Monthly Archives: April 2014

The Horror! The Horror! How dare we discriminate against men, by listening to women?

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For those of us who’ve learned to actually be aware of sexism and racism, it’s incredibly frustrating how the same stupid pathetic arguments about sexism keep getting regurgitated, over and over again, by clueless guys. It’s exhausting and frustrating to constantly answer the same stupid, bullshit arguments. But if it’s frustrating to a white guy like me, just imagine what it’s like for the people who are actually affected by it!

This rant is brought on by the fact that lately, there’s been a movement in the tech/engineering community to try to actually do something about the amount of sexism in the community, by trying to push conference organizers to include speakers outside the usual group of guys.

You see, it’s a sad fact that engineering, as a field, is incredibly sexist. We don’t like to admit it. Tons of people constantly deny it, and make excuses for it, and refuse to try to do anything about it. Many people in this field really, genuinely believe that technology is a true meritocracy: those of us who succeed because we deserve to succeed. We see ourselves as self-made: we’ve earned what we’ve got. Anyone else – anyone else – who worked as hard as we do, who’s as good as we are, would succeed as much as we have.

Unfortunately, that’s not reality. Women and minorities of all sorts have a much harder time in this community than the typical guy. But when you try to do anything about this, the meritocrats throw a tantrum. They get actively angry and self-righteous when anyone actually tries to do anything about it. You can’t actively try to hire women: that’s discriminating against the guys! You’re deliberately trying to hire inferior people! It’s not fair!

I am not exaggerating this. Here’s an example.

Summing it up, turning away qualified male candidates and accepting potentially less qualified female candidates just to meet a quota is not only sexist, but is a horrible face to put on the problem. My wife is a nurse. A female dominated field. I also own a construction company. A male dominated field. In neither field would you see a company or an organization suggest hiring one gender over the other to satisfy a ratio.

What exactly is being accomplished by limiting the speakers at an event, or your employee base to an equal male / female ratio? Turning away qualified candidates and hiring purely based on gender, or religion, or race? What does having a program like Women Who Code, PyLadies, Girls Who Code, Black Girls Code, etc… accomplish? The short answer; very little if anything. In fact, I believe groups like this have the potential actually do more harm than good. But I’m not here to debate the minutia of any of those groups so I’ll summarize;…

Seriously, I don’t think I’ve ever heard anyone seriously suggest that you should
turn away candidates based purely on gender, or race, or whatever.

What we do advocate is recognizing that there are people other than white guys in the world, and maybe we should actually include them once in a while.

The majority of people running tech conferences are white guys. When they think about inviting people to come speak, they’re naturally going to start off with a list of “Who do I know who’d be a good speaker?”. The nature of the tech community is that the vast majority of people that they immediately think of are going to be men – because the majority of people they’ve worked with, and the majority of people they’e seen speak at other conferences were men.

People love to handwave this, talking about how it’s just that the field is so male dominated. It’s true that it is, but there are many problems with that as an excuse:

  1. The male domination of the field isn’t meritocratic. We discriminate against people who aren’t like us at every level – from elementary school teachers discouraging little girls from being interested in math, to college classmates harassing women in their classes, to people conducting job interviews and making hiring decisions.
  2. The actual fraction of women who work in tech is much larger than the fraction of women in leadership, or who are invited to give talks at conferences.
  3. There are a shocking number of women who are driven out of technology and engineering by harassment by their male coworkers.

People get really angry when we say thing like this. Part of it is that old meritocracy thing: you’re saying that I didn’t earn my success.

Guess what? You didn’t. Not entirely. No one succeeds solely on the basis of their own merit. There’s always a huge amount of luck involved – being in the right place, having the right body, having the right parents, having the right opportunities. Yes, hard work, talent, and skill helped you get to where you are. It’s a very large, very important factor in your success. But if you were born to a different family, with exactly the same abilities, you might not have ever had any chance at succeeding, despite equally hard work.

The other side of this is actually a sign of progress. We’ve come to accept that racism and sexism are bad. That’s a really good thing. But in our black-and-white way of seeing the world, we know that sexism is bad: therefore, if I’m sexist, I’m a bad person.

