So it begins.

With a cup of coffee (with soy milk! I'm astounded.) on the food court terrace in Budapest Liszt Ferenc Airport. It always makes me giggle to see stores like Burger King and KFC so early in the morning -their bright colors and plastic signs seem so out of place at this time of day. The airport actually seems fairly empty - there's a fair scattering of people about, but they're quiet, subdued. I guess it still is rather early.

As predicted, the prospect of a day of travel made me all thoughtful. Somehow, everything got packed in my apartment. All the dishes got washed, the floor vacuumed, blankets folded and sheets cleaned. I seem to be leaving with almost exactly as much stuff as when I came - though a few books have traded places, new ones coming home with me and one or two staying behind.

I opted to take a taxi to the airport. The airport is about a half hour's drive from the city center where I live. Lived. By public transport (as I have done before), for me it's two stops on the bus that goes right by my front door, then about five stops on the tram on the main road to the metro station. Then, take the Metro (line 3) all the way to its end, then get on the 200E bus and take that to the end of the line at the airport. Takes about an hour, all told. Plus, with a backpack and a satchel in addition to my worn and weary pirate suitcase (to those of you who don't know, my big red suitcase has been with me on every big journey since I went to Thailand - almost seven years ago - and on its way back from one of the several trips to Germany, it lost one of its roll-y wheels. It was quite a cruel suitcase injury, it ripped it right out of the socket. It was like my suitcase had lost a tooth. And at that time, I only had about thirty-six hours at home in between trips and my mother, brilliant as ever, fashioned my suitcase a peg-leg so that it can still stand up on its own. So, it's a pirate. Thought you might need the backstory.), I decided a taxi was the way to go. In Budapest, even I am affluent enough for a taxi ride every now and then.

It was starting to snow. We zigzagged our way out of the city and to the highway as I thought about how little time I spent in cars in Hungary, or this semester at all. It's a thought that usually comes after some time in Europe - I spend plenty of time in busses, trams, trolleys and trains, but not cars. They seem like foreign little transport pods to me when I first use them again. (Speaking of foreign, my Hungarian got about as passable as one could hope for living in an extremely international city and not living with Hungarian people, nor needing to speak it on a daily basis, but it was good enough to read the little electronic sign on the cab driver's dash that said "Rottenbiller 27 to Airport, 7:30. Emily (foreigner)" That made me smile.) As we were drifting down the highway, I saw the 200E, the bus that I would normally have taken to the airport at the end of that public transport shuffle. I met the eyes of some of the people on the bus from my seat in the taxi, and decided, just for a second, to tilt my nose up just a bit and pretend to think I was better for being able to take a taxi. I saw a few eyes narrow. I wondered how many of them thought I was a pampered jerk, taking a taxi like that. And I wondered how many times that I've thought that of people in cabs was it actually just a game for the people on the other side of the glass.

I have a fair idea of what it will be like when I finally get home. It'll be 9:15 in the evening in Pennsylvania, where it's much snowier than here. And traditionally, it seems, someone in my family gets off the plane, there are hugs and smiles and questions about the flight, about the drive to the airport. Then, inevitably, the traveler will yawn and then one of the folks receiving the traveler will bunch up his/her eyebrows and calculate at which time in the destination timezone the traveler must have gotten up to make this journey. I'm always extremely giggly and talkative when I first get into the car to go home - and by the end of the drive (either 45 minutes or an hour and a half later, depend on which airport I've flown in to), I can barely keep my eyes open. Tradition is tradition.

As I was in line for this coffee here on the terrace, I heard somebody say my name - and I turned around to see one of my comrades from Algebraic Topology, one of the people who I got to know enough to call a friend this semester. His flight to Warsaw (getting to the states from here makes for an interesting journey) was just boarding and he dashed over to say a quick goodbye - it made me really glad to see him. I've said quite a few goodbyes, as you can imagine, over the last few days but he and I hadn't run into each other. The nice thing about the math community is, though, that once you get this far (and most of the people I've met here will go farther - grad school and beyond), the math community isn't all that large. I probably will see a lot of them again.

And, most importantly of all, I managed to get my snacks of choice through security today. I wasn't sure they'd let my carrot, cucumber, banana, and apple through - you never known with produce. I've seen a poor, confused German gentleman get very flustered in the Philadelphia airport for having an apple in his backpack (when an enormous dog and equally enormous security guard manhandle your backpack, you don't really expect it to be because of an apple, rather something more sinister) when he arrived in the states, but I think if you eat it before going through immigration, you're fine. So, I'll do that. I'm also interested in seeing the other passenger's faces when I do so. I miss fruits and vegetables while traveling - I always do, so I thought I ought to bring them this time. Still, I can't pretend that eating a cucumber as long as my forearm is a normal thing to do on a plane. Oh, well. I hope it makes some people smile.

And all of a sudden.

Leaving tomorrow. I'm sitting with a belly full of mulled wine and good, Hungarian salami, thinking about my flight tomorrow morning. This semester flew by.  I'm sure I'll write something tomorrow - traveling gives me some nice head-space to write. I'll check in then.

Just a tinge of pride.

I'm sitting in yet another lovely café in Budapest. The plethora of cafés with varying atmospheres is something that I will miss about living in a big city, especially a city with such a focus on café culture as Budapest. I am one damn proof away from finishing all my work for Spectral Theory, the course that has taken more of my time than any other in the history of my education. We met with Miklos one last time in this café this morning. I'm really happy to say that I got up the courage to ask him for a letter of recommendation for future endeavors- only one that is concrete in my mind right now, but you never know.  A strong letter of recommendation is invaluable, and I have a feeling he'll write a good one.

And I've learned so much in his course, in this semester here. It's almost as different from my math education at Mills as Germany was, though it's also different from Germany. Different areas of a spectrum. And though now I know what Hilbert Spaces are, can talk about tangent bundles, can think about the spectral radius of infinite graphs, the one thing I am most proud of: as I was typing my final writeup this morning (another amazing thing I've had to become proficient in this semester - using LaTex to type math) and going to check on the things I had typed yesterday, I found that one of my files hadn't saved. About an hour and a half's work of formatting enormous square roots (or 2n-th roots, rather) of vectors and matrix multiplication, limit suprema and infema, and all kinds of subscripts - GONE. And I looked through my files carefully to make sure that I hadn't just overlooked it, and after five minutes of fruitless searching, instead of pouting and griping and swearing, I just retyped it. This time, I remembered the formatting errors I had yesterday and it only took me about forty-five minutes. And it's all done now. I'm most proud of that reaction.

