Notes and reflections on the things I’ve learned while Doing Scholarship the last two week (i.e. studying math).
Mostly the past two weeks were on differential geometry (Lee):
Ch 4 (Submersion, Immersion, Embedding) comments:
Conceptually, by the Constant rank theorem, constant rank maps (smooth maps whose differential dFp:TpM→TF(p)N is constant rank at all p) are precisely the maps with a linear local coordinate representation (thus are maps well-modeled locally by its differentials).
Basically a nonlinear version of the linear algebra theorem that any square matrix can be expressed as [Ir000]. The proof is much more complicated however: basically a clever choice of coordinate transformation via the inverse function theorem.
The point of the chapter is to come up with various characterizations of submersion, immersion, embedding. For example, 1) smooth immersion iff locally smooth embedding, 2) smooth submersion iff every point is an image of a local section, 3) surjective maps ⇒ submersion & injective ⇒ immersion …
The proof of 3) is a very cool application of the Baire category theorem. Baire category theorem says the countable union of nowhere dense sets has empty interior; this is not very motivating, but reading Bredon[1] helped clarify its conceptual significance.
Namely, consider the more illuminating contrapositive statement: countable intersection of dense open sets is dense. Conceptually, the space is some configuration space, and dense open sets represent configurations that satisfy certain generically satisfied constraints (polynomial p(x) being nonzero is a prototypical example, which is a dense & open set). Then, the question is whether the property of a countable number of these constraints being satisfied at the same time is still generic, i.e. dense. The Baire category theorem says this is indeed the case (for locally compact Hausdorff spaces).
Sections are just right inverses, and their intuitive geometric content was a bit confusing until I read the wikipedia page: a section of f is an abstraction of a graph by viewing f as a sort of “projection map.” That makes sense! I’m sure this will come up later in the fiber bundle context.
The “figure-eight curve” and “dense torus map” as prototypical examples of smooth immersions that isn’t a smooth embedding, due to topological considerations.
Ch 5 (Submanifold) comments:
Similar to Ch 4, many useful characterizations of submanifolds and how to generate them. eg embedded submanifold iff locally a “slice” of the ambient manifold’s coordinate chart. embedded submanifold iff image of smooth embedding, immersed submanifold iff image of smooth immersion. Level sets of a smooth map at a “regular value” are embedded submanifolds …
Ch 6 (Sard’s theorem) comments:
Finally, one of the more fun chapters! Finally learned the proof of the Whitney embedding / immersion theorem that I’ve heard a lot about.
The compact case of the Whitney embedding theorem is much more conceptually straightforward:
Given a m (finite, possible since compact) chart of the n-dim manifold, literally just adjoin them while multiplying them with appropriate partitions of unity to get a M→Rnm map, and adjoin the m partitions of unity (a “chart indicator variable”) to get a M→Rnm+m map. This turns out to be an immersion, and thus an embedding since M is compact.
Apply the projection map RN→RN−1 with a 1-dim kernel Rv. By Sard’s theorem, this turns out to be an immersion (when restricted to M) for almost any choice of v, as long as N>2n+1. Repeatedly apply this to the massive codomain M→Rnm+m to get an immersion to R2n+1.
This projection map can in fact be promoted to an embedding, given that the original immersion of M to R^n is an embedding.
High-level takeaways:
The most dumb and obvious way of interpolating coordinate charts into a global map via partitions of unity, with slight modifications, gives a bona fide immersion of a manifold into RN!!
It was interesting to learn that there was a 1-2 decade period of foundational uncertainty (between the first proposal of the abstract manifold definition and Whitney’s above proof) where people didn’t know whether the abstract manifold definition was actually more general than RN or not.[2]
Partitions of unity really is used everywhere. I wonder how the theory of complex analytic manifolds ever do anything when analytic partitions of unity don’t exist.
Proof strategy of promoting a smooth map to a proper map (at the cost of increased dimensionality of the codomain) by literally adjoining a proper map next to it. Clever!
I presume this is the main motivation behind exhaustion functions (f:M→R s.t.f−1((−∞,c]) is compact ∀c∈R). It’s a proper map, it exists for any manifolds (again, shown by partitions of unity), and has codomain of dimension 1 so it minimally increases the function codomain dimension.
