Notes and reflections on the things I’ve learned while Doing Scholarship™ this week (i.e. studying math)[1].
I am starting to see the value of categorical thinking.
For example from [FOAG], it was quite mindblowing to learn that stalk (the set of germs at a point) can be equivalently defined as a simple colimit of sections of presheaf over open sets of X containing a point, and this definition made proving certain constructions (eg inducing a map of stalks from a map ϕ:X→Y) very easy.
Also, I was first introduced the concept of presheaf as an abstraction of a map that takes open sets and returns functions over it, abstracting properties like there existing a restriction map that composes naturally. Turns out (punchline, presumably) this is just a functor Open(X)Op−>Set!
Yoneda lemma is very cool. I recall seeing some of the ideas from Programs as Singularities (paper), where there are ideas of embedding programs (for the analogue in Yoneda lemma, the Yoneda embedding being a fully faithful functor from some category C … ) into a different space (… to the category SetC …) that contains “almost programs” ( … because the Yoneda embedding is not surjective, let alone essentially surjective), and that studying this enlarged space lends insight into the original space.
Rabbit hole: Yoneda lemma as expressing consistency conditions on Lawvere’s idea of Space and Quantity being presheaf and copresheaf??
I am also starting to more appreciate the notion of sheaf or ringed spaces from [FOAG] - or more generally, the notion that a “space” can be productively studied by studying functions defined on it. For example I learned from [Bredon] that a manifold, whose usual definition is a topological space locally homeomorphic to a Euclidean space, can equivalently be defined as a ringed space whose structure sheaf is valued in some subalgebra of continuous maps over a given open set. Very cool!
Started reading [Procesi] to learn invariant theory and representation theory because it came up quite often as my bottleneck in my recent work (eg). Also interpretability, apparently. So far I just read pg 1-9, reviewing the very basics of group action (e.g., orbit stabilizer theorem). Lie groups aren’t coming up until pg ~50 so until then I should catch up on the relevant Lie group prerequisites through [Lee] or [Bredon].
Also reviewed some basic topology by skimming pg 1-50 of [Bredon]. So many rabbit holes just in point-set topology that I can’t afford (time-wise) to follow, e.g.,
(1) nets generalize sequences and successfully characterize topological properties—I once learned of filters, and I do not yet know how they relate (and don’t relate constructively + why constructively filters are more natural) and especially universal net vs ultrafilter
(2) I didn’t know that manifolds are metrizable, but yes they are by an easy consequence of the Urysohn metrization theorem (second-countable & completely regular ⇒ metrizable). But I would like to study this proof in more detail. Also, how to intuitively think about the metric of a manifold?
I didn’t know that the proof to Urysohn metrization was this nice! It’s a consequence of the following lemma: recall, “completely regular” means given a point x and a closed set x∉C, there exists a continuous f:X→[0,1] s.t.f(x)=0 and f(C)=1. BUT adding second-countability to the hypothesis then lets you choose this f from a fixed, countable family F.
Then, mapping X under this countable family of functions (thus taking value in [0,1]N) turns out to be an embedding—and [0,1]N can be metrized, so X can be metrized as well.
(3) I learned about various natural variants / generalizations of compactness (σ-compactness, local compactness, paracompactness). My understanding of their importance is because:
(a) paracompactness implies the existence of partition of unity subordinate to any open cover (a consequence of paracompactness ⇒ normal, and Urysohn’s lemma, also paracompactness by definition allowing you to find the open refinement of the given open cover as required by the definition of partition of unity subordinate to an open cover.)
(b) for locally compact Hausdorff X, we can characterize paracompactness by “disjoint union of open σ-compact subsets,” which is much easier to check than the definition of paracompactness as locally finite open refinement of open covers.
e.g., from this, it is immediate that manifolds are paracompact: (1) locally Euclidean ⇒ locally compact. (2) Second-countable ⇒ Lindelof. (3) Lindelof & locally compact ⇒ σ-compact. (1) & (2) & (3) + above ⇒ manifolds are paracompact. From which other properties of manifolds immediately follow from that of paracompactness, eg manifolds always admit a partition of unity subordinate to any open cover.
But rabbit hole: recall, open sets axiomatize semidecidable properties. What is, then, the logical interpretation of compactness, σ-compactness, local compactness, paracompactness?