That’s not true. Sexism is a deeply ingrained attribute in our culture. It’s pretty much impossible to grow up in the US or in Europe, or in China, or in India, or in Africa, without being constantly immersed in sexist attitudes. You’re going to be exposed – and not just exposed, but educated in a way that teaches you to have the attitudes that come from your culture.

Recognizing that you’ve grown up in a sexist culture, and acknowledging that this had an effect on you and your attitudes doesn’t make you a bad person. It makes you human. Refusing to admit that your culture had any influence on you, and continuing to discriminate against people because you can’t admit that you might do something wrong? That’s what makes you a bad person.

I’ve told this story before, but it’s a damned good story, and it’s true, and it’s not hearsay: it’s my personal first-hand experience. It’s part of my own awakening to just how pervasive gender bias is.

A long time ago now, I worked for IBM Research. My third year working there, I volunteered to be the summer student coordinator for my department. The previous year, IBM Research had hired around 100 summer students, and exactly one of them was a woman. The vast majority were white guys, with a sizable minority of Chinese and Indian guys. The pool of candidates was nowhere near that skewed. It’s definitely true that men outnumber women in computer science by a sizable factor, but not 99 to 1. Our candidate pool was more like 5 men to 1 woman.

So, that year, the powers that be at the company decided that we needed to do something about it. What they decided to do was allocate a reasonable number of summer student slots to each department based on the departments budget, and they could use those slots to hire anyone they wanted. If they hired a candidate who was a woman or minority, they didn’t count against the budget. (Note that they did not reduce the number of students we were allowed to hire: they allocated based on the usual budget. They set up an additional budget for the extra students.)

My department was one of the smaller ones. We were allocated 5 slots for summer students. The day we started allowing people to request students, all 5 were gone within a couple of hours. The next day, the guy across the hall from me came to my office with a resume for a student he wanted to hire. Of course, it was a guy.

I told him that we couldn’t hire him – our budget was gone. But if he could find a woman, we didn’t need budget to hire her.

He threw a fit. It was the angriest I ever saw him. (Most of the time, he was a really nice, mellow guy, so he was really upset about his!) It was discrimination! Sexism! Unfair! He carefully went through the resume database looking for the best candidate, not the best male candidate. We were refusing to hire the most qualified candidate! On, and on, and on. I finally got rid of him, after pointing out at least a dozen times that I was just a lowly junior engineer, not someone who made the policy.

The next day, he was back in my office. He was practically bouncing off the walls: he’d gone back to the resume database, and he’d found a woman who was even better than the guy he’d wanted to hire.

This is the point of the whole story. He wasn’t some nasty, spiteful, misogynistic twit. He wasn’t being deliberately discriminatory. He wasn’t consciously screening out women’s resumes. But the fact is, when he went through the resume database without being forced to consider women, he’d eliminated the resumes of every single woman. He was going through a database of 1000s of resumes, and in that process of quickly skimming, he skipped over a more qualified candidate, because she had a woman’s name.

This is what happens in the real world. We don’t deliberately try to be sexists. We don’t act in a deliberately sexist or discriminatory way. But we’re part of a culture that has deeply ingrained sexist attitudes. We’re taught, by the way teachers treat boys and girls differently in school. We’re taught, by the way that society treats us differently. We absorb the message that when it comes to things like engineering, women are inferior. Most of the time, we don’t even really notice that we’ve absorbed that. But we have. It’s been hammered into us so many times, in so many ways, in so many settings – it would be shocking if we didn’t pick it up.

I’m using my experience at IBM as an example, partly because it’s such a vivid demonstration, and partly because it’s impossible to figure out the real names of anyone involved. But I’ve seen the same kind of thing happen in every job I’ve had where I’ve been involved with hiring. It’s not usually deliberate, but it’s very real.

The point of things like the pledges to not attend conferences that don’t have women and minorities as speakers and participants isn’t because we want to exclude the most qualified speakers. It isn’t because we want to force conference planners to include less qualified speakers. It’s because we know that it’s easy, without trying, to exclude some of the most qualified speakers, because the people running the conference don’t notice them.