More applications on Whitney approximation theorems and transversality arguments.
The latter, including the transversality homotopy theorem (actually learned this a year ago in my difftop class, though that class used Guillemin’s book where manifolds are always embedded in RN - so it’s good to learn them from a more intrinsic perspective) is very interesting.
It also ties to one of my motivation for all this math learning, backchaining from trying to do good alignment theory work, which is learning the math of structural stability and its role in the theory of forms (morphogenesis) cf Thom, Structural Stability and Morphogenesis (thank you Dan Murfet for explaining this perspective).
So much more elegant than the standard definition via charts and maximal smooth structures and such. Unsure of the utility of this characterization though, lol (read Lawvere’s paper).
“It is better to have a good category with bad objects than a bad category with good objects.”—Grothendieck (probably not). For example, the category of smooth manifolds is not nice, motivating smooth sets, diffeological spaces, and so on.
I found this intuition for adjoint functors illuminating. Specifically, note set maps f:X→Y and g:Y→X being inverses are equivalent to the condition that their graphs are mirrored along the diagonal, i.e. (x,f(x))=(g(y),y). Rephrase this using Kronecker delta, δ(x,g(y))=δ(f(x),y). Now δ can be seen as expressing a “relation” that could be exhibited by two elements of a set, i.e. equality (1) or inequality (0). But in general categories, objects can exhibit more relations—so replace δ by Hom - you get adjoint functors!
Why that long? The dimensionality reduction by projection is perhaps more nontrivial because of Sard, but the obvious gluing should have been sufficient to construct an immersion at least, albeit at the cost of inefficient codomain dimension. Maybe the historically difficult part was the concept of partition of unity and that it always exist in manifolds?
Becoming Stronger™ (Sep 28 - Oct 12)
Notes and reflections on the things I’ve learned while Doing Scholarship the last two week (i.e. studying math).
Mostly the past two weeks were on differential geometry (Lee):
Ch 4 (Submersion, Immersion, Embedding) comments:
Conceptually, by the Constant rank theorem, constant rank maps (smooth maps whose differential dFp:TpM→TF(p)N is constant rank at all p) are precisely the maps with a linear local coordinate representation (thus are maps well-modeled locally by its differentials).
Basically a nonlinear version of the linear algebra theorem that any square matrix can be expressed as [Ir000]. The proof is much more complicated however: basically a clever choice of coordinate transformation via the inverse function theorem.
The point of the chapter is to come up with various characterizations of submersion, immersion, embedding. For example, 1) smooth immersion iff locally smooth embedding, 2) smooth submersion iff every point is an image of a local section, 3) surjective maps ⇒ submersion & injective ⇒ immersion …
The proof of 3) is a very cool application of the Baire category theorem. Baire category theorem says the countable union of nowhere dense sets has empty interior; this is not very motivating, but reading Bredon[1] helped clarify its conceptual significance.
Namely, consider the more illuminating contrapositive statement: countable intersection of dense open sets is dense. Conceptually, the space is some configuration space, and dense open sets represent configurations that satisfy certain generically satisfied constraints (polynomial p(x) being nonzero is a prototypical example, which is a dense & open set). Then, the question is whether the property of a countable number of these constraints being satisfied at the same time is still generic, i.e. dense. The Baire category theorem says this is indeed the case (for locally compact Hausdorff spaces).
Sections are just right inverses, and their intuitive geometric content was a bit confusing until I read the wikipedia page: a section of f is an abstraction of a graph by viewing f as a sort of “projection map.” That makes sense! I’m sure this will come up later in the fiber bundle context.
The “figure-eight curve” and “dense torus map” as prototypical examples of smooth immersions that isn’t a smooth embedding, due to topological considerations.
Ch 5 (Submanifold) comments:
Similar to Ch 4, many useful characterizations of submanifolds and how to generate them. eg embedded submanifold iff locally a “slice” of the ambient manifold’s coordinate chart. embedded submanifold iff image of smooth embedding, immersed submanifold iff image of smooth immersion. Level sets of a smooth map at a “regular value” are embedded submanifolds …
Ch 6 (Sard’s theorem) comments:
Finally, one of the more fun chapters! Finally learned the proof of the Whitney embedding / immersion theorem that I’ve heard a lot about.