This week, I’ll start tracking the exercises I solve and pages I cover and post them in next week’s shortform, so that I can keep track of my progress + additional accountability.
[Procesi]: Procesi, Lie Groups: An Approach through Invariants and Representations
and I plan to do most of the exercises for each of the textbooks unless I find some of them too redundant. For this week’s shortform I haven’t written down my progress this week on each of these books nor the problems I’ve solved because I haven’t started tracking them, so I’ll do them starting next week.
Started reading [Procesi] to learn invariant theory and representation theory because it came up quite often as my bottleneck in my recent work (eg). Also interpretability, apparently. So far I just read pg 1-9, reviewing the very basics of group action (e.g., orbit stabilizer theorem). Lie groups aren’t coming up until pg ~50 so until then I should catch up on the relevant Lie group prerequisites through [Lee] or [Bredon].
Woit’s “Quantum Theory, Groups and Representations” is fantastic for this IMO. It gives physical motivation for representation theory, connects it to invariants and, of course, works through the physically important lie-groups. The intuitions you build here should generalize. Plus, it’s well written.
Also, if you are ever in the market for differential topology, algebraic topology, and algebraic geometry, then I’d recommend Ronald Brown’s “Topology and Groupoids.” It presents the basic material of topology in a way that generalizes better to the fields above, along with some powerful geometric tools for calculations.
Thanks for the recommendation! Woit’s book does look fantastic (also as an introduction to quantum mechanics). I also known Sternberg’s Group Theory and Physics to be a good representation theory & physics book.
I did encounter Brown’s book during my search for algebraic topology books but I had to pass it over Bredon’s because it didn’t develop the homology / cohomology to the extent I was interested in. Though the groupoid perspective does seem very interesting and useful, so I might read it after completing my current set of textbooks.
No worries! For more recommendations like those two, I’d suggest having a look at “The Fast Track” on Sheafification. Of the books I’ve read from that list, all were fantastic. Note that site emphasises mathematics relevant for physics, and vice versa, so it might not be everyone’s cup of tea. But given your interests, I think you’ll find it useful.
Becoming Stronger™ (Sep 13 - Sep 27)
Notes and reflections on the things I’ve learned while Doing Scholarship™ this week (i.e. studying math)[1].
I am starting to see the value of categorical thinking.
For example from [FOAG], it was quite mindblowing to learn that stalk (the set of germs at a point) can be equivalently defined as a simple colimit of sections of presheaf over open sets of X containing a point, and this definition made proving certain constructions (eg inducing a map of stalks from a map ϕ:X→Y) very easy.
Also, I was first introduced the concept of presheaf as an abstraction of a map that takes open sets and returns functions over it, abstracting properties like there existing a restriction map that composes naturally. Turns out (punchline, presumably) this is just a functor Open(X)Op−>Set!
Yoneda lemma is very cool. I recall seeing some of the ideas from Programs as Singularities (paper), where there are ideas of embedding programs (for the analogue in Yoneda lemma, the Yoneda embedding being a fully faithful functor from some category C … ) into a different space (… to the category SetC …) that contains “almost programs” ( … because the Yoneda embedding is not surjective, let alone essentially surjective), and that studying this enlarged space lends insight into the original space.
Rabbit hole: Yoneda lemma as expressing consistency conditions on Lawvere’s idea of Space and Quantity being presheaf and copresheaf??
I am also starting to more appreciate the notion of sheaf or ringed spaces from [FOAG] - or more generally, the notion that a “space” can be productively studied by studying functions defined on it. For example I learned from [Bredon] that a manifold, whose usual definition is a topological space locally homeomorphic to a Euclidean space, can equivalently be defined as a ringed space whose structure sheaf is valued in some subalgebra of continuous maps over a given open set. Very cool!
Started reading [Procesi] to learn invariant theory and representation theory because it came up quite often as my bottleneck in my recent work (eg). Also interpretability, apparently. So far I just read pg 1-9, reviewing the very basics of group action (e.g., orbit stabilizer theorem). Lie groups aren’t coming up until pg ~50 so until then I should catch up on the relevant Lie group prerequisites through [Lee] or [Bredon].