Alright. And now, we will continue what my close friends is calling the 'Spectral Theory Pub Crawl', though café crawl would be more accurate (yet most cafés in Budapest serve a host of alcohol as well, so really, it's pretty synonymous) - where we schlep ourselves and our laptops and notes to several cafés during the day, trying not to over-caffeinate and yet to stay engaged in the corrections, the typing, the formatting, the math - until we're all done!

Looking out of the window in Mainz.

Waiting for the bus in Budapest.

That tiny Christmas Market.

Hand-made crafts of all kinds for sale.

A fire outside to warm ourselves up.

And a dog with the perfect instincts for what to do in the cold winter.

Classic.

Thank you, xkcd.

I'm sitting right now in a very comfy chair in Mainz, trying to get my brain to focus on my second-to-last Spectral Theory assignment. Last Thursday, I finished everything (including the lectures) for Algebraic Topology. The research project is coming along, as is our paper summing up what we've done this semester (I still can't get over how exciting it is to write a math paper, and how giddy I get when I generate the nice PDF of our work - it looks like fancy math! Really, it's just typed math, which is not all that fancy, but still!)

And I've been absent here for a while. Yesterday, C and I went to a tiny Christmas market in a tiny town a very short train ride away from Mainz. We went with her neighbor and another friend to this village so small and cozy, it reminded me of Meadville, my home town, just a little bit. We got to see this amazing country house/farm that C's neighbor's parents own, to warm our cold toes next to a wood stove, have amazing soup with barley and pumpkin, and curl up next to the sweetest, oldest, most non-strangle looking Dachshund I've ever seen (I don't really like small dogs that much, but this one was positively lovely. At least thirteen years old, partly blind, and all sweetness- and with long, gray hair! That's right, not the strange, worm-like smooth-haired Dachshunds we're used to seeing yammering on corners.), pictured below:

Maxi couldn't really be bothered to get up from next to her favorite spot by the wall for a picture.

So, I went to her instead.

C's neighbor Miri has been known to take great pictures, and I hopefully will be getting some of the market from her. In the meantime, Erin, who came to stay with me last weekend in Budapest and enjoy some ex-pat Thanksgiving, wrote a lovely blog entry about her time here and her time in Edinburgh, where she is studying. Please do give it a read! Here it is.

Happy Budapestian Thanksgiving

Thanksgiving Week

There's several Thanksgiving/Hanukkah celebrations planned this week among the math students. I went to one yesterday with some folks - I knew the people who were hosting, but not many of the other people who came. There were about fourteen of us - not many in the group were close friends. It was several sets of roommates, a few classmates, some significant others of those classmates or roommates, etc - and something really neat happened. One of the hostesses said apologetically as we sat down with food (also, it must be said, cooking for a potluck Thanksgiving is a lot less work than doing it all by yourself! Turns out we can cook pretty well, too.), "Guys, do you mind if we briefly go around this circle and everyone says something that they're thankful for?"

The cheesiest, most traditional family-style Thanksgiving thing to do. We all giggled a bit awkwardly, because what's more stressful in a social situation than being honest about your own gratitude? Who wants to be that authentic with people they barely know? Plus, you also want your gratitude to sound cool, for people to go "oh, wow! What a good thing to say..." when you say your piece. So, this is buzzing in all of our heads, and then the hostess offered to go first. And we went around the circle - these crazy students, all of us slightly weird 'cause we're math and computer science people, some having Thanksgiving away from home for the first time - and everyone took this seriously. Everyone was sweet, honest, even made eye contact with other people in the circle while talking. A hush fell over the room as we went, and it was such a lovely moment. I really hadn't been expecting it. I guess we really are all people with real feelings, no matter how often we feel that we, ourselves, are the only ones really alive. It was enough of a moment to drive thoughts of Number Theory, Combinatorics, and grad school applications out of the minds of the people present, at least for those few minutes all together.

I couldn't make it to swim practice because I spent too long at acapella rehearsal.

I never thought I would say that sentence. But it's true! Yesterday, I got together with some other students in this program who were interested in forming an acapella group - well, if only for the last four weeks of the semester. I haven't sung in a group since chorus in elementary school, not even middle or high school - and back then, the sopranos were all the popular girls who thought I was weird, and my sister (who was SO COOL) was an alto, so I, of course, decided I was an alto. I have figured out since then that I am actually a soprano, which was kind of a fun discovery.

And I've also grown up watching my sister become a more and more elegant and talented actress and singer, from the second I saw her perform - well, not for the first time, but for the first time when I felt I was conscious as a person, if that makes sense - back in our hometown. She was the evil orphanage owner Ms. Hannigan in "Annie", and tore up the stage all by herself during the terrifying and hilarious "Little Girls" song, and I remember being so proud of my sister, so proud to be my sister's sister, during the applause at the end of her solo song that I felt like I was going to float over the audience. Since then, of course, she went to major in Theater at college and has since been in countless plays, and has directed some (including an entire musical in our hometown, and she did the set design, EVERYTHING). Okay, where is this going? Basically, this is going to the fact that I've always felt like I had the least talent when it came to music in my family, but given that my family is so crazy good at music, this was a skewed opinion of myself. It is, however, also true that of my family, I am the one who has spent the least time actually working at music of any kind. I did practice diligently and extremely resistantly when I had saxophone lessons in elementary school, and then perhaps a bit more willingly when I played in the jazz band in high school, and I could here that I sounded better if I practiced more! Still, that's nothing to the hours my sister has spent in rehearsals all her life, or the ones my parents have attended - both for voice and various instruments.

So, after avoiding the scary sopranos in chorus in elementary school and deciding that Rachel was the singer and the best I could hope for was people NOT wincing if I sang next to them in church or at birthday parties, then I figured out that I was a soprano. Then I started playing guitar. And then I got quite happy and content and even a bit proud of how I could sing when I was alone with my guitar. But I still don't really like to sing in front of people. I've never physically shaken so much in my life as when I am ever asked to sing in front of people. Talking? Oh, that I love. I'll tell jokes, stories, explain proofs - I'm fine with that. But singing?? So, part of me got the better of the rest of me yesterday, and I turned up at this acapella rehearsal, hoping that at least there would be a lot of other sopranos and other people in general so I wouldn't be heard so much.

Eh, I was the only soprano. There were only six of us there at all. And do you know what? We completely lost track of time and sang straight for an hour and a half and I had such a blast! We were working on 'Because' by the Beatles  - you know that haunting song:

Aaaah... because the world is round it turns me onnnnnn.... Aaaah... because the wind is high, it blows my miiiiiind....