They’re just like my friend at IBM: they’re not deliberately trying to exclude women. But if they don’t actively try to think about people outside the usual pool of guys like them, they won’t include any. And if they don’t, then they’re priming the next round of conference planners to do the same: if everyone you’ve seen give a great talk at a conference is a guy, then when you’re planning a conference and you try to think of some great speakers to invite, then who’s going to come to mind?

I’m particularly annoyed at the snipe that the author of the quote up above takes at “Girls Who Code”. GWC is a great organization. If you actually take the time to listen to the people who run it, you’ll hear some appalling true stories, about things like young women who go to college to study computer science, and on their first day in class, have classmates telling them that they’re in the wrong classroom: this is a programming class, not a class for chicks.

We have a community where we treat women like that. And then we rant and rave about how horribly unfair it is to do anything about it.

Roasted Vegetable Recipes

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One more post of recipes, and then I’ll get back to math, I promise!

One of my proudest accomplishments that somehow, I successfully taught my children to not just eat, but to love vegetables. I think part of that is genetic – neither of them are supertasters. But the other part of it is a combination of training and cooking.

The training side is, I think, simple. Most adults are convinced that vegetables are icky but necessary. They’re wrong. But they actively teach that to their children. They make eating vegetables, even when they’re delicious, into a chore.

The other side is that because most adults think that veggies are icky, they cook them in ways that don’t taste good.

Take one of my favorite vegetables as an example: brussel sprouts. My children will actually fight over who gets the last bite of brussel sprouts. When we’re talking about what to make for dinner, they beg me to make them! But when I mention this to most people, they act like I’m absolutely insane: brussel sprouts are gross!

If you take a bunch of brussel sprouts, and throw it into boiling water for 20 minutes, what you get is a stinky, pungent, bitter, mushy, revolting little ball of green awfulness. But, if you slice them thin, and saute them in a smoking hot pan with olive oil, salt and pepper, and a bit of garlic, until the edges start to turn brown, what you get is absolutely amazing: sweet, crisp, and wonderful.

So, what I’m going to share here is a couple of vegetable side dishes I made in the last week, which were fantastic. All of them are roasted vegetables – for some reason, people don’t think about roasting veggies, but it’s often one of the easiest and tastiest ways to cook them.

First: simple roasted brussel sprouts.

  1. Preheat your oven to 450 degrees.
  2. Take a pound of brussel sprouts, and cut them into quarters.
  3. Toss them with enough olive out to coat them, but not drench them.
  4. Sprinkle generously with salt and pepper.
  5. Spread them out over a baking sheet.
  6. Put them into the hot oven, for around 10 minutes. After ten minutes, take them and and turn them. If they look really well done and brown on the edges, then they’re done; if not, put them in for up to another 10 minutes.
  7. Take them out, and toss them with a teaspoon or two of balsamic vinegar.

That’s it – and they’re amazing.

Next: Chili Glazed Roasted Sweet Potatoes

This one I’m particularly proud of. I absolutely love sweet potatoes. But normally, my wife won’t touch them – she thinks they’re gross. But this recipe, she actually voluntarily had multiple helpings! It’s sweet, salty, and spicy all at the same time, in a wonderful balance.

  1. Take a couple of pounds of sweet potatoes, peel them, and cut them into cubes about 2 inches on a side.
  2. Toss them with olive oil to coat.
  3. Mix together 1 teaspoon of salt, 1/4 cup of brown sugar, and one generous teaspoon of kochukaru (korean chili powder).
  4. Sprinkle it over the oiled sweet potatoes, and toss them so they’re all coated.
  5. Spread onto a baking sheet, and cook at 350 for about 30 minutes, turning them at least once. They’re done when the outside is nicely browned, and they’ve gotten soft.

Finally: roasted cauliflower

  1. Preheat your oven to 450.
  2. Take a whole head of cauliflower, and break it into small florets. Put them into a bowl.
  3. Take a half of an onion, and slice it thin. Toss the onions with the cauliflower.
  4. Coat the cauliflower and onions with olive oil – don’t drench them, but make sure that they’ve got a nice coat.
  5. Sprinkle generously with salt and pepper.
  6. Spread onto a baking sheet, and into the oven.
  7. After 10 minutes, take them out, and turn them, then back in for another 10 minutes.

