My journey.

After four months of living in Budapest, I'm finally leaving. I'm writing this on the airplane over the Atlantic Ocean, although by the time I post it online I will be in America. Now the past semester feels like a wonderful dream: everything was so different in Europe, but now it's all going back to normal. I'm waking up.

Budapest's airport is on the outskirts of the city, so reaching it was a challenge. There is public transportation that could take me to the airport, but the metro doesn't start running quite early enough for me to catch my flight at 7:00 AM. There is also a taxi service we could use, but a taxi could be unreliable. I didn't want to risk a taxi without a back-up plan. (In fact, a few of my friends arranged to go by taxi, but their taxi didn't come; it was quite a scare, but they got to the airport.)

In light of this, a friend and I decided to get a hotel by the airport and stay there the night before our flights, so that getting to the airport in the morning wouldn't be a hassle. We ended up getting to the airport without any problems. On the other hand, getting to the hotel the night before was quite a terror.

The main problem was that each of us had multiple heavy bags. Getting them on and off the metro was hard enough—in fact, it was no trivial task to fit them on the metro at all. Then we had to go from the metro to the bus stop, which involved a long staircase with several landings. We couldn't carry our bags down the staircase all at once, so we took one or two at a time, setting them on each landing in turn and going back for the others. Eventually some kind onlookers came and helped us. They must have thought we looked positively ridiculous, scrambling up and down the stairs with so many bags.

After we finally conquered the stairs and arrived at the bus stop, we noticed the elevator.

On the bus, my heaviest bag would fall over every time the bus made a sudden movement. So would my friend. After getting off the bus, we had some walking to do. This was the worst part, because for some reason one of my bags kept falling. No matter how many times I positioned and re-positioned it, after thirty seconds it would be slipping again. I was getting increasingly frustrated, and my friend couldn't help but laugh at me. Part of our walk was on a gravelly road with no sidewalk, where we had to hope that the oncoming cars saw us in the dark. (Apparently they did, because I survived.) Also, we had to cross something similar to an expressway exit, as well as a railroad.

After that journey, we were positively bushed. We fell asleep soon after arriving at the hotel, only to wake up four hours later for the early airplane. That's where I am now: on a nine-hour flight, writing this journal entry and intermittently sleeping. Was the overnight hotel really a better plan than an early-morning taxi would have been? We can't know for sure, but it was certainly a better story.

That night, when I was walking from my apartment to the metro, it hit me that that it was the last time I would ever make that walk. It was the last time I would see my neighborhood. I almost cried. Budapest was home to me for four months, and I'll never forget it. I've grown as a person, though the change was so gradual that I didn't really notice until it was time to leave. My experience in Budapest will stay with me for the rest of my life, and hopefully Budapest's fatty foods won't do the same.

Things I miss and will miss.

I am going back to America in two days! I am excited to be home, but also sad to be leaving Budapest. I decided to make two lists: one list of things I miss about America and another list of things I will miss about Budapest.

Things I miss about America:

• Milkshakes.
• Understanding what people say.
• The tutoring center at my college.
• Professors who are always available outside of class.
• My friends and family.
• Movies.
• Going places without having to deal with copious amounts of smoking.
• TGI Friday. This restaurant exists in Budapest, but it is ridiculously overpriced, even by American standards.
• My college.
• Free tap water at restaurants without even having to ask.

Things I will miss about Budapest:

• The excellent-quality low-cost food.
• The public transportation system.
• The panini press that came with my apartment.
• The colorful money.
• Being able to go somewhere like Austria for a weekend, just on a whim.
• My new friends.
• The beauty and history all around the city.
• Túró Rudi.
• The thermal baths.
• The palacsinta restaurant.
• Goulash soup.

These aren't in any particular order, other than just the order in which I happened to think of them. Also, after writing this up, I noticed that there are a lot of food ones. I like food.

I have at least two more journal entries left. First, my Logic class. Second, some of the sights that I have seen around Budapest itself.

Salzburg, Austria.

I went here just for a weekend, with one of my friends. Due to a failure to plan ahead, we arrived in the town Friday night without having reserved a place to stay. Fortunately we had a map with a few places we could try; the place we ended up staying the first night had room for only two people for one night; it's a good thing there were only two of us. The hostel was really small, and we had to share a room with two strangers. Also, the door to our room didn't lock from the inside, so we had to sleep with our door unlocked in a strange place in a foreign country. It was pretty unnerving, but none of our things were stolen. Or maybe something was stolen and I just haven't noticed yet. It must not have been important. By the way, I took a picture of our hostel from far away:

Other than the brief lodging panic, we had a great time. Salzburg is known for two main things: it is Mozart's hometown, and it is the setting and the filming location of The Sound of Music. I went to the house where Mozart resided as a young man; it is now a museum with various artifacts from the life of Mozart, including instruments and letters. Since I'm a pianist, it was really exciting for me to see the piano that Mozart used. The experience felt kind of like a pilgrimage.