 Und so weiter and so forth.  One other guy offered to sing the soprano part with me, but an octave lower, and it turns out - I can sing. After only working on proofs with some of these people and never having really seen some of the rest of them, to somehow make music was so exciting. Now I understand why Rach does it all the time. Next week, we're gonna start working on 'Finite Simple Group of Order Two', like the grade-A nerds we are. Google it. :)

--------------------

In other news, I'm consistently surprised at how brains work with languages. I'm sitting in Starbucks again, waiting to work on my research project with my friend Dan, and when I came in, I was a pro at ordering coffee. I not only can distinguish between the words for 'push' and 'pull' now, so I don't look like an idiot trying to get into cafés anymore, but I also had the entire conversation in Hungarian:
--
"Hello!"
>"Hello."
"What can I do for you?"
>"I'd like a drip coffee, please."
"For here or to go?"
>"For here."
"What size?"
>"Small."
"With milk or black?"
>"With milk, please."
Etc, etc.
--
(this is also particularly funny because of all the cafés in this city, Starbucks is the one where they don't even bat an eyelash if you speak in English)

 But yesterday at the gym, I had forgotten my water bottle and was feeling like my throat and body could give the Sahara a run for its money. I was standing in the locker room when I decided that it was worth it to buy the overpriced water at the counter in the gym because I was sure I was going to evaporate into nothingness before I made it home. And I stood there, half changed into my real person clothes, trying to remember the word for water. And the only word that would come to mind for the full four minutes I stood there was the one for butter!!! I didn't want to ask for that and didn't feel like doing the pointing and waving dance to convey what I wanted, so I just went home. I didn't evaporate. But still, I have no idea how my brain works.


---------------------

Also, lastly - I was approached yesterday out of the blue by a fellow student with whom I have had very few interactions. We're in Algebraic Topology together and that class isn't one where the students have to present things very often - but still, he came up to me very kindly and asked me if I had anything planned for next summer. I said that I didn't yet, and told me that he thought I would make a fantastic counselor at a proof-writing and math summer program for high school students interested in math. He said he thought I was a very competent, engaging, and kind presenter, and encouraged me very much to apply for a position. It made my day.

Sometimes, reality is more important than ultrafilters and manifolds.

So, another thing I have been up to lately is swimming. As some of you might remember, I started to actually learn how to swim (I mean, I knew how to not drown by the time I was about six or seven, but I didn't really swim) in my last semester at Mills - it was part of my extremely academic final semester (i.e. Abstract Algebra and Intermediate Programming, but then Model UN, Tango, and Swimming. :D). I had a class where I went swimming twice a week and during which I not only properly learned three of the four strokes but also made the incredible realization that I can swim at different speeds. If you ever have been not a runner and then have decided to go for a run or start running, you know what I'm talking about. You think you only have one speed. For me, that speed was enough for me to end up gasping at the other end of the pool, wondering how the hell I was supposed to do that for the entirety of the fifty minute class. The equivalent of this, of course, is sprinting for the first two minutes of your run and then ending up with a stitch in your side and wondering how the hell anyone could go for runs of twenty or thirty minute length. The answer, of course, is to jog. Or, to do the swimming equivalent of jogging.

I was extremely impressed by my own progress in that swimming class. As someone who runs and works out frequently, it was great to have something that is so low-impact as far as exercise is concerned. But also, swimming is brilliant in how it engages the entirety of your body - arms, legs, core, and it does wonders for your endurance and wind. The only other sport I've ever done that is so equal-opportunity when it comes to the body is rowing, and swimming is even nicer on the body than that. So, I was extremely proud of my progress and it got to the point that (with a few breaks in between) in a class period, I could swim a mile, and was very in tune with how my body moved in the water. It was fun!

Now, zoom forward six months from my last swimming class. I've been in Budapest for a while and have some friends here who invite me to come swimming with them with a 'very low-key team' - this means that on the team, the ages of the swimmers vary from 21 (the youngest ones are the students I'm going with) to the wizened old 60-something Hungarian men who are all wirey muscle and dry humor, with all kinds of sizes and shapes in between. I decided to go with them because, hey, I enjoyed swimming when I did it before and it's a good activity to do with friends.

Let me note: of the three people who I go with, 1. Was on the swim team all through high school and played other sports throughout college. 2. Was on the swim team in high school and plays water polo in college. 3. Was on the swim team all four years in high school and all four years in college.

Remember the amount of experience I have? Yeah. :)

I got my first swim cap from the team yesterday!

So, the fun thing is also that the pool we swim at is large enough but there are a lot of people on the team, so you are frequently smooshed in a lane with other folks. The lanes are organized by speed. Also, of course, my friends are in the second to fastest lane 'cause they're all young and spry and experienced. Now, I could either bump into the rotund Hungarian ladies I don't know in the slow lane way on the other side of the people and have awkward hand-gesture-y encounters, OR I could swim in the lane with my friends who can probably swim an entire length of the pool in the time I get myself organized enough to push properly off the wall. My friends are nice and told me that as long as I understood pool lane etiquette (i.e. let people pass you if they are going faster) that I could stay with them. One lap in the lane goes like this:

Finish: <--------------<

                                ^

Start :  >--------------> 

So that if you look at the lane head-on from the start position, it looks like a road. You drive on the right, people pass on the left.

So, yesterday, the workout involved a lot of IM's (individual medley) - where you swim the strokes in this order: Butterfly - Backstroke - Breaststroke - Freestyle . The distance varies depending on the exercise - sometimes 25m each, sometimes 50, sometimes it varies from stroke to stroke. 

So, first of all, they're all faster than me. Second of all, I can't do Butterfly, but I try to at least do a baby version of whatever they're doing - either giving myself more rest time than they need or doing a little shorter distance, etc. I let each of them take off from the wall (I say "take off" because boy, they really fly!) at the beginning of this exercise last night and I generally do freestyle when they are swimming Butterfly. However, everyone else in the pool is doing the same workout, so everyone else (except for the one or two other people who, for their own reasons, are doing something different -- this is very low-key, as you recall) is swimming Butterfly. Have you watched people swim Butterfly? You know the huge dolphin kicks they do? And the big arm swoops? 