The compact case of the Whitney embedding theorem is much more conceptually straightforward:
Given a m (finite, possible since compact) chart of the n-dim manifold, literally just adjoin them while multiplying them with appropriate partitions of unity to get a M→Rnm map, and adjoin the m partitions of unity (a “chart indicator variable”) to get a M→Rnm+m map. This turns out to be an immersion, and thus an embedding since M is compact.
Apply the projection map RN→RN−1 with a 1-dim kernel Rv. By Sard’s theorem, this turns out to be an immersion (when restricted to M) for almost any choice of v, as long as N>2n+1. Repeatedly apply this to the massive codomain M→Rnm+m to get an immersion to R2n+1.
This projection map can in fact be promoted to an embedding, given that the original immersion of M to R^n is an embedding.
High-level takeaways:
The most dumb and obvious way of interpolating coordinate charts into a global map via partitions of unity, with slight modifications, gives a bona fide immersion of a manifold into RN!!
It was interesting to learn that there was a 1-2 decade period of foundational uncertainty (between the first proposal of the abstract manifold definition and Whitney’s above proof) where people didn’t know whether the abstract manifold definition was actually more general than RN or not.[2]
Partitions of unity really is used everywhere. I wonder how the theory of complex analytic manifolds ever do anything when analytic partitions of unity don’t exist.
Proof strategy of promoting a smooth map to a proper map (at the cost of increased dimensionality of the codomain) by literally adjoining a proper map next to it. Clever!
I presume this is the main motivation behind exhaustion functions (f:M→R s.t.f−1((−∞,c]) is compact ∀c∈R). It’s a proper map, it exists for any manifolds (again, shown by partitions of unity), and has codomain of dimension 1 so it minimally increases the function codomain dimension.
More applications on Whitney approximation theorems and transversality arguments.
The latter, including the transversality homotopy theorem (actually learned this a year ago in my difftop class, though that class used Guillemin’s book where manifolds are always embedded in RN - so it’s good to learn them from a more intrinsic perspective) is very interesting.
It also ties to one of my motivation for all this math learning, backchaining from trying to do good alignment theory work, which is learning the math of structural stability and its role in the theory of forms (morphogenesis) cf Thom, Structural Stability and Morphogenesis (thank you Dan Murfet for explaining this perspective).
Rabbit holes that I could not afford to pursue:
The category of smooth manifolds is an idempotent-splitting completion of the category of open subspaces of findim cartesian spaces?!?!?! My mind is blown.
So much more elegant than the standard definition via charts and maximal smooth structures and such. Unsure of the utility of this characterization though, lol (read Lawvere’s paper).
There is a duality between the category of smooth manifolds and the category of R-algebras. Fascinating how such dualities between algebra and geometry seem to be a very common motif throughout different fields, I’m sure this will come up in Vakil’s book later. Also curious about Gelfand’s duality on this for topological spaces.
“It is better to have a good category with bad objects than a bad category with good objects.”—Grothendieck (probably not). For example, the category of smooth manifolds is not nice, motivating smooth sets, diffeological spaces, and so on.
Dichotomy between nice objects and nice categories: in the context of alignment theory, maybe I can view Programs as Singularities as enlarging an instantiation of this idea by enlarging the class of Turing machines.
I found this intuition for adjoint functors illuminating. Specifically, note set maps f:X→Y and g:Y→X being inverses are equivalent to the condition that their graphs are mirrored along the diagonal, i.e. (x,f(x))=(g(y),y). Rephrase this using Kronecker delta, δ(x,g(y))=δ(f(x),y). Now δ can be seen as expressing a “relation” that could be exhibited by two elements of a set, i.e. equality (1) or inequality (0). But in general categories, objects can exhibit more relations—so replace δ by Hom - you get adjoint functors!
Example of how reading books in parallel improves learning efficiency.
Why that long? The dimensionality reduction by projection is perhaps more nontrivial because of Sard, but the obvious gluing should have been sufficient to construct an immersion at least, albeit at the cost of inefficient codomain dimension. Maybe the historically difficult part was the concept of partition of unity and that it always exist in manifolds?