I also went on a The Sound of Music–themed bus tour of Salzburg. We got to see a lot of the landmarks from the life of the real Trapp family, as well as some important locations at which the movie was filmed. I really like The Sound of Music, so this tour was really fun.

This road appeared in the movie twice: first, Maria danced and sang down this road with her bag and her guitar in hand as she went to meet the Trapp family for the first time; second, the family pushed their car down this road as they were attempting to flee the country.

This is the real-life gazebo that the movie's "Sixteen Going on Seventeen" gazebo is based on.

The villa that was used for filming.

The backyard lake that was used for filming.

The abbey that real-life Maria was from.

The bus tour also took us by beautiful lakes and mountains.

I tried a Topfenstrudel (strudel pastry thing with sweet cheese in it). Strudel is a traditional Austrian food. It was delightfully awesome. This is what it looked like.

I am really glad I was able to visit Salzburg. It was a beautiful mountainous small town, which many would call "charming". I will close with some photographs that I took from Salzburg's fortress.

FUNctional analysis does not live up to its name.

Last summer I learned some quantum mechanics and thought it was pretty interesting. Some of the theoretical parts use a lot of functional analysis, so I thought that functional analysis would be a fun class to take. After all, its name contains the word "fun", so how bad could it be?

While some of the topics in the class were fairly interesting, some other topics were boring and complicated and pointless. The homework problems were extremely difficult, which is not bad by itself. But, more importantly, upon reading the problem I would rarely see why this is a problem that needs solving. There would be so many steps, each one being a small result that I don't care about, which when taken together prove a larger result that I also don't care about. So, since I wasn't motivated to do the homework, halfway through the semester I decided to audit the class.

I did learn some cool stuff, though the specifics often eluded my understanding. I learned the basics of Banach spaces and Hilbert spaces. I also learned about bounded linear transformations; the term "bounded" here means that there is an upper bound on the growth of a vector when it is put through the transformation. In finite-dimensional vector spaces (that is, Rⁿ), all linear transformations are bounded; however, in infinite dimensions this is not always the case, and often it is necessary to consider only the bounded transformations. 

Another cool thing is about eigenvalues. In finite dimensions, all the values of λ for which A–λI is non-invertible are eigenvalues of A; you have probably used this fact many times if you have ever taken linear algebra. With infinite-dimensional linear transformations, there may be some values of λ for which A–λI is non-invertible and yet λ is not an eigenvalue. The former set is called the spectrum, and the spectrum always contains the eigenvalues—but not everything in the spectrum is an eigenvalue. The professor showed us an example of a transformation whose spectrum includes the real numbers in the interval (–2, 0) but which has no eigenvalues at all. 

There are several phenomena like these: two distinct concepts which always happen to coincide in finite dimensions don't always coincide in infinite dimensions. I'm glad that I learned what I did; it gave me a new perspective on some parts of mathematics. I also learned something about myself: analysis isn't really my cup of tea.

Verona photographs.

 

The ancient Roman arena in the early morning.

 

Some ancient Roman ruins.

 

Gnocchi (dumplings) with meat sauce.

 

Real Italian pizza.

 

A bridge from the side at night.

 

Every side street was beautiful.

 

The river is called the Adige (pronounced like a DJ).

 

An awesome bridge, some buildings, and the Adige.

 

The city at twilight, as seen from the San Pietro castle.

 

Candy and more candy.

Verona.

I went to Verona for two days. I do not know the language of Italian, but I may as well: it seems nearly identical to Spanish, which I do know. Most of the written Italian around the city was mildly comprehensible to me. I feel like I know Italian better than I know Hungarian.

This city is so old. I visited an ancient Roman arena, an ancient Roman theater, and two medieval castles. (It seems every European city has to have a castle; Verona has two.) One of the castles now contains an art museum, in which I saw a large collection of paintings, sculptures, and frescoes (although, in my opinion, the theme of Jesus was overdone). I also went to a photography museum located in an ancient Roman ruin. It was surreal to be able to see a crumbling stone archway and photos from the filming of Planet of the Apes at the same time.