Imagine this: you are swimming up the right side of the lane, very close to the lane marker because you know you will be passed very soon. You are thinking about when to breathe in your stroke and think: 'if I turn my head to the left, I'm going to get a mouthful of water because they're swimming back to the wall on that side of me."And as you turn your head to the right to take a breath, it just so happens that the swimmers in the OTHER lane (which is very close to you, given that you're a handsbreadth from the lane marker) are RIGHT THERE coming back to the wall because you are slow enough that they've been down to the other end and have already turned around and are making their way back. So, you turn to breathe, manage to sneak a breath and not get too much water in your mouth from the splash from their feet, put your face back in the water and watch the stormy seas in front of you, and plan how you will turn around at the wall (given that you can't do flip turns) without causing a collision!

This is very exciting.

I have never swum in such close proximity to other people before, especially people who were so much better than I am. When I would come up for air during breaststroke and see the waves of water around me from people on either side, though my foggy goggles, it looked so very much like the stormy seas that I had to sing the Pirates of the Caribbean theme to myself in my head while I was under the water. I generally recite poetry to myself while I swim, it helps me stay in a rhythm and keep me motivated when I'm tired. Some favorites are old poems I somehow memorized from childrens books, including the Sorting Hat songs from Harry Potter. But just as I was feeling dead last night in the pool, having a hard time pushing myself for the last ten minutes, I remembered that I had memorized the wonderful V-speech from V for Vendetta, and that got me through the last 100m. It was a good time.

But the point is, last night, there were times when I just wanted to make sure that I was going to be able to breathe, and compared with the incredibly theoretical proofs that plague my mind so often, it was nice to have a problem so rooted in reality that I needed to solve for a change.

Where have I been? What have I been up to?

These questions were posed to me yesterday, and my blogging absence was pointed out, so I thought I would jump back into it this morning!  In answer, I have been spending my time, as ever, with math. In Spectral Theory, we're talking about crazy beasts called 'Ultrafilters' (honestly, the names for things in math just get sillier and sillier as we go on. We proved a theorem in Algebraic Topology called the 'Hedgehog Theorem', and I know that a certain mathematical object called a 'Super Algebra' also exists. Good grief!)

But some of the rest of the time, I've been getting to know people - having potlucks and get togethers, and encountering things like this most magnificent pie:

This pie (which is a good 3.5 inches deep) sparked a lively discussion at this potluck, about how cruel it is that cakes are always allowed to be as tall as they please, yet pies are reduced to tiny heights by design. So, my friends threw the design away and made an enormous pie.

I've also been wandering around Budapest and finding lovely little markets that are popping up in this cold weather. The last one I was at had several antiques booths where old Hungarian money, ancient bottles and tools, aged china and silver, and a whole manner of old photographs and paintings could be found. I strolled around with a friend, each of us with a glass of mulled wine, and had a lovely Saturday.

My friends from back in California have been in communication lately as well, including the lovely Matilda and Kamaji, as you might remember from my summer adventures in Berkeley.

Is there anything more lovable?

Have you ever seen such wise, knowing eyes??

Receiving those pictures from my old friends made me miss that life there. Somebody asked me what I had done over the summer and before I knew it, I was back with the chickens and the lemon tree, work in downtown SF and walks in the Berkeley hills with the dogs and my friends. It really was a great summer.

And things are winding down here for the semester. If I'm not mistaken, there are only four more full weeks of the semester! Then I'll be dashing quickly to Pennsylvania, to finally see my mom and sister and MY lovely dog again (I'd having her living with me here if I could), then to Washington, DC to see the rest of the family, and then some time with friends in Germany and Scotland over New Years.

And speaking of Scotland, my good friend Erin is coming here for Thanksgiving, which means I'll see her in less than two weeks! I can't wait to wander around the city with her - it's such a marvelous thing to have become an expert within the place that you live, so that you can show it to others. So much fun. :)

Alas, ultrafilters and ultralimits await. I was talking to a friend about my Algebraic Topology course the other day and how our professor doesn't really lean towards the incredibly rigorous proofs (I'm talking variables with several layers of indices on them, etc) because so much in the subject can be easily shown by a picture, but that would take ages to write out precisely. (If you think about it, this makes sense - think about drawing a plane perpendicular to a line in three dimensions. You can see just how perpendicular it is, but expressing that mathematically takes some crunchy numbers and variables. Now, imagine that we're talking about manifolds in n-dimensions... Yeah. Pretty hairy stuff.) However, understandably, some students don't like the lack of rigor in our proofs, because it makes them feel a bit afloat and not as in charge of the material. It can seem a bit 'hand-wave-y', as the phrase goes. And I was thinking about it during our conversation because my friend was particularly opposed to this kind of hand-waving (imagine someone waving a hand at a picture on a blackboard, saying "can't you see this this, if you pinch the corners together, looks like this??" - that's what I mean) and I couldn't help but compare her to Hermione talking about Divination, and then I had to compare my professor to Trelawney, and then my Algebraic Topology lectures became very amusing for a few days.

Yesterday

I went and saw an incredible musician - Amanda Palmer (please, don't judge her by her eyebrows. Just watch the damn video.) The link I'm giving here is to a Ted Talk that she gave - the first place I ever saw her, and only a few weeks ago. (If you are interested, do check out her songs - in particular, I think 'Want It Back' is a great one. She describes the music in her talk as a cross between Punk and Cabaret, which is a pretty cool combo.) And as it turned out, she was performing in Budapest ( a rescheduled show - it wasn't supposed to be happening during the time that I'm here). It was fucking amazing. There isn't any other way to say it.

The concert was on a boat on the Buda side of the river. It was such a small venue that you could see everything no matter where you were, and the energy in the audience was just amazing. She really is an entertainer.

Leaving the concert at around midnight, looking back at one of the many Budapest bridges. 

 I went to the concert with my good friend Kaia, and we giddily talked about how amazing it was for the entire trip home - both the concert and Run Lola Run, which we watched just before we went to the show. It was such a spur-of-the-moment decision to go to the concert - I only knew about four of her songs, Kaia knew none - and it was such a good one.

Heading home.

Beautiful Day for a Cemetery Walk

Some things don't need words to be poetic.

Graphs and sheep, continued.

This is what I do instead of my new spectral theory homework. Oh, well!

Graphs and Sheep

I want to give you a bit of a taste of what we're doing in Spectral Theory, and finally, I have a problem that can be explained in normal, real-world terms. Almost. Kinda. :)

So, imagine that you have a herd of sheep - a very large herd. Say there's 10,000 sheep. And you know that 10% of the sheep are sick, and you need to run a test on this illness. However, you can't tell if a sheep is sick from the outside- you have to put it in the fancy testing machinery to figure it out. So, how do you find a sick sheep? Well, you probably start by picking one at random. And you have a 1 in 10 chance that you guess right. So, the question that I want to pose is: how likely is it that you choose, say, 100 sheep at random and STILL do not find a sick one?

