Math(s)! Currently: Summer of Math Exposition

Explaining to a friend who was trained as an engineer that very few fields actually look like the real numbers and very few rings look like the integers.

 

As a mathematician, this is the worst thing I've seen in quite a while, especially since I spend a lot of time these days arguing with a computer software developer about abstractions of various degrees of leakiness.

 

I should probably stop saying that various things are just versions of the Fourier transformation unless I actually have a reason to back it up. Not all dualities are Fourier transforms.

 

Summer of Math Exposition 3 results!

 

Is only talking about commutative groups abelist?

 

For commuting variables, you can have polynomials of any degree. When all of your variables anticommute, including with themselves, you can't get anything of higher degree than the number of variables you have, so the space of polynomials for anticommuting variables is finite.
This leads to some extra objects that you don't hear about as often in the commuting-variable case, because in the commuting variable case the corresponding objects are infinite-dimensional and therefore badly behaved.

 

Something fun: The median second-smallest prime factor of an integer is 37, in that if you take a random integer, about half the time its second smallest prime factor will be < 37, and about half the time its second smallest prime factor will be > 37, and the rest of the time its second smallest prime factor will be equal to 37.

 

....Let me guess, is this related to the weird shenanigans seen in quantum field theory? It feels like a similar flavor to that... What objects do you mean, exactly?

 

Yeah, fermion wavefunctions are examples of anticommuting variables. This is why we get things like electron orbitals only being able to hold a finite number of electrons.

The thing that prompted this post is that I have been reading about superalgebras and Lie superalgebras. A supervector space has two types of basis vectors, degree 0 vectors and degree 1 vectors. A supercommutative algebra is a supervector space equipped with a multiplication operation. Degree 0 elements commute with everything, degree 1 elements anticommute with each other.
A Lie superalgebra similarly has two types of generators. Here the Lie bracket is antisymmetric if one of the inputs has degree 0, and is symmetric if both inputs are degree 1. The Jacobi identity also gets modified.
We can look at the simple Lie superalgebras. There are a bunch that are generalizations of the simple Lie algebras. For instance, an orthosymplectic form on a supervector space is a bilinear form that is symmetric on degree 0 vectors and antisymmetric on degree 1 vectors, and the corresponding orthosymplectic Lie superalgebras are the natural extensions of the orthogonal Lie algebras.
But there are some weird ones. For instance, consider the set of polynomials in n variables. As mentioned, if the variables all commute, then the space of polynomials in n variables is infinite-dimensional, and so is the set of derivations, i.e. the linear, first-order differential operators with polynomial coefficients. The set of derivations form a Lie algebra, and I think it's simple, but it's infinite-dimensional so we don't talk about it much.
In contrast, if all of the variables anticommute, the superalgebra of polynomials is finite-dimensional, and so is the Lie superalgebra of derivations. So we have a family of finite-dimensional simple Lie superalgebras that doesn't look like the ABCDEFG classification. This family is called W(n), or W(0, n).
If some of your variables commute, you get W(m, n), which is again infinite-dimensional. Lie algebraists don't talk about W(m, n) or W(m, 0) much because a lot of our favorite theorems require finite dimensions, for instance if you need to use some form of the pigeonhole principle or descending chain arguments and rely on eventually running out of basis vectors. But W(n) as mentioned is finite-dimensional.
There is also a "special" sub-Lie superalgebra of W(n), called S(n), which is very distinct from the special linear Lie algebra, and a sub-Lie superalgebra, called H(n), that preserves a symmetric bilinear form, and is quite different from the orthogonal Lie algebras. Again, these exist for commuting variables, but are infinite-dimensional.

 

Note: the super in supersymmetry also involves superalgebras and Lie superalgebras, but the relevant one is not simple. The degree 0 part is the Poincare Lie algebra so(n-1,1)+Rⁿ, where Rⁿ acts as translations, and the degree 1 part is a spinor representation with a symmetric Rⁿ-valued bilinear form. It's the extra degree 1 stuff that makes the superpartners of particles have a different spin than regular particles.

 

Also apparently superalgebras can be queer.

 

Some proofs you can sort of figure out the ending once you know the main ideas.
And some of them end in a tedious computations that you just have to grind out.
And then there are those that end like "and then we take dualize three times in a row and the answer pops out" but the author thinks that that's too obvious to say explicitly.

 

Speaking of proofs, here's Irme Lakatos' 1976 paper Proofs and Refutations, which argues that mathematics, as a human activity, proceeds not from axioms and definitions to conclusions and proofs, but in a back-and-forth manner between counterexample, proof-adjustment, and definition-adjustment. I like his discussion of local counterexamples and global counterexamples. As a formalist, however, I think he conflates a little the human endeavor of studying mathematical objects, and the mathematical objects themselves. Perhaps this is due to more recent logic-as-mathematics developments via type-theory and the like.

Also, the main body of the paper is presented akin to a Socratic dialogue, only instead of two or three characters it's a teacher completely losing control of a dozen-student math class.