One-Woman Math Squad

Summer

It’s summer! I’m not teaching or taking classes. I get to do whatever I want!

So far I’ve been doing math, making summer clothes, working in the garden, seeing friends, and doing outside/physical things. Plus a category that I refer to as “internet chores” which—I’m not exactly sure what that consists of, but it takes a lot of time. And I don’t even mean wasting time online I mean dealing with important stuff. I don’t have a computer at home anymore so now I have to plan for this and it amazes me how much of it there is to do.

I’m also planning to facilitate a math self-study community support group thing that I’m totally excited about. We’re calling it the UnMathClass. Publicizing it is one of my internet chores.

In conclusion, I love summer. Also I went swimming in a lake yesterday! It was cold enough that the bony parts of my body just ACHED the whole time, but warm enough that the fat-covered parts of my body got used to it instead of getting colder and colder. So I guess it’s not exactly be summer yet, but we’re getting there.

Things I know so far

So what’s the point of higher category theory, and weak n-categories and (weak?) infinity categories? What do I know so far?

Well category theory is a concise way of describing a bunch of seemingly different mathematical ideas. A category has:

-objects (these could be sets, or numbers, or dots on a paper)

-morphisms, (these could be functions between sets, or arrows between dots on a paper)

-composition of morphisms (do one function, then do another. this gives you a third function)

-identity morphisms (when you compose this with another morphism, you just get that other morphism itself, like multiplying by 1 or adding 0)

-strict associativity of composition of morphisms (like (a+b)+c = a + (b+c))

And this framework describes a lot of the stuff I would want to do math on: sets and groups and fields and topological spaces. It means I can talk about all these things at once and I LOVE talking about a lot of things at once. Yay, category theory is great.

And it gets even better: you can define functors, which are like functions between categories. And then you can make a category whose objects are categories and whose morphisms are functors! This is bliss.

But then, you realize you can define these things called natural transformations, which are—wait for it—like functions between functors. So you can define a category that has functors as its objects and natural transformations as its morphisms! So now you have this thing called a 2-category, where you sort of have two categories stacked on top of each other with the morphisms of one category forming the objects of the next category. And you can stack these as high as you want. 3-categories, n-categories. This, truly, is bliss.

BUT there was a problem all along that I was hiding from you: some things that I would want to do math on do not form a category in any nice, sensible way. I need to study this part more to get clear on exactly why, but I think it has something do do with considering things as equivalent even though they don’t have an isomorphism between them. It definitely has something to do with associativity breaking down.

So sometimes instead of (a+b) + c and a + (b+c) being exactly equal, you get that those two things are relatedin some nice way without being the exact same object. You can describe this relationship by a morphism, or a morphism between morphisms, or a morphism between morphisms between morphisms.

But somehow then your associativity laws for those higher-dimensional relationships are also never strict, so you keep needing to jump up a dimension, and in fact you need to know that you’re able to jump up to every dimension and never stop.

Aaaand you end up with something called an infinity category. And it’s pretty cool. And I need to do more reading to pin down these last 4 paragraphs.

Tumblr will you be my accountability buddy?

So funny thing about math grad school—at some point a switch flips and after a LIFETIME of being told what to do, academically, you’re supposed to become an independent researcher who finds her own problems and figures out what’s important and how much depth to go into things and so forth. 

This is, you can probably guess, really really really hard.

Hilarious!

So as an attempt at self-accountability, I’m committing to produce SOMETHING to convey to SOMEONE an very minimal overview of higher category theory—why it’s interesting, what the different ways of constructing its definitions are and what advantages/disadvantages they have, and what it can do for a topologist.

And I’m going to publish that something, or a record of it, right here, by Wednesday at 5 pm!

Maybe I’ll give a 10-minute math talk to any grad students I can round up, or maybe try to do it over skype for friends and family? Grad students sounds easier. 

Accidental math poetry for Geddes

From one of my professors, talking about homotopy theory and higher category theory:

“At each stage, you’re never asking for an identity. You’re always asking for a witness for some fact.”

Math Crush

I have the world’s biggest math crush on Eugenia Cheng of The Catsters.

https://twitter.com/DrEugeniaCheng

http://cheng.staff.shef.ac.uk/

Prelim Prelim Prelim

I can calculate fourier series, but I can’t yet use them to compute infinite sums unless I’m using the fourier series of x^2 to compute the sum of 1/(n^2).