So, you might wonder why is this question relevant. Let me put it in computer science terms - not graphs yet, but we'll get there. The reason this question exists is because doing things "randomly", generating truly random numbers is a lot of work for a computer. Sure, it can still do it quite fast, but it's costly in terms of running time and memory. So, if you pick one purely random sheep, then another purely random sheep, then another and another --- this gets costly. The probability that in 100 steps you still do NOT find a sick sheep (or a certain file, or whathaveyou in computer science terms) is fairly easy to calculate.

So, where does Spectral Theory come into this? We aren't computer scientists - or even theoretical computer scientists (which is a field! I had no idea, but it sounds fascinating.) Well, it turns out that a much nicer way in terms of computer power to do this searching for sick sheep is to put a 'graph structure' on the herd. What does that mean? Just give all the sheep numbers - 1 through 10,000. And then decide how you want the sheep to be connected to each other - one simple way is to say that sheep 2 is connected to sheep 1, sheep 3, and sheep 10,000, for example. Let every sheep be "connected" to three others. (Bear with me, sounds weird, I know)

In doing this 'connecting' thing, we are turning the herd of sheep into a "3-regular" graph. That simply means that every vertex (or dot) in the graph has exactly 3 neighbors - three edges (or lines) coming to it. Here are some examples of three regular graphs - the first, second, and fourth graphs are 3-regular. The third and fifth are not. Notice that the first, second, fourth, and fifth are all graphs on 6 vertices, yet they are very different:

There's a lot of different kinds of graphs out there. If 'friendships' are edges between people (vertices), imagine what the graph of facebook looks like.

So, the sheep are now a 3-regular graph. In fact, we can also require that the graph is 'connected', which means there's always a way to get from one vertex to another by walking on the edges - so, we don't have any completely isolated sections of the graph. No sheep is separated from the rest of the herd!

Okay. So, let's pretend we have this structure on the sheep. Then, I pick my first random sheep to test for the illness. That's still a completely random number, so still some work for the computer - but I only have to do it once. After I've picked my first sheep, then if it wasn't sick and I need to keep looking, I pick one of that sheep's neighbors. That's easy for a computer to do - almost no work at all. And I search through the herd in this manner. This process is called a random walk.

So, now you can hear the original question as it was framed on my last Spectral Theory homework: Given a d-regular connected graph G on n vertices (n is the size of the herd - don't be intimidated by letters standing for numbers) and a subset of the graph H with c vertices, what is the probability that a random walk of length k that begins in G\H (you read this notation "G without H"  - so starting at one of the vertices of G that isn't in the subgraph H) will stay completely in G\H?

Does the translation make sense? H is the sick sheep - there are c of them in there. What's the probability that if I pick a sheep at random from the healthy ones and then pick k sheep all together by choosing neighbors of the sheep before - what are the chances that I don't wind up with any sick ones in this bunch of sheep I've picked?

Oh, math. It's a funny thing. In Algebraic Topology the other day, we proved a theorem called the Hedgehog Theorem. Maybe I'll take the time to try to explain that in a little while.

Back in Pest

Not a bad place to work for an afternoon - a café above the Alexandria Bookstore on Andrassy, near the Oktogon in Pest.

So, I'm back from Germany. I can't even quite explain how good it was to be back there. Familiar town, familiar people, familiar language. I know which busses to take to get where, I could get to the university from my old apartment with my eyes closed (probably), and it was amazing how connected I felt to my surroundings when I could understand the people. I didn't realize how much I was missing that here. My Hungarian is certainly more than it was when I got here, but that's hardly remarkable. In fact, it would be a shame if anything but that were the case!  It was neat to realize (as I did a few times this weekend) how my vocabulary has been growing without my knowing it, but still - it's nothing to actually speaking the language of the country one is in.

And now, I'm sitting in this beautiful cafe, working on a presentation for tomorrow about Stochastic Calculus, Portfolio Theory, and the crazy models that my research partner/friend and I have been working on for the past -- two months? Is it that much already? Miklos mentioned this morning in Spectral Theory that we're more than halfway through the semester. I don't quite know how that happened.

On the runway again.

I'm staring out at the almost sunset (it will be setting in maybe an hour - which is ridiculous, because it's 3 in the afternoon) over the wide tarmac field in front of the Vienna Airport. I came on a bus from Budapest and in about two hours, I'll be hopping on a plane to Frankfurt. Even though here, the German is accented in an unfamiliar way, I still can understand people and I can read the signs around me - it's amazing how much more connected that makes me feel. Still, even as I was on the bus, I noticed my progress in Hungarian that had been going on without my knowledge - the kind of progress that happens when you live in a language, you know?  - for example, I got on the bus and the passport control folks came on. I saw their uniforms - "Rendőrség". And I thought, "That's funny. I wonder why the police uniforms are in German? We're still in Hungary." - the reason being, of course, that I understood 'Rendőrség' too quickly, I assumed it must have been German. So, that's only one word, and that is negligible progress, but I'll take what I can get. This is a strange language!

And so far, the Vienna airport is endearing itself to me. There are little pod-desk complexes set up every few gates. About four desks arranged in such a way that four are clumped together with a little screen in between each, two facing "north" if you will, a right handed one and a left handed one, and two facing "south" (also left handed and right handed) - but they're made out of couch-ish material. Not very squishy, but more comfortable than the hard plastic chairs. And each has some outlets, which is always nice for travelers, and the wifi is free. I have enough to work on that requires no internet that I would have been completely fine without it, but with the wifi, I can continue my tradition of checking in while traveling.

Even though the cup of tea and bottle of water I just bought definitely (I just checked on google) cost as much as my lunch out with a friend yesterday in Budapest, I'm excited to get out for the weekend. People have been going insane with stress about midterms. Students around me were dropping like flies - when asked how his weekend was, one of my friends replied, "I'm considering going into the countryside here and chopping wood for a living." Another replied in only sounds, not words. And all of us are wondering whether or not we are too young to have mid-life crises.