I also visited a house which was supposedly inhabited by Juliet (from the Shakespeare play). I'm pretty sure Romeo and Juliet is a work of fiction, but apparently the house still manages to attract quite a lot of visitors. The house was made into a museum about Romeo and Juliet. There was a statue of Juliet outside the house, and I guess there is a tradition whereby people grope the statue's right breast for some reason. Juliet's tomb is also somewhere in Verona, but I didn't go see it.

I love Italian-cuisine food in America; now I know that I also love Italian-cuisine food in Italy. With the exception of one pasta dish that had a weird sauerkraut sauce, everything I ate here was unbelievably delicious. I had gnocchi (potato dumplings) with meat sauce one night, and I had beef tortellini the next night; both were exquisitely soft, almost fluffy, but definitely not mushy. I enjoyed the pizza very much; it was crunchy and thin, and I think it's incomparable with American pizza. The risotto with red-wine cream sauce is now the second purple food that I have tried and liked. Horse meat is a traditional Veronese dish; I never ordered any for myself, but I tried it when my friend got it. It tasted mostly like beef, but a little tougher.

I love ice cream, so of course the gelato was to die for (figuratively). It seemed like every street corner had a gelateria, and I just couldn't help myself. I ate chocolate gelato, tiramisu gelato, and nutella gelato. Another dessert I tried and loved was panna cotta, which was kind of like crème brûlée without the blow-torching.

The city's architecture was the most breathtaking I've ever seen. I saw astonishingly beautiful buildings every time I turned a corner. Verona's river, the Adige, was gorgeous and stunning and calm and blue. I love looking at water, and the Adige is the most beautiful water I've ever looked at. I still have to organize my photographs, and I'll post some of them as soon as I do.

Combinatorics.

Combinatorics is what BSM is most renowned for. For decades Budapest has been the mathematics hub of the world, especially for combinatorics. Paul Erdős, the greatest mathematician of the twentieth century, revolutionized the field of combinatorics; by the way, he was one of the founders of BSM.

Combinatorics is the mathematics of arrangements of things. An example of a fact from combinatorics is "The number of ways to tile a 2-by-n board with dominoes is given by the nth Fibonacci number." Another example is that there are no more than five regular (Platonic) solids.

I took a combinatorics class at my college last year, and I absolutely loved it to bits; thus I am taking a level-2 combinatorics class here in Budapest. The class focuses on hypergraphs: things that are like graphs, except that each edge, rather than consisting of two points, can be a set of any number of points. In the course of studying hypergraphs, other topics also come up.

One of my favorite topics was the following: Given a set of n objects, consider the subsets of size k, and partition them into families so that, within each family, every pair of subsets has an object in common. What is the smallest number of families for which this is possible? This is known as the Kneser problem. This can be stated in terms of hypergraphs, and also in terms of traditional graphs. It turns out that the answer is n–2k+2, but proving it is not easy. The proof takes a huge detour into the land of topology, considering the n objects as points on a sphere and covering the sphere with the subset families. Or something.

Another topic I especially liked was the probability method. This method, pioneered by Erdős, is used to prove the existence of a given kind of structure. Using principles from probability, you show that the probability of a random structure being of the desired type is greater than zero; this means that at least one such structure must exist. For example, Erdős used the probability method to prove the existence of a certain edge-coloring on the complete graph. [More specifically, this gave a lower bound for the Ramsey number R(n).] In general, the flaw of this method is that it cannot construct the thing for you; it can only tell you that it exists somewhere.

McDaniel College and other things.

My math program (BSM) takes place at McDaniel College Budapest, which is an international college campus. McDaniel hosts a variety of international and study-abroad programs, one of which is BSM.

I don't really know anything about the other programs McDaniel hosts. I do know that we share the building with a bunch of international students who appear to spend all their time smoking right outside the front door. They don't step aside when people are trying to enter or exit the building; in fact they often don't move at all. So every day, before and after class, I have to wade through a pile of oblivious smoking college students.

McDaniel makes up for this infestation by having a piano in the basement. It's not a very good piano, but I've never been one to care about that. It's a good stress relief for me to go downstairs after class and play around on the piano. On the overnight train to Prague two months ago, I wrote myself a lullaby because I couldn't sleep; I've recently been spending some time at the piano working out the details of the lullaby. It's coming along nicely.

I'm really happy in Budapest, but I'm also starting to miss home. It will be nice to be back in America in a month.

Taking pictures of quadrilaterals.