Also reviewed some basic topology by skimming pg 1-50 of [Bredon]. So many rabbit holes just in point-set topology that I can’t afford (time-wise) to follow, e.g.,
(1) nets generalize sequences and successfully characterize topological properties—I once learned of filters, and I do not yet know how they relate (and don’t relate constructively + why constructively filters are more natural) and especially universal net vs ultrafilter
(2) I didn’t know that manifolds are metrizable, but yes they are by an easy consequence of the Urysohn metrization theorem (second-countable & completely regular ⇒ metrizable). But I would like to study this proof in more detail. Also, how to intuitively think about the metric of a manifold?
I didn’t know that the proof to Urysohn metrization was this nice! It’s a consequence of the following lemma: recall, “completely regular” means given a point x and a closed set x∉C, there exists a continuous f:X→[0,1] s.t.f(x)=0 and f(C)=1. BUT adding second-countability to the hypothesis then lets you choose this f from a fixed, countable family F.
Then, mapping X under this countable family of functions (thus taking value in [0,1]N) turns out to be an embedding—and [0,1]N can be metrized, so X can be metrized as well.
(3) I learned about various natural variants / generalizations of compactness (σ-compactness, local compactness, paracompactness). My understanding of their importance is because:
(a) paracompactness implies the existence of partition of unity subordinate to any open cover (a consequence of paracompactness ⇒ normal, and Urysohn’s lemma, also paracompactness by definition allowing you to find the open refinement of the given open cover as required by the definition of partition of unity subordinate to an open cover.)
(b) for locally compact Hausdorff X, we can characterize paracompactness by “disjoint union of open σ-compact subsets,” which is much easier to check than the definition of paracompactness as locally finite open refinement of open covers.
e.g., from this, it is immediate that manifolds are paracompact: (1) locally Euclidean ⇒ locally compact. (2) Second-countable ⇒ Lindelof. (3) Lindelof & locally compact ⇒ σ-compact. (1) & (2) & (3) + above ⇒ manifolds are paracompact. From which other properties of manifolds immediately follow from that of paracompactness, eg manifolds always admit a partition of unity subordinate to any open cover.
But rabbit hole: recall, open sets axiomatize semidecidable properties. What is, then, the logical interpretation of compactness, σ-compactness, local compactness, paracompactness?
This week, I’ll start tracking the exercises I solve and pages I cover and post them in next week’s shortform, so that I can keep track of my progress + additional accountability.
I am self-studying math. The purpose of this shortform is to publicly write down:
things I’ve learned each week,
my progress through the textbooks I’ve committed to read[2], and
other learning-related reflections,
with the aim of:
allocating some time each week reflecting on what I’ve learned to self-improve,
be kept socially accountable by publicly committing on making progress and actually sticking through the textbooks,
and write / think in public which I enjoy doing so.
I am currently reading the following textbooks:
[Lee]: Lee, Smooth Manifolds
[Bredon]: Bredon, Topology and Geometry
[FOAG]: Vakil, Foundations of Algebraic Geometry
[Procesi]: Procesi, Lie Groups: An Approach through Invariants and Representations
and I plan to do most of the exercises for each of the textbooks unless I find some of them too redundant. For this week’s shortform I haven’t written down my progress this week on each of these books nor the problems I’ve solved because I haven’t started tracking them, so I’ll do them starting next week.
Woit’s “Quantum Theory, Groups and Representations” is fantastic for this IMO. It gives physical motivation for representation theory, connects it to invariants and, of course, works through the physically important lie-groups. The intuitions you build here should generalize. Plus, it’s well written.
Also, if you are ever in the market for differential topology, algebraic topology, and algebraic geometry, then I’d recommend Ronald Brown’s “Topology and Groupoids.” It presents the basic material of topology in a way that generalizes better to the fields above, along with some powerful geometric tools for calculations.
Both author’s provide free pdfs of their books.
Thanks for the recommendation! Woit’s book does look fantastic (also as an introduction to quantum mechanics). I also known Sternberg’s Group Theory and Physics to be a good representation theory & physics book.
I did encounter Brown’s book during my search for algebraic topology books but I had to pass it over Bredon’s because it didn’t develop the homology / cohomology to the extent I was interested in. Though the groupoid perspective does seem very interesting and useful, so I might read it after completing my current set of textbooks.
No worries! For more recommendations like those two, I’d suggest having a look at “The Fast Track” on Sheafification. Of the books I’ve read from that list, all were fantastic. Note that site emphasises mathematics relevant for physics, and vice versa, so it might not be everyone’s cup of tea. But given your interests, I think you’ll find it useful.