And I don't think the stress is worth it. Yesterday, my body made the shift from stress (which is a very active, tense feeling) to weariness. And I didn't like that one bit! Yeah, these classes are hard, but they aren't life. I have to remember to breathe. Hopefully, the other students will realize that, too - one of my friends said that he was going to take care of himself this weekend "no matter the cost". I think it's a shame that that has to be a big deal, that taking care of ourselves is something we do rarely. I think it's just because we're not the best with moderation - either in work or relaxation.

Speaking of work, though - I have to go and correct all of my previous homework assignments for Spectral Theory. That is our midterm - turning in everything again. It's a lot of work, but it's also rather neat to see how much we've learned in the semester so far. And I think that the things we're talking about in Spectral Theory and Algebraic Topology are about to meet in very unexpected way! If you are interested in some of the insane stuff we get up to in Algebraic Topology, check out this video.  This is what's called the 'Alexander Horned Sphere' - you don't need to know anything about it. Just know that from a topological standpoint, what is interesting about this is that it is simply a very, very strange sphere. (By sphere, I mean something like a basketball - not solid in the middle. Just the outside. The surface ((the outside)) is only two dimensional, but because of how it is connected, it has to "live" in three dimensional space.) Anyways, it's kinda cool!! Check it out. 

Eger

Wondering what the day would bring.

Peeking out the window...

To see the lovely main square of Eger.

Give me an old wooden door and a crumbling stone wall any day. 

We knew there was supposedly a place nearby where one could go horseback riding - only five minutes or so by car, so the books said. So, we decided to walk - and our walk took us outside the city limits on country roads with no sidewalks and thick brush on either side. This was when we, the intrepid travelers, stopped for some trail-mix sustenance.

My friend Dan ahead of me, as we reached the summit of the hill that was basically our entire walk out of the city. It was our 'if it's not here, we'll turn back' point on the road.

And this was what we discovered there. To the right on another hill, we could see many, many horses - and we knew we were on the right track.

Kaia even made a friend at the top of the hill.

Dan has worked at vineyards before and taught us a bit about harvesting grapes with the aid of a few that had been left on the vine.

We made it to the riding place just around lunchtime, where there was a fabulous restaurant waiting for us.

Dan and Kaia, with two glasses of wine. The waiter/restaurant owner didn't speak any English, so I was speaking to him in German. I asked for two glasses of wine for the other two and just for some water for me, and he looked at me and simply said, "Why?" To which I had to smile and order a glass also for myself. It was some of the best wine I've ever had. And as you can see, in Hungary, a glass is a proper glass!

Some furry friends that weren't horses, but that I was very, very glad to see.

This one reminded me so very much of my own Abby. It was good for my soul to play with a dog again!

And that was our day in Eger. We made our way back to town, saw the castle, had some dinner, and piled on to a train, sleepy and happy, to come back to Budapest. We did end up riding at this fabulous place surrounded by vineyards. We each got the chance to ride and at first, I thought would be boring - on the website, it had seemed like trail riding would be the thing, but then it ended up being more in a paddock and with the horse on a lead. And, exciting for all of us, it was English style riding, not western. I thought I would be bored, like I said - but when my turn came, it was like a proper (and lovely!) lesson! I only walked and trotted for the whole time, but they taught me a lot about my posture while riding, had me do various things with my arms while riding to show that you don't need them to hold on at all - even to twist and touch my right hand to my left foot and vice-versa while trotting rather quickly! It was beautiful to be on a horse again, even though English riding is like a dusty memory for me. I hope I get the chance to do it again soon.

And it was fun to get to use my German, fun to have a weekend (or even just a day) with no math in it. I gave myself the day completely off, and I'm very glad to have done so. Tomorrow things can begin again, as I suppose they must. But this was a grand, grand day out!

Morning in Eger.

A real bed instead of a couch-bed for me last night, and this view out of my window. :) I'm looking forward to today!

Trains will always make me think of Germany.

But this weekend, I’m leaving from Keleti Pályaudvar, not from Mainz Hauptbahnhof , I have a ‘jegy’, not a ‘Fahrkarte’. But when I get the shiny, square-shaped ticket, when I settle into mostly-inflexible inevitably blue-fabric-covered chairs, when I see the conductor coming with his strange clicking device to check my ticket – I’m always in Germany in my head. I miss it.

This weekend, though, I am going to Eger – a small town about an hour and a half away from Budapest (not quite sure in which direction yet- though, if I think about how the sun looks outside of the window – I’m guessing North? Northeast? I’ll check.) Along with an apparently lovely church and town square, this place is also host to an amazing national park with lots of caves (complete with thrilling stalactites, etc.) , and a valley called ‘Valley of the Beautiful Women’ (okay, in some translations, ‘Siren Valley’, but I like the first one) which is the home of many tiny shops bearing the fruit of the many vineyards in the area. 

And it’s not like I don’t have homework this weekend. On the contrary – there is as much as there always is, if not a bit more (midterm season – also that ‘you-wanted-to-apply-to-gradschool-right-then-now’s-the-time’ season). And I know I will be spending some of this weekend doing that homework. But I’m excited to sit in a different café, to have a different view out of the window. To walk somewhere different in the mornings… And, yes, to have some fun. I realized how little of the city/country I had seen earlier this week – it was an intense moment as I realized I had never even been to the Bascilica, only been on the Buda side of the river three times, only ever leave my tiny neighborhood (which includes the BSM school) at most once per week. And sure, I’ve got a lot of fun math to prove for it. But man, I’m here. I need to actually be here. 

Which is why last week, I went to see a friend of mine perform with some fellow musicians that he found when he moved here and has been playing with ever since. This was their first gig that wasn’t on a sidewalk in front of the Buda castle, and it went well! And now, a trip to Eger.

And next weekend? A trip to Germany. This math is like a treadmill. I have to keep going or I will fall off, but it’s not like I can actually finish it all, fast as I might go, hard as I might push. So, why not have some fun?

What else can I say.

It was a great lunch!

He's done it again.

So, I spent most of today trying to force my brain back into work mode. It didn't go well. My head was so determined to relax since I had fought the fight, defeated the test, and I had been promising it the award of relaxation since I registered for the test in June -- and at the same time, even if I hadn't wanted to relax, it was so dead after all the anxiety and stress that it couldn't function well anyway.

So, I spent a long time today all but actually banging my head into my Spectral Theory homework. I have been hanging on in this class, but certainly for the last two weeks, my head wasn't in it completely, since I was studying for my exam. And while I have been shifting my focus to other things, the material has been getting more and more complex, and my overview of the material was fading. I felt like the pieces were slipping apart and it culminated in the assignment today, - I really felt like I didn't know the big picture anymore. I wrote several emails to Miklos asking small questions about the assignment, and finally, a few minutes ago, I wrote to him and mentioned what I just said above - that I've lost the big picture and just feel adrift in general.