When I wrote about my Geometry class in a previous entry, I said, "I don't know anything about projective geometry." Well, today we started projective geometry. In a nutshell, it's the geometry of perspective drawings. If lines are parallel in real life, you draw them so that they actually meet on the paper, at the "vanishing point". It is also the geometry behind photography, and one of my homework problems is:

"Show that any convex quadrilateral can be photographed in such a way that in the picture we see a parallelogram."

This sounded like it would be fun to do in real life. Here are the results:

The opposite sides are parallel, right?

Wrong. This is the same drawing, head-on. Not quite a parallelogram.

Heidelberg, Germany.

I had a long weekend, so I visited a friend in Heidelberg, Germany. (She's an American studying abroad there for a year.) I stayed for three days and saw some cool things and some beautiful things.

Heidelberg is a relatively small city. It doesn't have tons of big attractions to go see. Rather, I get the impression that it's the kind of city where you just go explore and see what little wonders you can find. So that's exactly what I did. This is what I found:

We climbed up a big hill with lots of pretty autumn trees. I can't remember if we got to the top, but this is what our view was like:

I really really like that bridge. It looks like it comes straight out of an Impressionist painting. By the way, the river in Heidelberg probably has a name, but my friend and I called it the Slurpy river, on account of the sound it was making while we were walking along it one night.

The highlight of my trip was seeing the student jail. Heidelberg is a major university city in Germany, and back in olden tymes the university had legal jurisdiction over its students. When the students broke the rules, they would sometimes be put in the jail. Eventually it became a rite of passage for the students, and they would harass officers or behave rowdily, with the specific intention of going to jail for a few days. Nowadays, the jail is open to the public. Graffiti have accumulated over the centuries; it's really quite a sight.

I don't know German, so I couldn't read the graffiti. My friend, who does know German, told me that a lot of what we were seeing was really weird poems.

I leave you with another photo of the majestic Slurpy:

Geometry.

Geometry is my favorite class this semester. In the geometry class I'm taking, the main focus is not on circles, triangles, and so forth. Our objects of study are isometries: space transformations that keep distances the same. For example, in plane geometry, some isometries are translations, rotations, and reflections.

We can combine two isometries of a space by applying one after another, resulting in a third. The result is that the set of isometries forms a group of transformations that act on the space. We can then study the abstract algebraic properties of the group of isometries on a given space: how isometries interact, the general form each one takes, what its fixed points are, whether it reverses orientation, and more. In learning about isometries, we find out about the geometric properties of the space.

For the first half of the semester, we studied Euclidean space. Rather than confine ourselves to two dimensions or maybe three, our class covered general n-dimensional Euclidean space (characterized by the well-known Pythagorean distance formula). Each Euclidean isometry has a part that "rotates" or "reflects" (kind of), followed by a part that translates (slides) the space. The rotational or reflectional part can be represented by a matrix, so we can deal with the Euclidean isometry group through linear algebra. (For those of you who know linear algebra: remember orthogonal matrices? The linear part of an isometry is an orthogonal matrix, so the algebra of orthogonal matrices comes up a lot here. Your knowledge is useful!)

The class applied these ideas to learn about symmetry groups of three-dimensional solids (polyhedra); a symmetry group consists of the ways that a solid can be rotated or reflected. In particular, each of the five Platonic solids has a rotation group which can be characterized by permutations of certain parts of the polyhedron. For example, the rotations of the cube are identified with permutations of the four long diagonals, so we find that each cube rotation is equivalent to a reordering of a set of four things. This was one of my favorite parts of geometry, because of these ideas; I find them exceedingly nifty and compelling.

We just started with spherical geometry, and this seems like it's going to be really interesting, too. The geometry of the sphere follows naturally from the geometry of Euclidean space, because we can consider the sphere as an object in Euclidean space. Later, we will learn about hyperbolic geometry and projective geometry. I have had some exposure to hyperbolic geometry before (it is pretty much the opposite of spherical geometry), but I don't know anything about projective geometry. I am looking forward to these exotic new lands with exotic new distance formulas!

P. S. The following math video is not quite relevant to geometry, but I figured I'd post it anyway because it's so beautiful. There are two parts.

More on graph theory: List colorings.

For those of my readers who know some combinatorics and have seen graphs before, I'd like to say a little bit more on one topic from graph theory that I've been learning about. If you aren't so interested in math and don't care about graphs, then you can skip ahead and read the last two paragraphs. The topic I want to tell you about is list coloring, which is a generalization of coloring a graph.

