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. :)

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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.


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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.