And he wrote back, - classic Miklos: "Hi Emily. Of course you lost it somewhat! The main thing is to have some kind of balance and to enjoy it. :) " Word for word, that is his email to me. I think I'm going to go to bed and try it again with a fresh brain and that perspective in the morning.

The deed is done.

I made it through. The exam was easier than I expected, though not easy, and I have a feeling I did as well as I needed to do. At a price, though. I have no brain left. I actually just heard the sound of a mosquito buzzing and saw a flicker of movement near my eye and, without thinking-- slapped myself in the face. I wish I were kidding. The worst part is that I don't even know if I caught the damn bug, since I can't see my own face...

And after the exam, I came home, and sort of stared at the walls for a while. I had some food (breakfast at around 6:30, being in the exam building starting at 8, only leaving at 12, I was hungry!) and zoned out for a while, and then I went over to Miklos' house!

That's right. He hosted a dinner for the Spectral Theory class today, because he wanted to get to know us better. It was a perfect event for a post GRE afternoon. No one else in the class took the exam (though several other students at this program did), so no one there wanted to talk about the test (amen), and it was the weekend, so we were relaxed enough not to talk about math all the time - but also dorky enough to make lots of jokes about it. Miklos made us "liba maj" (fois gras) in the traditional Hungarian way - gently cooked and served thinly-sliced on bread with raw onions, or various pickled vegetables -or even with his wife's homemade raspberry jam (sounds weird, but it was delicious). Then, there was some wine, some math jokes, some talking, some storytelling - eventually gnocci and then dessert got added to the mix, and all in all a lovely evening.

And now, I'm going to sleep. I still have quite a bit of work to do - after all, my classes didn't stop while the GRE happened. But my head is so much clearer. The underlying anxiousness is slowly abating as I realize the test is over.

I'm now writing this while lying on the couch, and my head is actually on my arm, not even propped up anymore. Now one eye is closing. I do believe it's time for bed.

Things That Are Much Worse Than the GRE That I Have Already Weathered And Survived: A Pep Talk

It's tomorrow morning at 8:30 a.m. At 11:30, I will walk out of there, with the pressure of the last four months off of my shoulders, one way or another. It's just a damn exam. I'm so excited to be done.

And, as I go to sleep here tonight, I want to give my brain something fun to think about, as mentioned in the title. So, below are some harrowing experiences I have had that were both a) more painful than the GRE Math Subject Test will be tomorrow and b) lasted longer than the aforementioned test will. I don't know if it's for me or for you, but it's here.

1. Surviving through the morning after the day that I hiked Half Dome with my family in Yosemite - when I thought none of my bodyparts belonged to me anymore.
2. Getting my wisdom teeth out.
3. Sitting through graduation practice at high school and college in stuffy rooms with not enough air.
4. Bowing incorrectly to a monk in Thailand and sitting through the rest of a silent, 2-hour service in the temple when people kept intermittently turning around to stare at me.
5. That one time I was on a plane from Frankfurt to San Francisco, and 12 hours into the flight we circled and circled and circled around San Francisco, then had to land in Oklahoma City, they barely had enough fuel for such a big plane, then we had to fly back and circle and circle and circle around Denver before landing and I had to fly to SFO from there.
6. Having strep throat a few weeks ago.
7.The day of crew practice when I almost fell out of the boat due to incorrect form and was thoroughly embarrassed/shamed and also very sore from all of the rowing.
8. Studying for this damn test for the past several months.

I was on the exercise bike today with my GRE book and I was listening to the Pirates of the Caribbean soundtrack - and as I turned to the page about the integration of complex functions, the music swelled with the first appearance of the main theme in that film, and I all of a sudden was out of myself and observing the situation, realized the dramatic music and me, sitting on a STATIONARY BIKE with a book about MATH, so worried about one damn test. And I started laughing so hard I had to stop pedaling. Life cracks me up sometimes.

So, my plan. Go to sleep. Wake up. Have some breakfast. May or may not involve bacon. And treat myself as awesome. That's what I did all through my undergrad - it worked for me there. No reason for it not to now.

"Hm. Let's do something simple - consider the infinite 3-regular graph ."

Mathematicians are weird.

The thing is, you have to wait around to figure out what the things are that make sense to you without thinking, which are the ones you have to take a few seconds to figure out, and which are the ones that feel beyond you when you first encounter them, but really you need an hour with a blank piece of paper with an afterthought of follow-up questions. For example, something "simple" today had me simply reeling as I couldn't get it into my head (having to do with matrix multiplication, which is something I learned in high school - though not in such an abstract way) and then, later in the day (at the colloquium talk I just left) something that came to my mind without thinking: picture a lattice, like the kind on top of a pie. Basically, looks like a checker board or chess board - vertical and horizontal lines with squares in between. Now, there can be other kinds of lattices - ones that don't have little squares, but rather, little triangles, or parallelograms, or -- hexagons. In the talk today (about percolation - the way liquids move through particles, like how far water gets if it has to seep through a bunch of densely-packed sand), it turns out that in percolation theory, many results can be proved when using hexagonal lattices but not with the traditional square ones. I asked why. "Because," the speaker responded, "The faces of the hexagons are the vertices of triangles." And I had the picture in my mind immediately.  (Grab a piece of paper - draw a dot on the center of each hexagon you draw, and connect them - you'll see what I mean. This result is not all that deep or anything - I was just happy to visualize it so quickly.)

Okay, okay, so the more I think about it, the more pathetic a victory that tiny moment was. However, I've been thinking today. Quite frequently, in my head, I think about how I don't feel like I'll ever be worthy of the title 'mathematician' because there are so many people who know so much more about math than I do - so very, very much more. But all of a sudden as I was feeling sorry for myself like this this morning, I thought of a tiny fish saying that it can't swim because it doesn't go as fast or as deep as the mighty narwhal, for example. What a stupid thing to say. Of course it can swim! Damn well, too!

Hold on, though. Maybe, in this metaphor, I might actually be a sea lion- because, I can swim - and in some areas of the ocean, I may even be a force to be reckoned with - but I like to spend a lot of time on land (a.k.a. playing music, reading, thinking about philosophy, psychology, economics, etc) and not in the water (i.e. the murky and seemingly infinite depths of math). In fact, I can't be in either place indefinitely.... Oh dear. Metaphor carried me away. I'll swim back to the point.

