One-Woman Math Squad

Things I know so far

Mon, 06 May 2013 11:49:03 -0400

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?

Fri, 03 May 2013 17:44:44 -0400

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

Fri, 03 May 2013 17:11:21 -0400

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

Mon, 22 Apr 2013 23:25:58 -0400

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

Wed, 10 Apr 2013 13:52:54 -0400

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.

One-Woman Math Squad: More Lipschitz continuity

Tue, 09 Apr 2013 01:29:00 -0400

One-Woman Math Squad: More Lipschitz continuity:

thingsendinginphysics:

mathsquad:

physicsphiends:

mathsquad:

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…

Compact Metric Space

-

It turns out that I need two things:

continuous real-valued “separatness points”

and X has to be compact (or maybe you need compactness?).

If you have all that, you get a couple of nice things.

-

I can add a constant.

If your closure is the whole space,

That’s the same as being dense.

And they separate points,

And I was given that X.

-

It took me way too long to learn that,

and I was intimidated by the words.

Distinct points have disjoint neighborhoods

And if your closure is the whole space,

you get a couple of nice things.

-

-Geddes

I have the best best friend of anyone in the world.

You are both wonderful. This seems like something one or both of you might enjoy? http://quomodocumque.wordpress.com/2013/01/29/katos-lecture-notes-are-like-a-modernist-novel-about-commutative-algebra/

Oh that’s great! I have professors who say funny things sometimes, but this is way better.

Also we went to college with the person who took those notes!

One-Woman Math Squad: More Lipschitz continuity

Tue, 09 Apr 2013 00:45:02 -0400

One-Woman Math Squad: More Lipschitz continuity:

physicsphiends:

mathsquad:

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…

Compact Metric Space

-

It turns out that I need two things:

continuous real-valued “separatness points”

and X has to be compact (or maybe you need compactness?).

If you have all that, you get a couple of nice things.

-

I can add a constant.

If your closure is the whole space,

That’s the same as being dense.

And they separate points,

And I was given that X.

-

It took me way too long to learn that,

and I was intimidated by the words.

Distinct points have disjoint neighborhoods

And if your closure is the whole space,

you get a couple of nice things.

-

-Geddes

I have the best best friend of anyone in the world.

More Lipschitz continuity

Mon, 08 Apr 2013 19:38:00 -0400

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

Mon, 08 Apr 2013 01:32:58 -0400

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.

Math joke

Fri, 05 Apr 2013 10:21:36 -0400

I’m going to the graduate student topology conference this weekend. I’m excited! And I have high hopes that this could be the first time I go to a conference and actually understand some of the talks.

But I’ve been studying so much analysis with the prelim coming up…what if I don’t fit in? I’ll be like “how far is it to the banquet?” and everyone will look at me weird and eventually someone will say, “…you can walk there without leaving campus”

(This is a joke because distance is an analytical property of a space that topologists don’t care about. Path-connectedness is a topological property, though. So if the topologists know there’s a path to get to dinner, they don’t care how long or short it is.)

More studying

Thu, 04 Apr 2013 13:50:42 -0400

Yesterday and today I’ve been trying to prove that Lipschitz continuous functions from a compact metric space X to the real numbers are dense (in the uniform sense) in the space of continuous functions on X.

There are three different notions of distance hiding in what I just said—the distance between real numbers, the distance between points in X, and the uniform-norm distance between functions—and I think the point is to get them to work together. Also compactness is about having infinitely many of something but only needing finitely many of them, and I expect to have to use that. Maybe I’ll take the minimum of some distances, because you can always take the minimum of finitely many things.

But that’s not very visual or intuitive. What I’m trying to prove is that, if I have this function that’s continuous (which is a pretty nice property. It means the graph of the function doesn’t jump around), I can approximate it as closely as I want by a Lipschitz continuous function (which is an even nicer property. It means that the slope of the function is never greater than this given bound). So sqrt(x) is continuous but not Lipschitz continuous on the closed unit interval [0,1], because it gets infinitely steep near 0.

So I’m basically trying to prove that I can approximate sqrt(x) as closely as I want by a function whose derivative is bounded. How can that be? it’s so steep. But I guess the interval on which it’s arbitrarily steep is arbitrarily small. Hmm. Maybe I can draw a picture of this.

pegghetti: dream-passi0nately: my current mentality is “im sad and i hate myself but i have to...

Wed, 03 Apr 2013 10:29:48 -0400

pegghetti:

dream-passi0nately:

my current mentality is “im sad and i hate myself but i have to get good grades”

that’s how I feel

is there any other way to feel
to quote an original song from the cinematic masterpiece, School of Rock, “got good grades but don’t got no soul”

I don’t feel this anymore, but I remember it from high school, and I bet a lot of my students are in this place. School is terrible! Everyone drop out!

Grandma Got STEM!

Tue, 02 Apr 2013 23:16:29 -0400

So cool, so cool! Stories of grandmas and grandma-like people who are or were badass scientists, engineers, and mathematicians: Grandma Got STEM.

More scheming

Tue, 02 Apr 2013 13:35:39 -0400

I had a birthday and am now in my late 20’s! Which means a lot of great things, but also that Grace Llewellyn didn’t write a handbook for my liberation. So I have to write my own. First step, finding people out in the world who are moving in the direction I want to go. Have some links!

exco

The Open Master’s Program

Scheming and Studying

Wed, 27 Mar 2013 13:27:30 -0400

I’m studying for prelims again—real analysis this time. It’s fun except when I get discouraged because it seems like too much to learn (my analysis background is limited).