All three of these got eaten not just by adults, but by kids. The brussel sprouts and sweet potatoes were eaten not just by my kids, but by the kids of other people too, so it’s not just the crazy Chu-Carroll’s who thought they were delicious!

A Recipe for Gefilte Fish

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My mom died last friday night, a little bit after midnight. I’ll probably write something about her, when I’m feeling up to it, but not yet. Being a jewish dad, when I’m depressed, what do I do? I cook.

Last year, for the first time ever, I made homemade gefilte fish for Pesach. If you didn’t grow up Jewish, odds are you’re not familiar with gefilte fish. It’s a traditional ashkenazi (that is, eastern european jewish) dish. It was originally a ground fish mixture cooked inside the body cavity of a fish. It evolved into just the stuffing mixture, simmered in a fish stock. These days, most people just buy it in a jar. If you grew up with it, even out of the jar, it’s a treat; if you didn’t, and you’ve been exposed to it, it looks and smells like dog food.

Honestly, I love the stuff. In general, I’m not a big fan of most traditional Jewish foods. But there’s something about gefilte fish. But even as I enjoy it, I can see the gross side. It’s crazy overprocessed – I mean, come on – it’s fish that will keep, unrefrigerated, for years!

But made fresh, it’s a whole different kettle of fish. This stuff is really good. You’ll definitely recognize the flavor of this as gefilte fish, but it’s a much cleaner flavor. It tastes like fish, not like stale overprocessed fish guts.

So this year, I’m depressed over my mom; after the funeral, I sent my wife out to buy me a bunch of fish, and I made up a batch. This time, I kept notes on how I did it – and it turned out even better than last year.

It’s got a bit of a twist in the recipe. I’m married to a chinese woman, so when the Jewish holidays roll around, I always try to find some way of putting an asian spin on the food, to reflect the nature of our family. So when I cooked the gefilte fish, instead of cooking it in the traditional simple fish broth, I cooked it in dashi. It’s not chinese, but it’s got a lot of flavors that are homey for a chinese person.

So… here’s the recipe for Mark’s homemade salmon dashi gefilte fish!

Ingredients

  • 2 whole pike, gutted and cleaned, but with skin, head, and bones
  • 2 whole red snapper, gutted and cleaned, but with skin, head and bones
  • 2 pounds salmon filet
  • 3/4 to 1 cup matzoh meal
  • 3 eggs
  • salt (to taste)
  • 2 sheets of konbu (japanese dried kelp)
  • 2 handfulls dried shaved bonito
  • 4 or 5 slices of fresh ginger, crushed
  • 2 onions
  • 2 large carrots

(For the fish for this, you really want the bones, the skins, and the head. If you’ve got a fish market that will fillet it for you, and then give you all of the parts, have them do that. Otherwise, do it yourself. Don’t worry about how well you can fillet it – it’s going to get ground up, so if you do a messy job, it’s not a problem.)

Instructions

  1. First thing, you need to make the stock that you’ll eventually cook the gefilte fish in:
    1. If the fish store didn’t fillet the fish for you, you need to remove the filets from the fish, and then remove the skin from the filets.
    2. Put all of the bones, skin, and head into a stock pot.
    3. Cover the fish bones with with water.
    4. Add one onion, and all of the garlic and ginger to the pot.
    5. Heat to a boil, and then simmer for two hours.
    6. Strain out all of the bones, and put the stock back into the pot and bring to a boil.
    7. Add the kombu to the stock, and let it simmer for 30 minutes.
    8. Remove from the heat, and strain out the kombu.
    9. Add the bonito (off the heat), and let it sit for 15 minutes.
    10. Strain out the bonito and any remaining solids.
    11. Add salt to taste.
  2. While the stock is simmering, you can get started on the fish:
    1. Cut all of the fish into chunks, and put them through a meat grinder with a coarse blade (or grind them coarsely in batches in best food processor you can get your hands on.
      r.)
    2. Cut the onion and carrots into chunks, and put them through the grinder as well.
    3. Beat the eggs. Fold the eggs and the salt into the ground fish mixture.
    4. Add in maztoh meal gradually, until the mixture holds together.
    5. Refrigerate for two hours.
  3. Now you’re ready to cook the gefilte fish!
    1. Heat the stock up to a gentle simmer.
    2. Scoop up the fish into balls containing about two tablespoons of the fish mixture, and roll them into balls.
    3. Add the fish balls into the simmering stock. Don’t overcrowd the pot – add no more than can fit into the pot in a single layer.
    4. Simmer for 10-15 minutes until the fish balls are cooked through.
    5. Remove the balls from the simmering liquid. Repeat until all of the fish is cooked.
    6. Put all the cooked fish balls back into the stock, and refrigerate.