So, the point is, as you can clearly see, the GRE is in less than 48 hours and I'm on my last mental legs. But at least, I understand how hexagons can be like triangles. Guess that's a small victory.

Humor is one of my favorite parts about humanity.

Read this! (It'll take 20 seconds).

Those unexpected words: "Undress the manikin!"

A few weekends ago, I went with some of my friends to an 'Escape Room' here in Budapest. This is a very local thing, a phenomenon that makes total sense with a culture so steeped in (and skilled at) mathematics, logic, and problem solving*.

It goes like this: you get a group of 2-5 friends and find a little hole-in-the-wall Escape Room (sometimes called Trap Room) place. The person at the desk welcomes you in quietly, asks which room you want to visit or asks for your reservation number. You answer, and the group files into a dusky room, usually in a basement, usually with strange things painted on the walls - dusty, rusty, strange things hanging around the room. Barrels. Rope. Broken mirrors. Piles of clothing. Old photographs. And the man from the counter sees everyone is inside, says "You have one hour," and shuts the door. You hear it lock. You have one hour to figure out how to open the door again.

And you find your way out! It might be small mathematical calculations from strange diagrams painted on the walls, getting a cork out of a wine bottle with only a rag, finding a marble in a dusty corner that then becomes a vital part of a contraption to free a tiny key from a glass box -- clues are hidden around the room and hopefully after an hour, you have opened every possible part of the room (in the case of the one that I did, there were several rooms contained in one, and you had to thoroughly explore all three to find the key code to re open the original door). And our group - fantastically mixed, by the way. It's good to have people who are good at different things, and we had exactly that. Math people and non math people, tall people, small people, creative people, determined people. It was perfect. Everyone did something to help the group. And boy, did it teach you to think of everything as important and worthy of investigation.

At one point, the person who could watch our attempts to get out through the camera in the ceilings of room noticed our time was running out and that we were attempting something fruitless and said those words over the loudspeaker:"Undress the manikin!" We had rifled through the manikin's clothes, found various clues in them, but none of us had dared to undress him completely! I will never look at manikins the same way again. A few of the people in this group are fellow students and when we're stuck on a problem, we've taken to saying that to each other as an expression of being at a loss for what to do next.

I hope to go back and do that again sometime soon, maybe with some different people. I can build my team with care. :) I think it's a brilliant idea, this type of thing. If it were perfect for me, it would also involve a rope climb, or a complex pulley system where we had to lift our team members up - something physical. I feel it's a shame that we don't really use our bodies anymore. Not for real things. Even the gym feels fake and mechanical. Helping someone move, helping in a garden, raking the lawn - I love those things. I just wish they could be incorporated into my daily life more - in a natural way. (I think I'll be one of those wirey sixty-year-olds who bikes to work even in pouring rain...)

All in all, though, a brilliant thing to be able to do. I hope to participate in several more and get myself out of many other locked rooms before this semester is over.

--

*I realized tonight after listening to some fellow students that "problem solving" is a misleading phrase when spoken by mathematicians (or wanna-be mathematicians) to mixed audiences. Here are a few "problems" the way I (we?) use the word: 1. Consider a square constructed by 8x8 smaller tiles, each square in shape (i.e. like a chess board). If one tile is removed from the upper left hand corner of the board and one from the lower right hand board, is it possible to cover the rest of the board in dominos (with no overlaps or pieces hanging off the edge), given that a domino covers two adjacent tiles? 2. How many strictly increasing integer sequences begin with 1 and end with 1000? As you can see - not real life problems. We're as good at those as the rest of you.

And it's October!

I'm not quite sure how all of a sudden, it's the 5th week of the semester. I was checking on my flight itinerary for Christmas, and at the top of the webpage, it says "75 Days Until Departure". Wow.

In other news, Miklos is at a conference this week. This means various things to me, but really one important thing - we didn't have a NEW assignment due this week in class. Instead, it was suggested that we spend the weekend typing up revisions of our old assignments. As I may have mentioned before, in Spectral Theory, we write up proofs for an assignment, turn them in via email (fancy math programming - well, not all that fancy, but seeing as I'm still quite new to it, I feel fancy whenever I do it) a few days before class and then during class, get them handed back and see what we did wrong, and those who had the proofs right (or at least mostly) go up and present them for the rest of us. So, the tech copies we have so far are the uncorrected versions, and this kind of correction takes a while. For example, to get the humble vector brackets to appear around something modest, like: <x/2, 5y>, in LaTex (the programming language), one has to type:

\langle \frac{x}{2}, 5y \rangle

 

As you might imagine (with computers being as brilliant as they are) every time I accidentally typed 'angel' instead of 'angle', I spent twenty minutes finding the error that was keeping my document from compiling. Basically, once you are fairly fluent at it, this language makes creating typed math documents much, much easier (for articles, journals, books, - and homework, sometimes). And look how nice it ends up!

(who knows if I will have readers this picky, but about two thirds down, YES 'a' can be negative - it should read a=/=0. Haven't fixed that bit yet. :) )

I'm still new at it, so I'm not the greatest - my spacing is still sometimes wonky, and I know there are things you can do to make everything that much more neat, or fancy, or whatever- but still! Five weeks ago, I couldn't do this at all, so progress has been made!

As I was going back and editing my assignments, I really did see how much I had learned. The pace of my courses here is such that I never feel like I'm coasting. Back at Mills, I certainly didn't feel like I was coasting the whole time, but I always felt like I was standing on solid ground by the end of a lecture, even if the beginning of a lecture had me feeling a bit wobbly with new concepts. (Well, a few times even at the end I didn't feel like all was well, but the rest of my foundation was so solid that it didn't matter) Here - I don't feel that stability. And until you take the moment to go back and think over what you've done, it feels like you haven't made any progress, if you are used to measuring progress in your own confidence. But I have learned things.

But right now, quite honestly, the main thing on my mind is the GRE. The big bad exam. It's a week from Saturday - on the 19th. And right now, the main thing I'm thinking is that I want to take it now. As I sit and work through ages of tedious integrals, breaking my brain to remember trigonometric identities and recall how to find the equation of a surface in three dimensions - I really realize how much more interesting the math I am doing now is. But studying is important, I realize, which is why I do it. And even if I feel a bit like I'm going mad in the process, it'll be done soon.

I can barely imagine the free time I will have when it's over!

Speaking of free time, I should get back to work now, but I'll leave you with a bit more math humor. Check it out here. Good ol' xkcd. Always good for some fun.