Don't let your schemes be dreams

[math-memes]

[wassupblucoat]

Ok let’s have a go and see if I can remember my algebraic topology.

Grapes = 1.

Cookie = 2.

Bread = integers.

Hamburger = Cyclic group of order 2.

Hotdog = projective space of dimension n over the real numbers.

Then the (singular?) cohomology groups of P^n over Z/2Z has a ring structure (given by cup product).

Pizza(A, B) = Hom(A, B).  This is a contravarient functor in the first variable.

Banana = Ext, the derived functor of Hom.

The exact sequence after that is the universal coefficient theorem for cohomology.  In my copy of Hatcher this is Theorem 3.2.

Then the problem is to describe the cohomology ring of projective space, with coefficients in Z/2Z.  Theorem 3.19 in Hatcher answers this question.  It’s (Z/2Z)[x]/(x^{n+1}).

So that degree in pure math is useful for something after all.

[the-real-numbers-deactivated202]

Do you have any idea where I could find an introduction to exterior products with concrete examples I can visualize? And/or maybe a way to map (some useful subset of) their results to an array of real numbers? I love the idea of the generalized Stokes theorem and of being able to define torque and magnetism without cross products, but apparently I need to learn about exterior products to understand, and the Wikipedia articles on this just make me feel lost.

I’m gonna be real with you I googled “exterior product” and this is one obtuse operation, I can’t seem to find a basic example or simple definition on Wikipedia or mathworld or anywhere else

[the-real-numbers-deactivated202]

I mean there’s a Khan academy video

[wassupblucoat]

OK below is way more than I intended to write.  TL;DR: exterior products are “natural” if you think in terms of abstract algebra, so maybe it would be best to do a bit of abstract algebra first and then come back to it.

The way that makes most sense to me (warning: I’m an algebra person) is the universal property way.  In my opinion this is way has the most natural motivation, and no formulas are “dropped from the sky.”  Because of the abstraction, this is not the easiest/quickest approach if you haven’t seen abstract algebra, but if you’re just satisfying a curiosity it may be worthwhile to learn some algebra anyway.  I won’t go over the construction here, just say how the parts of it can be motivated.

The exterior product is an example of a multilinear, alternating map (definitions below).  It is in fact the “most generic” multilinear, alternating map, in a precise sense described below.  So to understand exterior products, we should first understand these maps.

Lets use torque as an example.  Before we give a definition for what torque is, we list some properties we think it ought to have.

We have some vector space V = R^3 representing space.

We have some vector space W = ??? whose elements are all possible torques.

Whatever a torque is, we should be able to get one from a pair of vectors r and F in V, representing force and displacement.  That is, we should have a function from V x V to W.

This function should be bilinear: if we scale either the force or displacement, the resulting torque should scale by the same amount.  If we hold displacement constant and add two forces (as vectors in V), the resulting torque should be the sum (as vectors in W) of the individual torques.  Same for if we hold force constant and add displacements.

This function should be alternating: if the force and displacement are in the same direction, the resulting torque should be zero.

So we know torque should be an alternating bilinear map out of V x V.

The exterior product comes in here: Any alternating bilinear map f : V x V -> W factors uniquely as a linear map g : ∧²V -> W composed with the exterior product V x V -> ∧² V.  That is, the wedge product turns alternating, bilinear maps into just plain old linear maps, and vice versa.  By abstract nonsense, this property completely determines the space ∧²V up to unique isomorphism. 

So if we don’t have any more assumptions to make, we just decide that a torque should be an element of ∧²V.  One can show that in this case ∧²V is three-dimensional (this is where the work is), and so this agrees with the classical definition.

This is done in many “standard” books on abstract algebra (I know at least Dummit & Foote and Lang do it this way; Aluffi has a more modern book I haven’t read which has a lot of exposition on universal properties as well.)  Unfortunately because of the scope of these books this is always covered after ~500 pages of other material, but the only hard prereqs needed from earlier chapters are definitions and the general “algebra vibe.”

[p-locally]

Suppose you have an action of a group G on a group H and an action of H on another group K, in other words we have two homomorphisms  ρ : G→Aut(H)   μ : H→Aut(K)

is there really no way to make G act on K given all this information?? This is driving me crazy, I can’t seem to piece together such a morphism

[model-theory]

probably 1/|H| sum(g.h.k) summing over h in H

if H finite, that is

[model-theory]

@wassupblucoat is right sum(g.h)=sum(h), so that doesn’t work. You can of course produce arbitrary actions such as g.k=g.(e_H).k, but this is part of a larger structure: the action of GxH on K via (g,h).k=g.h.k

Then my above suggestion would be the restriction of the representation of GxH to Gx{e}

[wassupblucoat]

I think that’s just the identity, since g(e_H) = e_H.

My gut says there is no non-trivial natural (functorial?) way to do it, but I haven’t thought about it enough to formulate that precisely.

I can think of at least one case where this setup happens “in nature.”  Let F be a field.  Let K = (F, +), H = (F\{0}, *), and G = Aut(F), i.e. field automorphisms of F.  H acts on K by multiplication, and G acts on both K and H by definition.

It seems this is a very special case in that (1) we can identify H with a subset of G (how do you identify S_3 with Z/2 x Z/2 ?), (2) we basically defined G to be compatible automorphisms of K and H (the map a -> 1/a is an automorphism of H, but not K), and (3) K and H are abelian.

Perhaps what is true is that you can always take quotients/subgroups where this happens, degenerating to something trivial when there are no “compatible automorphisms”?

