Sometimes I see a math cartoon and wish I had thought of it first. This is one of them:

Of course I doubt I could have captured the moment so well.

The best part of it is that someone emailed it to me with the comment, “Hey Courtney, This cartoon reminded me of you. -Joe”

Sweet. It’s a fact: you don’t really know me until you’ve seen me do my Summation by Parts dance.

Dear Courtney:

Here are your result(s) from the June 2008 qualifying exam:
825-6 Q
871-2 Q.

The Graduate Advisory Committee (GAC) notes that you now have 2 Q scores, one of which is in 825-6. Therefore you have passed the Qualifing Exam and are now admitted to the Ph.D. program.
Congratulations!

Well folks, it’s true! After countless hours of studying and weeks of agonized waiting, I know for sure: I Q’d my exams! EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE!

This means that I only have to pick a research area, form a committee, pass my comprehensive exams, spend 2-4 years working on my dissertation, and then defend my research to a panel of professors…and then I’ll be done and have my PhD!

In all seriousness, the quals are some of the most challenging exams I’ll have had to take in grad school–and now I’m done! HOORAY!

Recently, my little sister Brittney graduated from college. No big deal–she only graduated summa cum laude from NYU, having been inducted into Phi Beta Kappa (a liberal arts honor society) and Gamma Sigma Epsilon (a chemistry honor society), with an embarrassingly high GPA, and really cute shoes.

I’m very proud of my sister. I never would have imagined that she would grow up to be a cosmopolitan smart-ass who can walk two miles in stiletto boots without a grimace. Nor would I have guessed that we both would end up loving math despite our parents (although Brittney does love chemistry, too, despite her older sister).

I’d like to think that I had a little part in shaping my sister into the wise-cracking woman she is today. It’s more likely that she taught me lots of lessons in sharing, humility, and grace.

And now, just so she doesn’t get too much of a swelled head, I present a clip from an old home movie. I’ve decided to call it Sisters - a duet.

If nothing else, I definitely taught her how to hog the camera.

Congratulations on your innumerable achievements, kiddo!

Some of you who read my rambling blog might occasionally wonder what it is that I actually do as a math grad student. I’ve decided that it’s high time I tried to give a reasonably coherent explanation, so here goes.

Most of math, or at least the math I’m interested in, spends a lot of time redefining what it means for two things (numbers, groups, rings, spaces, and other technical math constructions) to be the same. That’s right–I spend all day deciding what “=” means, depending on the context.

Question 1: What does 2*2-2 = 4+6-8 mean to you?

For example, suppose I’m interested in numbers that have a remainder of 1 when you divide them by 3. If that’s the only property I really care about, then I might say something silly like [1] = [4], where those brackets remind me that I’m working with a special equivalence class, i.e., all the numbers that have remainder 1 when divided by 3.

Question 2: Can you find all the equivalence classes with respect to division by 3? (Hint: take a look at the numbers on a touch-tone phone.)

Other times, I’m looking at a space. Is a donut sitting flat on a table the same as a donut balanced on its edge? To me they are. Is a donut’s surface the same as a coffee cup’s surface? Not usually, but sometimes all I care about mathematically is the fact that there’s a big hole in the space I’m looking at. Geometrically, of course, they’re very different–a coffee cup has a large, well, cup in it, and a donut certainly doesn’t. So from the perspective of one branch of math (algebraic topology), “donut = coffee cup,” while in other branches (geometry, calculus), “donut ≠ coffee cup.”

Question 3: Can you think of two objects that might be the same with respect to certain conditions, but different with respect to others?

So that’s part 1 of what I do all day. It’s a small part, but it’s the first question a mathematician asks when thinking about some type of mathematical object: “How do I define what X = Y really means?”

Part 2 of what I do will discuss different ways to put together things that mathematicians know to get new information–that’s right, we’re talking about proofs! As a primer, I suggest you go play some sudoku, because I’m probably going to make a lot of analogies to a sudoku game.

–Answers–

Answer 1: That’s up to you!

Answer 2: The equivalence classes are [0] (everything that 3 divides evenly, i.e., with no remainder), [1], and [2] (everything that 3 divides with a remainder of 2). Formally, we can define these as
[0] = {0,3,6,…} = {3n : n is an integer}
[1] = {1,4,7,…} = {3n+1 : n is an integer}
[2] = {2,5,8,…} = {3n+2 : n is an integer}.

Answer 3: The first thing that came to my mind was “Apples and Oranges!” They’re the same in that they’re both fruit, of course, but they’re also the same in that they are “homeomorphic” to a solid ball–that means I could smush them around to look like a 3D ball. They’re different in the sense that an apple has “saddle points” while an orange (ideally) is completely spherical. And apples taste better.

Well, I just handed in my last qualifying exam, hopefully the last one I’ll hand in ever! I feel pretty good about my performance. There were things that could have gone better, but hey. In a month or so, I hope to tell you all that I’m now a qualified math-nerd.