The Heartbleed Bug

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There’s a lot of panic going around on the internet today, about something called the Heartbleed bug. I’ve gotten questions, so I’m giving answers.

I’ve heard lots of hype. Is this really a big deal?

Damn right it is!

It’s pretty hard to wrap your head around just how bad this actually is. It’s probably even more of a big deal that the hype has made it out to be!

This bug affects around 90% of all sites on the internet that use secure connections. Seriously: if you’re using the internet, you’re affected by this. It doesn’t matter how reputable or secure the sites you connect to have been in the past: the majority of them are probably vulnerable to this, and some number of them have, in all likelihood, been compromised! Pretty much any website running on linux or netbst, using Apache or NGINX as its webserver is vulnerable. That means just about every major site on the net.

The problem is a bug in a commonly used version of SSL/TLS. So, before I explain what the bug is, I’ll run through a quick background.

What is SSL/TLS?

When you’re using the internet in the simplest mode, you’re using a simple set of communication protocols called TCP/IP. In basic TCP/IP protocols, when you connect to another computer on the network, the data that gets sent back and forth is not encrypted or obscured – it’s just sent in the clear. That means that it’s easy for anyone who’s on the same network cable as you to look at your connection, and see the data.

For lots of things, that’s fine. For example, if you want to read this blog, there’s nothing confidential about it. Everyone who reads the blog sees the same content. No one is going to see anything private.

But for a lot of other things, that’s not true. You probably don’t want someone to be able to see your email. You definitely don’t want anyone else to be able to see the credit card number you use to order things from Amazon!

To protect communications, there’s another protocol called SSL, the Secure Sockets Layer. When you connect to another site that’s got a URL starting with https:, the two computers establish an encrypted connection. Once an SSL connection is established between two computers, all communication between them is encrypted.

Actually, on most modern systems, you’re not really using SSL. You’re using a successor to the original SSL protocol called TLS, which stands for transport layer security. Pretty much everyone is now using TLS, but many people still just say SSL, and in fact the most commonly used implementation of it is in package called OpenSSL.

So SSL/TLS is the basic protocol that we use on the internet for secure communications. If you use SSL/TLS correctly, then the information that you send and receive can only be accessed by you and the computer that you’re talking to.

Note the qualifier: if you use SSL correctly!

SSL is built on public key cryptography. What that means is that a website identifies itself using a pair of keys. There’s one key, called a public key, that it gives away to everyone; and there’s a second key, called a private key, that it keeps secret. Anything that you encrypt with the public key can only be decrypted using the private key; anything encrypted with the private key can only be decrypted using the public key. That means that if you get a message that can be decrypted using the sites public key, you know that no one except the site could have encrypted it! And if you use the public key to encrypt something, you know that no one except that site will be able to decrypt it.

Public key cryptography is an absolutely brilliant idea. But it relies on the fact that the private key is absolutely private! If anyone else can get a copy of the private key, then all bets are off: you can no longer rely on anything about that key. You couldn’t be sure that messages came from the right source; and you couldn’t be sure that your messages could only be read by an authorized person.

So what’s the bug?

The SSL protocol includes something called a heartbeat. It’s a periodic exchange between the two sides of a connection, to let them know that the other side is still alive and listening on the connection.

