it to mean something along the lines of a Lie group. Maaaaaybe you might then be able to define manifolds and rings “over” some weird category (though “the Lagrangian graph category” doesn’t feel like it could be the right sort of category, even if there were such a thing) but never “in” it: a thing “in” the “Lagrangian graph category” would surely have to be Lagrangian graphs, or graphs understood in some Lagrangian way, or something like that. I have heard “flat” applied to kinda-algebraic-kinda-geometrical objects so maybe there could be “flat manifold rings” and even “universally flat” ones, but I’d be a little surprised. Usually anything of the form “An X group is a Y ring” is going to be nonsense; rings have extra structure that groups don’t, and if you have some special kind of ring you’re not going to call it an anything group; but in mathematics a red herring is not necessarily either red or a herring, and e.g. a “quantum group” is a kind of algebra, and an algebra is among other things a kind of ring, so meh. But “Riemannian group” feels very strange indeed. I am too rusty to be more than, say, 95% confident that this can’t all be made to fit together somehow, but it’s very much not the way I’d bet.
Well, you might be able to. :)
Just confirming that I, another mathematician (and a pretty rusty one; I’ve been working in industry for decades) would also look at that and think “that sure looks like word salad to me” in a way that I wouldn’t for “an excellent Cohen-Macaulay domain is normal”.
Unpacking the thought process a little, though I still expect this to be incomprehensible to people who aren’t mathematicians: I don’t know what the “Lagrangian graph category” might be, but I couldn’t swear there isn’t one; but if there is, it doesn’t seem like a thing that would have things called “manifold rings” in it. “Manifold ring” is not a phrase I have ever heard but I could imagine someone using it to mean something along the lines of a Lie group. Maaaaaybe you might then be able to define manifolds and rings “over” some weird category (though “the Lagrangian graph category” doesn’t feel like it could be the right sort of category, even if there were such a thing) but never “in” it: a thing “in” the “Lagrangian graph category” would surely have to be Lagrangian graphs, or graphs understood in some Lagrangian way, or something like that. I have heard “flat” applied to kinda-algebraic-kinda-geometrical objects so maybe there could be “flat manifold rings” and even “universally flat” ones, but I’d be a little surprised. Usually anything of the form “An X group is a Y ring” is going to be nonsense; rings have extra structure that groups don’t, and if you have some special kind of ring you’re not going to call it an anything group; but in mathematics a red herring is not necessarily either red or a herring, and e.g. a “quantum group” is a kind of algebra, and an algebra is among other things a kind of ring, so meh. But “Riemannian group” feels very strange indeed. I am too rusty to be more than, say, 95% confident that this can’t all be made to fit together somehow, but it’s very much not the way I’d bet.