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.

Some days are just like this, I suppose.

Every cup I reach for falls over, ever fork slips out of my hand, every idea fades as soon as it's been formed - emails are sent with typos and even my whiteboard won't stay on the wall. And since this is my physical and mental environment, you can imagine how I am reacting to everyone around me today: specifically, I feel like slapping everybody in the face, hunching my shoulders, and going to take a nap for the rest of the day. But it's okay. As soon as I realized that today was just going to be like that no matter what I did, it got a bit easier, and I could giggle a bit at just how ornery I was and how uncooperative the rest of my immediate universe was.

And so now I'm home, I've managed to carve out some headspace for math with some beverage that I have not yet spilled and I think I'll be able to concentrate a bit.  I wanted to tell you first, though, about something that happens sometimes in Algebraic Topology. So, Algebraic Topology (ALT for short) is every Tuesday and Thursday from 12-2, that mystical time of day when everyone is either lost in thought because they are daydreaming about lunch, or falling asleep because they just had lunch, or bothering everyone by eating lunch right there and then in class. In general, the rule in Hungary is to start the class actually at a quarter after whichever hour was on the schedule, and then at the halfway mark to take a fifteen minute break. Quite civilized. So, we get started around 12:15. The gears start turning in a woefully slow manner, we brush off the dust of the weekend and start to think about higher-dimensional donuts and loops and other mind-boggling things. By the time the break rolls around, the professor has usually revealed something so similar to what can only be described of as a thought-twister (akin to the familiar tongue-twisters) and has promised us a proof after the break that will resolve our worries and pain in comprehension of the topic at hand.

We shuffle out to the hallway, the open window, the bathroom, or the coffee machine in a daze. We all show up back in the classroom before the break is over - little lost sheep, don't know where else to go. And usually, some kind of deep discussion ensues for the last five minutes of the break before we actually do more math. Today, it started this way - a comment from a friend of mine who sits in the back row and waits for the math to impress him, but in a somehow charming way. "Professor," he says, lounging in the corner. "What if I just... don't believe you?"

I turn around at the question. And I see that instead of the typical sassiness on his face, he actually looks a bit like he just lost a battle for the comprehension of the last topic. (Let me just say that that last concept involved assuming that one can visualize a cube in n dimensions and then hold that in mind while you do weird morphy math to it and glue it somehow to an n+1 - dimensional cube...) "Believe me about what?" The professor asks. This professor is about 34, named Boldizsar (effing badass mathematician name, if you ask me), and seems like he breathes this infinite-dimensional business and could read our ALT textbook the way I do Calvin and Hobbes.

My friend in the back row continues. "Just -- everything. What if I think you're making this all up??" His utter confusion is so plain on his face that this question echoes and skitters around the room and we all laugh, including Boldizsar. For the next ten minutes, we follow along this slightly meta route and talk about mathematics and the larger world, how some it relates to physics, how it's all just a model anyway, how some mathematicians can't even talk to any other mathematicians because their particular area of study is so specific that it doesn't relate to anything else.

And even though I'm about to go and labor over the technical and rigorous definitions we don't do in class but that I need for understanding ("That's the worst and most boring part of mathematics." - quote Boldizsar, when we asked him why we don't do these technical things in class), and even though it hurts and I don't really want to do it on some level - at the end of class, I had another moment when I remembered why I do this. A concept was presented that was so mind-bogglingly abstract for me -- well, let me first briefly clarify. (Am I just avoiding researching the definitions by writing this? I'll let you decide. ) Some/most of what we do in ALT is mind-bogglingly abstract by any normal standards. But when you don't really understand something that is abstract, it doesn't really seem all that insane. That not knowing, that confusion, fits rather easily in my head. For example, I don't know what a black hole is, and when I think about black holes, I think about how I don't understand them, and that's about it. But if someone were to talk to me about what a black hole is really like in terms that I understand, the more I understand what goes into that concept, the harder it is to really think about because the idea is becoming more well-formed.

So today, among all the n-dimensional cubes and infinitesimally knotted pieces of 1-dimensional rope, one concept came up where I knew all of the bits that came into the definition of this new thing, but I never knew they could be put together in that way before. I felt like my jaw actually hit the desk when the idea crashed into my brain. (I write with my face so close to my notebook paper that this is actually quite possible - confirm with old classmates of mine). I understood each of the pieces so well and put them together and it felt like having the idea fully formed was stretching the inside of my brain. That's what math does to my head. It's kinda fun.