[wassupblucoat]

authors of C macros 🤝 my calculus students

putting parentheses everywhere, just in case

(Do NOT tag lisp.)

[the-real-numbers-deactivated202]

Anarcho-Algebraist

[wassupblucoat]

Those actually exist?

https://en.wikipedia.org/wiki/Alexander_Grothendieck#Political_activism

[twocubes]

there keeps being (and i sometimes enjoy contributing to) discourse/jokes about systems of numbers, but, realtalk? anytime some things form a rig, those are numbers

[twocubes]

anytime you can interpret a polynomial with some things as arguments those are numbers

integers? numbers. 

integers modulo something? numbers.

square matrices? numbers. 

polynomials? numbers. 

sets with intersection as multiplication and symmetric difference as addition? numbers.

[wassupblucoat]

So basically calculus professors have been lying when they say “dx” is not a number.  This is dual number erasure.

[model-theory]

everybody gangsta until they pull out \mathbb{G}

[the-real-numbers-deactivated202]

Baby mapping

[model-theory]

How dare you be funnier than me

[the-real-numbers-deactivated202]

Infant-morphism

[wassupblucoat]

Fun fact: Serre deliberately named his famous paper Géométrie algébrique et géométrie analytique (Algebraic geometry is analytic geometry) so that everyone would call it GAGA.  He chose this name because after introducing the main trick of the paper, the rest of the results were “like child’s play.”

[just-shower-thoughts]

Learning more advanced math in school basically unlocks more buttons of the calculator.

[wassupblucoat]

Wow Lockhart was right people do think like this.

(If you do think math is like this, go read “A Mathematician’s Lament,” now.)

[wassupblucoat]

Discussions on the mathblr discord

[rebrobindoesmath]

tip: use \, to add a small space between characters in LaTeX

[wassupblucoat]

Better yet, redefine `\forall` and `\exists` to be operators, using `\DeclareMathOperator` from the amsmath package.  Then you don’t have to change your formulas at all, semantically it makes sense, and LaTeX takes care of the spacing for you:

```

\let\oldforall\forall % Remember the old version of \forall

\let\forall\relax % Free up the name \forall

\DeclareMathOperator{\forall}{\oldforall} % Make \forall an operator

```

[wassupblucoat]

One taught me love.

One taught me patience.

One taught me pain.

[the-real-numbers-deactivated202]

Limits are one of those things that are “easy” for an instructor to explain intuitively but just absolutely fuck with inexperienced students.

[the-real-numbers-deactivated202]

There are a lot of factors at play. I think all of them can be used as lessons on how to teach a better math class for more general audiences.

[the-real-numbers-deactivated202]

As a calculus tutor for the past 2 years and a TA this year, I’ve seen students consistently struggle with them.

Part of it is conceptual, they don’t understand the difference between “behavior leading up to” and “behavior at”.

Part of it is they don’t know why they’re doing it and they can’t put it in context.

Part of it is just general discomfort with algebra and mathematical terminology.

Part of it is the fact that despite it being the backbone of calculus, it gets a week or two at most. Maybe you’re thinking it’s way too much already. I don’t know how to make that practical. I just know it goes fast.

Part of it is the theory/practice divide when calculus is done poorly. I can explain to you the concept of a limit all day and how it’s defined, but the definition doesn’t really tell you how to find it like a derivative does. And you sometimes don’t directly tie finding limits to the idea of limits very well. You just send the students off to do it and they have no clear way of translating definitions to answers.

Part of it is that half the students struggle to do algebra anyway so it’s pretty much guaranteed that they don’t get it fast enough. 🤷‍♂️

It’s true, some percentage of students also just don’t care enough. But it’s early on in the semester. You can still reach them.

I think if you considered all of these factors when designing your class, your class would be the best class ever. Not that you have to consider them to be a good class at all, but they do help.

[wassupblucoat]

This is my first semester TA’ing calculus, and this post sums up my experience as well.  Another huge problem arises if your class decides to have the students do epsilon-delta proofs.  The majority of freshmen in an intro calc class have never proved anything before (aside from HS geometry maybe), and a subtle definition with three quantifiers and a conditional is not an easy place to start.  That’s not to mention the million “tricks” that you have to use when working backwards (e.g. “well, as long as delta < 1 we can assume delta^2 < delta, so we’ll make a note of that and pick delta = min(1, ...)”).

Because of this, epsilon-delta ends up being probably the hardest part of the course, even though it comes so early in the semester.  After it’s over, we’re going back to “normal” plug-n-chug calculations with derivatives and integrals, and it’s never seen again.  This leaves students confused and the point of working with the formal definition gets completely lost.  Maybe it would make more sense if the formal definition was used later, like to understand the proofs of Rolle’s theorem or the intermediate value theorem, or to highlight subtleties about uniform convergence.  But at any rate that’s way too much to cover given the typical background students have.  (But apparently epsilon-delta isn’t!  Who’s drawing this line?  It’s like we want to get the point across that being precise about limits is hard, without the payoff that comes with doing it correctly.)

[math-memes]

[wassupblucoat]

[everythingstarstuff]

Crumb - Locket [Official Video]

[wassupblucoat]

So apparently the thing that you’re supposed to do before you’re allowed into the algebraic geometry club is to do every exercise in Atiyah and MacDonald.  I didn’t want to do my actual homework today so I just worked on that.  At least the book is short so I might actually finish it.

[wassupblucoat]

so uh I never really got into this tumblr thing but maybe there are some dank math memes on here.

[wassupblucoat]

when you’re so tired and out of memes but you have to stay up to reset your sleep schedule