I would type more, but my hands are killing me from all the frantic writing. And I really need a nap!

I read an article this morning that I found a bit disheartening:

The Freedom to say No, Boston Globe:

The Boston Globe
The freedom to say ‘no’
Why aren’t there more women in science and engineering? Controversial new research suggests: They just aren’t interested.

Now two new studies by economists and social scientists have reached a perhaps startling conclusion: An important part of the explanation for the gender gap, they are finding, are the preferences of women themselves. When it comes to certain math- and science-related jobs, substantial numbers of women - highly qualified for the work - stay out of those careers because they would simply rather do something else.

I wonder how much the atmosphere of academia has to do with the choices women are making. I love math, but I do think that there’s a certain baseline testosterone that makes it hard for me to feel completely at home in a math department. Moreover, there’s a battery of exams and other hoops to jump through that seem antiquated, at least from my limited perspective.

So ladies, if you’re reading–why do you do what you do? Did you ever like math? Do you still? I’m really curious. I have this gut feeling that the article fails to touch upon the fundamental issue.

I know of several quiet ways to spend a Sunday afternoon–reading, birdwatching, napping with Moo.

Being next to the window and watching a tree fall on the house is not one of them.

Nobody was hurt, and the house is 100 or so years old, which means it is built better than a modern tank. Now it’s just a matter of the landlord and his chain saw taking it down.

But still–eek! I don’t know that I’ll ever forget the sound of the trunk snapping or the feeling of the house shaking when it made impact.

Next year I start teaching, which leads me to ask: what should I do with this blog?

My gut feeling is that I’ll close it down, give my parents a password to access the archives, and start writing more “professionally” (read: less personal stuff). This includes picking through older entries, polishing, and republishing them.

What would you do in my shoes?

I made it through year one of graduate school without any major incidents! I still have quals in less than a month, but for now, it’s time to leave math aside for a few days while Tim and I head East to see my little (6′ tall) sister graduate. It will be a whirlwind visit (we arrive tomorrow afternoon and leave Monday morning, with every minute scheduled), but at least it won’t involve any mathematical analysis!

A new qualifying exam policy passed at UNL in time for my class: we only have to take two exams (one of which must be analysis), instead of three. For my second exam, I decided to take–drum roll–topology! I know we were all expecting me to take the algebra qual, but I think I might end up becoming a topologist. I like homology, a lot. I like covering spaces. I like drawing homotopy pictures (though I do dislike parametrizing things). In short, I like algebraic topology a lot. I still like algebra, but this year was sort of a “meh” year for algebra in comparison to topology. I’m not the biggest fan of Galois theory (although I did like the Galois correspondence with respect to covering spaces and fundamental groups!). I don’t get giddy at the thought of Sylow subgroup problems. Modules are lovely, though, and I do like the linear algebra we did.

Next year I’ve decided to throw caution to the wind and NOT take the measure theory class that seems to be standard fare for the second year. Instead, I’m taking the second year algebra sequence, plus a course in commutative algebra, and a course in differential topology. It’s going to be such a fun year! Mostly, I decided to take these courses so that when push comes to shove in the Spring of next year, I’ll have an idea what I want to specialize in. Commutative algebra or low dimensional topology. Or something else entirely–who knows?

Add teaching to the mix and I think I’ll have a full plate next year. But in a good way. I think that with a year break from analysis, I’m going to be a lot happier!

Looking into the future, I think I’ll probably take the measure theory class my third year, and possibly the complex analysis class…sometime. I’ll definitely take homological algebra, because homology is just that exciting! I love the snake lemma! I love chains and boundary maps and long exact sequences induced by short exact sequences! I don’t know what other topology classes are offered at UNL, but I imagine I’ll take some more topology, too.

Also on the horizon: TAing for the IMMERSE algebra section this summer, along with a reading course in knot theory (i.e., in practice, ha, ha) because I’ve never actually seen any before. I also have some non-UNL math to do: revising some papers so that they can finally be published (cross your fingers!).

And of course, I have to spend time with my husband and my cat, who were beginning to think they would never see me again. I did laundry today. I got dinner all ready to go as soon as Tim gets home. I did dishes, I brushed Moo until she ran away and hid, and I even contemplated washing the kitchen floor. I did wifely things.

Did I blog this yet? Tim got me a Yamaha full-sized keyboard for my birthday. I was shocked and overwhelmed and awestruck. Tim is the best! I’ve been playing almost every day, and it’s been really helping me–mentally and emotionally. It’s also nice bicycling weather, and riding my bike makes me very happy. So all in all, though this year was really hard, though I may have yet to find the balance between “good wife” and “good student,” I think that I will consider this a success.

Besides, it’s not a good program if you don’t think about dropping out at least every other week. Right?

I think this video really captures how I feel about my tea:

Elemental - Cup Of Brown Joy from Moog on Vimeo.