I’m pretty sure I can show the supremum of the fourier transform of f is the L-1 norm of f, and that it’s acheived only at x=0. Some classmate was trying to tell me a really fancy way of doing this yesterday, but today I’ve convinced myself I just need to show something is positive for certain values of a variable and negative for others. 

I think I know all permutations of compactness, local compactness, and completeness implying or not implying each other for metric spaces.

I can tell you all about the Cantor-Lebesgue function and how it’s not absolutely continuous aka it’s WACKY.

I need some more practice on my monotone convergence theorem and dominated convergence theorem. And on everything about integrals. Like Fubini/Tonelli. I did a pretty good job one one Fubini/Tonelli problem the other day, but now I can’t remember which one it was.

I know Hoelder’s Inequality and the Stone-Weirstrass theorem.

I know that the fat cantor set is closed and nowhere dense but has positive measure.

More Lipschitz continuity

Geddes told me that she reads my mathiest posts as poetry, so here is an update on Lipschitz continuity, for your literary pleasure:

It turns out that I need two things to do this proof: the Stone-Weierstrass theorem, and compactness of X. The Stone-Weierstrass theorem is about when you have a collection of functions that form a sub-algebra of C(X, R), the continuous real-valued functions on X, that “separates points”. Which means that for any distinct points, you can find a function that maps those points to distinct values. Oh and X has to be compact and Hausdorff.

If you have all that, you get a couple of nice things, one of which I’ll use for my prelim problem.

The Lipschitz continuous functions form a sub-algebra when the domain is bounded (or maybe you need compactness?). And they separate points since f(x) = x is Lipschitz and separates points. And I was given that X was a compact metric space, and all metric spaces are Hausdorff. So I have all that. And then this theorem tells me that the closure of my algebra is X if the algebra (or its closure) contains the constant functions, which it does in this case. Or alternately it tells me that the closure is X unless there’s a point x_0 in X on which every function in my algebra takes the value 0. Which there can’t be because I can add a constant to any Lipschitz function and get another Lipschitz function that takes a different value on each point.

SO the closure of the Lipschitz functions is X, the whole space.

And if your closure is the whole space, that’s the same as being dense. So I’m done!

Vocab time for math majors: an algebra is  a vector space where the vectors form a group under multiplication! It took me way too long to learn that and I was intimidated by the words “an algebra” for a long time.

Ok technically I should say “module” instead of “vector space” but don’t worry about it. A module is just like a vector space but over a ring instead of a field. Thaaaaaat’s another vocab word that used to scare me.

Hausdorff means “distinct points have disjoint neighborhoods” and is a topological property. If you want to know what’s not Hausdorff, google “bug-eyed line.”

Drinking from the fire hose

I went to that conference and it was great, because I like being overwhelmed with way more math than I could possibly understand. I like the discipline of trying to pay attention and seek for understanding even when I’m not finding it. It feels like really important practice for life: just being alive in the world is often Too Much of a Good Thing for me and I try to shut out some of my experience by eating cookie dough and watching TV. (Also life is sometimes just too much of a bad thing.) But I prefer the times when I’m able to be open to more bandwidth of my life.

So the conference was great, and hard. People kept wanting to tell me about math and I kept wanting to tell them to stop, or to just stop paying attention, but I mostly rolled with it and it was great. I got help studying for my analysis prelim and inspiration to keep reading about higher categories.

Also I learned that one of my classmates has run a marathon and one was homecoming king of his high school! And there was some mild sexist bullshit, and the thing where none of the boys say anything to challenge sexist bullshit, even though they probably wouldn’t have said it themselves. I didn’t engage with that particular comment except to sing “this is ridiculous” over and over until the subject got dropped. But later I had some hopefully productive but still uncomfortable conversations with classmates about identity and privilege. I guess I was challenging myself to be open to my classmates as much as to be open to math. I like them and I’m glad we were forced to hang out in a car for sixteen hours this weekend.

Oh and one more fire hose: somehow it’s still the work week and always has been and always will be and I don’t understand why I don’t have a day off.