And at the same time, I’m scheming about alternatives to grad school for people (me) who want to learn math at the highest level and do math research, but don’t want to participate in the university system. It seems like the hardest part would be finding other people who wanted that. Especially people who know more math than me to serve as mentors/tutors/advisors. I bet most of us are in universities. But I also bet there are people out there who’d be interested in something different.

Anybody know anybody?

Secrets of Circles at Minnesota Children's Museum

Thu, 14 Mar 2013 16:17:00 -0400

Happy Pi Day folks!

Yesterday I went to the Minnesota Children’s Museum with two of my housemates. They have an exhibit, called Secrets of Circles, that’s visiting from the Children’s Discovery Museum of San Jose.

My favorite part of the exhibit was the Exploratorium-style turntable, but that thing is my favorite part of any exhibit it’s in. It has all the play value of silly and surprising careening rings and balls, gives visitors lots to experiment with, and brings out some deep physical concepts so visitors can see them and tinker with them.

The three-year-old who was with me liked the Vietnamese round boat best. This kid is into playing pretend, especially pretending to be in different environments, like houses, stores, boats, or the park. Climbing in was a doable challenge, and the paddles were kid-sized. We had to pretend to swim away in order to convince her to leave it.

There were some cars with different-shaped wheels in a round case and as they rolled they were supposed to be tracing a kind of graph of the car’s height over time on a cylinder, but the lines were too faint to see and I mostly wished the cars had just been out for kids to play with on the floor. Ditto with the piece that had two carts, one with wheels and one without, each laden with a cinder block.

There was a real wood lathe! I didn’t get to try it because there were kids using it the whole time I was there. It looked fun but maybe like the file/carving tool that was provided was kinda ineffective. Probably they were erring on the side of safety. It would have been cool to have more powerful setup that was staffed.

There were a lot of fabrics printed with circles. That made me wonder about science museum or children’s museum exhibits about textiles. But that’s a topic for another day.

Happy Pi Day! Eat something round!

Lisa Simpson is my hero.

Tue, 12 Mar 2013 14:07:37 -0400

Lisa Simpson is my hero.

Logic puzzle with Geddes

Mon, 11 Mar 2013 23:56:44 -0400

You have 3000 apples that you want to transport over a distance of 1000 miles, but your truck only holds 1000 miles AND whenever it has any apples on it, it loses 1 apple per mile. How many apples can you get to your destination?

We solved this problem over the phone and it was fun! We talked more about how much to interrogate the boundaries of the universe described by the problem (e.g. the whole apple-falling-off mechanism), which reminded me of how our prelim exams have a notice on the top that part of what we’re being tested on is understanding the context in which it makes sense to ask a particular problem. This is a hard thing to learn!

We also talked about the way that math problems are a good thing, but problems in the rest of your life are a bad thing. But maybe they’re not.

And I thought it was great how much she wanted to know the answer, that she felt like she couldn’t stop until she figured it out. I remember feeling this way back when hard math problems were rare and thrilling. Now I probably walk away from 10 unsolved math problems/questions a day, and if I didn’t have the ability to forget about them, I wouldn’t survive.

Set

Mon, 11 Mar 2013 01:46:03 -0400

Set:

My super-cool big brother is starting a website about games, and the first step is a newsletter. This issue talks about Set, which is a great and mathy game. See if you can spot me in the text!

Analysis and algebra, nature and nurture

Sun, 10 Mar 2013 01:30:33 -0500

There are two broad regions of the universe of math that are called analysis and algebra. I don’t know what they are or how to explain the difference between them, but they’re different. Part of the difference is that anaysis deals with things that are continuous, like ramps, and algebra deals with things that are discrete, like stairs.

The discipline seems to think that each mathematician is oriented toward one or the other of these broad areas; each of us is either an analyst or an algebraist. And I don’t love being forced into binaries like that, so I’ve spent an inordinate amount of time trying to convince myself that I don’t have an orientation toward either, or wondering how I came to have my orientation and what I can do to unlearn it and give the other side a fair chance.

But, at least for now, I’ll say it, I am an algebraist. I’ve been exposed to A LOT more algebra than analysis. I can count on one hand the number of differential equations I’ve solved. And a lot of the analysis topics I know I first learned in an algebraic context. There’s something called point-set topology that most people learn in a real analysis class, then maybe use in a topology class, and maybe then, if ever, they apply it to some algebraic or number theoretic setting. But my education has been all jumbled up and backwards, so the first time I ever did point-set topology, it was in a number theory class. I learned about the completeness of the p-adic numbers before the completeness of the real numbers! It was confusing and not the best way to learn things, but I eventually figured it out and I think it made my brain different from the brains of people who learned real analysis first.

I know this isn’t meaningful to people who don’t do math. But the point is: there are two things, and I’ve learned drastically, ridiculously more about one of the things than the other, both by chance and by choice. And this imbalance has persisted for so long that it’s fundamentally shaped the way I see both of the two.

And now I’m learning about analysis almost for the first time, and it’s cool! It’s really cool! I finally learned what the gram-schmidt process is today and I was like, “yeah, that’s a great fucking process and is just what I hope I would come up with if asked to turn any old basis into an orthonormal basis.” And distributions are like functions but WAY better. And sigma algebras, which are part of the formal mathy way of looking at probability theory, finally make sense to me and they are SO THE RIGHT WAY of looking at probability theory. It’s delightful.

But these moments of satisfaction aren’t as forceful as the ones I talked about in my last post, when I learn a new way of looking at something algebraic. I just don’t care as much about these analysis-type questions. I don’t know if that’s because I haven’t been thinking about them for as long, or something more intrinsic, and I guess it doesn’t matter.