One of the options in the heartbeat is an echo request, which is illustrated below. Computer A wants to know if B is listening. So A sends a message to B saying “Here’s X bytes of data. Send them back to me.” Then A waits. If it gets a message back from B containing the same X bytes of data, it knows B was listening. That’s all there is to it: the heartbeat is just a simple way to check that the other side is actually listening to what you say.

heartbeat

The bug is really, really simple. The attacker sends a heartbeat message saying “I’m gonna send you a chunk of data containing 64000 bytes”, but then the data only contains one byte.

If the code worked correctly, it would say “Oops, invalid request: you said you were going to send me 64000 bytes of data, but you only sent me one!” But what the buggy version of SSL does is send you that 1 byte, plus 63,999 bytes of whatever happens to be in memory next to wherever it saved the byte..

heartbleed

You can’t choose what data you’re going to get in response. It’ll just be a bunch of whatever happened to be in memory. But you can do it repeatedly, and get lots of different memory chunks. If you know how the SSL implementation works, you can scan those memory chunks, and look for particular data structures – like private keys, or cookies. Given a lot of time, and the ability to connect multiple times and send multiple heartbeat requests each time you connect, you can gather a lot of data. Most of it will be crap, but some of it will be the valuable stuff.

To make matters worse, the heartbeat is treated as something very low-level which happens very frequently, and which doesn’t transfer meaningful data. So the implementation doesn’t log heartbeats at all. So there’s no way of even identifying which connections to a server have been exploiting this. So a site that’s running one of the buggy versions of OpenSSL has no way of knowing whether or not they’ve been the target of this attack!

See what I mean about it being a big deal?

Why is it so widespread?

When I’ve written about security in the past, one of the things that I’ve said repeatedly is: if you’re thinking about writing your own implementation of a security protocol, STOP. Don’t do it! There are a thousand ways that you can make a tiny, trivial mistake which completely compromises the security of your code. It’s not a matter of whether you’re smart or not; it’s just a simple statement of fact. If you try to do it, you will screw something up. There are just so many subtle ways to get things wrong: it takes a whole team of experts to even have a chance to get it right.

Most engineers who build stuff for the internet understand that, and don’t write their own cryptosystems or cryptographic protocols. What they do is use a common, well-known, public system. They know the system was implemented by experts; they know that there are a lot of responsible, smart people who are doing their best to find and fix any problems that crop up.

Imagine that you’re an engineer picking an implementation of SSL. You know that you want as many people trying to find problems as possible. So which one will you choose? The one that most people are using! Because that’s the one that has the most people working to make sure it doesn’t have any problems, or to fix any problems that get found as quickly as possible.

The most widely used version of SSL is an open-source software package called OpenSSL. And that’s exactly where the most bug is: in OpenSSL.

How can it be fixed?

Normally, something like this would be bad. But you’d be able to just update the implementation to a new version without the bug, and that would be the end of the problem. But this case is pretty much the worst possible case: fixing the implementation doesn’t entirely fix the problem. Because even after you’ve fixed the SSL implementation, if someone got hold of your private key, they still have it. And there’s no way to know if anyone stole your key!

To fix this, then, you need to do much more than just update the SSL implementation. You need to cancel all of your keys, and replace them new ones, and you need to get everyone who has a copy of your public key to throw it away and stop using it.

So basically, at the moment, every keypair for nearly every major website in the world that was valid yesterday can no longer be trusted.

Manifold are the Manifolds

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In the stuff I’ve been writing about topology so far, I’ve been talking about topologies mostly algebraically. I’m not sure why, but for me, algebraic topology is both the most interesting and easy to understand bit of topology. Most people seem to think I’m crazy; in general, people seem to think that algebraic topology is hard. It must say something about me that I find other things much harder, but I’ll leave it to someone else to figure out what.

Despite the fact that I love the algebraic side, there’s a lot of interesting stuff in topology that you need to talk about, which isn’t purely algebraic. Today I’m going to talk about one of the most important ones: manifolds.

The definition we use for topological spaces is really abstract. A topological space is a set of points with a structural relation. You can define that relation either in terms of neighborhoods, or in terms of open sets – the two end up being equivalent. (You can define the open sets in terms of neighborhoods, or the neighborhoods in terms of open sets; they define the same structure and imply each other.)

That abstract definition is wonderful. It lets you talk about lots of different structures using the language and mechanics of topology. But it is very abstract. When we think about a topological space, we’re usually thinking of something much more specific than what’s implied by that definition. The word space has an intuitive meaning for us. We hear it, and we think of shapes and surfaces. Those are properties of many, but not all topological spaces. For example, there are topological spaces where you can have multiple distinct points with identical neighborhoods. That’s definitely not part of what we expect!

The things that we think of as a spaces are really are really a special class of topological spaces, called manifolds.

Informally, a manifold is a topological space where its neighborhoods for a surface that appears to be euclidean if you look at small sections. All euclidean surfaces are manifolds – a two dimensional plane, defined as a topological space, is a manifold. But there are also manifolds that aren’t really euclidean, like a torus, or the surface of a sphere – and they’re the things that make manifolds interesting.

The formal definition is very much like the informal, with a few additions. But before we get there, we need to brush up on some definitions.

  • A set {\mathbf S} is countable if and only if there is a total, onto, one-to-one function from {\mathbf S} to the natural numbers.
  • Given a topological space (T, \tau), a basis \beta for \tau is a collection of open sets from which any open set in \tau can be generated by a finite sequence of unions and intersections of sets in \beta. What this really means is that the structure of the space is regular – it’s got nothing strange like an infinitary-union in its open-sets.
  • A topological space (T, \tau) is called a Hausdorff space if and only if for any two distinct points p, q \in T, there are at least one open set o_p where p \in o_p \land q \not\in o_p, and at least one open set o_q where q \in o_q \land p \not\in o_q. (That is, there is at least one open set that includes p but not q, and one that includes q but not p.) A Hausdorff space just defines another kind of regularity: disjoint points have disjoint neighborhoods.

A topological space (T, \tau) is an n-manifold if/f:

  • \tau has a countable basis.
  • (T, \tau) is a Hausdorff space.
  • Every point in T has a neighborhood homeomorphic to an open euclidean n-ball.

Basically, what this really means is pretty much what I said in the informal definition. In a euclidean n-space, every point has a neighborhood which is shaped like an n-ball, and can be separated from any other point using an n-ball shaped neighborhood of the appropriate size. In a manifold, the neighborhoods around a point look like the euclidean neighborhoods.

If you think of a large enough torus, you can easily imagine that the smaller open 2-balls (disks) around a particular point will look very much like flat disks. In fact, as the torus gets larger, they’ll become virtually indistinguishable from flat euclidean disks. But as you move away from the individual point, and look at the properties of the entire surface, you see that the euclidean properties fail.

Another interesting way of thinking about manifolds is in terms of a construction called charts, and charts will end up being important later.

A chart for an manifold is an invertable map from some euclidean manifold to part of the manifold which preserves the topological structure. If a manifold isn’t euclidean, then there isn’t a single chart for the entire manifold. But we can find a set of overlapping charts so that every point in the manifold is part of at least one chart, and the edges of all of the charts overlap. A set of overlapping charts like that is called an atlas for the manifold, and we will sometimes say that the atlas defines the manifold. For any given manifold, there are many different atlases that can define it. The union of all possible atlases for a manifold, which is the set of all charts that can be mapped onto parts of the manifold is called the maximal atlas for the manifold. The maximal atlas for a manifold is, obviously, unique.

For some manifolds, we can define an atlas consisting of charts with coordinate systems. If we can do that, then we have something wonderful: a topology on which we can do angles, distances, and most importantly, calculus.

Topologists draw a lot of distinctions between different kinds of manifolds; a few interesting examples are:

  • A Reimann manifold is a manifold on which you can meaningfully define angles and distance. (The mechanics of that are complicated and interesting, and I’ll talk about them in a future post.)
  • A differentiable manifold is one on which you can do calculus. (It’s basically a manifold where the atlas has measures, and the measures are compatible in the overlaps.) I probably won’t say much more about them, because the interesting thing about them is analysis, and I stink at analysis.
  • A Lie group is a differentiable manifold with a valid closed product operator between points in the manifold, which is compatible with the smooth structure of the manifold. It’s basically what happens when a differentiable manifold and a group fall in love and have a baby.

We’ll see more about manifolds in future posts!