As you may know, a graph's vertices can be colored such that no two adjacent vertices are the same color. Now let's restrict how we can color the graph: given a graph G, at each vertex v of G, put a list of colors L(v). We now properly color G such that each vertex v has a color from its list L(v); this is called a list coloring of G.

The chromatic number χ(G) is the smallest number of colors with which G can be properly colored, without list restrictions. We can define a similar notion for list colorings, considering the size of each color list. The list-chromatic number, or choice number, is the smallest r such that G can be list-colored for every possible system of lists of length r. We denote this as ch(G). Note that this number gives the length of each list, not necessarily the number of colors we use; this confused me at first.

The choice number and chromatic number are related: for any graph G,  χ(G) ≤ ch(G). Do we ever have χ(G) = ch(G)? In my graph theory class we learned that, if G is the line graph of a bipartite graph, then χ(G) = ch(G). (By the way, the proof of this theorem uses the stable-marriage existence theorem!)

This theorem is applicable to an important combinatorial problem: Given an n-by-n matrix where each position carries a list of length n, is it possible to fill each position with an item from its list in such a way that each row and each column has n distinct items? If we have {1, …, n} at each position, then it is possible; such a matrix is called a Latin square. But can this be done for any system of lists?

This problem can be solved by constructing a graph with the entries of the matrix as vertices and drawing edges between two vertices if they are in the same row or column of the matrix. Then the problem becomes: is χ(G) = ch(G) for this graph? The graph is actually the line graph of the complete bipartite graph. So, by the theorem I mentioned earlier, there is a list coloring; the answer to our question is yes, we can always select items for the matrix, no matter what the lists are.

One problem on my graph theory homework this week was to prove that, if all of the positions of the matrix have length-n lists except one list which is length-(n/2), then a proper choice is not always possible. This means that we have to construct such a system of lists on the n-by-n matrix, such that it is impossible to choose n distinct items in each row and column.

This is perhaps the hardest problem that I had so far been given in graph theory, and I didn't really know where to start on this one; fortunately, neither did my two roommates. So last night, we sat down together for a few hours and tried to construct the list-system. We made lots of sketches and guesses, most of which didn't work. But over time, we got to know the problem and understand parts of it, and eventually one of us had a breakthrough. One of my roommates found a list system on the 4-by-4 matrix that worked; then the other roommate found a way to prove why it worked (using the pigeonhole principle!); and then I was able to extend the construction and the proof to the general n-by-n case. We had solved the problem—but more importantly, we had fun with it. It was exhilarating to try different approaches and finally find one that worked, and I was really happy that I had friends that shared this feeling with me.

That's what I love most about the BSM program: I always have math to do, I always have people to do the math with if I want, and we always have tons of fun doing the math. I feel like I'm swimming in math, and every day I am even more sure that this is what I want to be doing for the rest of my life.

Graph theory.

Everyone thinks they know what a graph is, but most people don't know what a graph is. Do you know what a graph is? You will definitely know what one is after I tell you. A graph is not the thing you had to draw in high-school algebra. A graph is also not a giraffe, contrary to popular belief. A graph is something like this:

The Petersen graph: one of the most famous graphs. (Image from Wikipedia.)

A graph is a bunch of dots connected by lines or curves. The dots are called vertices, and the lines and curves are called edges. This is all there is to a graph. In particular, a graph is generally not concerned with distances or orientations. The above graph, the Petersen graph, is exactly the same graph as this one:

Also the Petersen graph. (Image from Wikipedia.)

But it looks different! It doesn't matter, though; it's still the same graph. The only thing that matters for a graph is which vertices are connected by edges. It turns out that, in the pair of graphs in this paragraph, you can label each vertex with a number in such a way that the same numbers have edges between them in both versions of the graph. That's how you know they're the same graph. In fact, this was one of my first homework problems in my graph theory class.

I think graph theory is really fun. The problems I get in the class are challenging and interesting—the best possible combination. The professor gives us information in lecture, but then on the homework we have to use what we learned to discover new things, in a way that isn't always immediately clear.

The class has spent a lot of time on Hamiltonian cycles. A cycle is a path through the edges of a graph that doesn't repeat any vertices and ends where it began. A Hamiltonian cycle is a cycle that reaches each vertex once. Not every graph has a Hamiltonian cycle, and an interesting and important part of graph theory studies the question of which graphs have Hamiltonian cycles and which ones don't.

Another really cool thing I've learned about in class is graph coloring. To properly color the vertices of a graph, you assign each one a color in such a way that there is no edge between two same-color vertices—that is, the endpoints of an edge are different colors from each other. Any graph can be properly colored (just put a different color on each vertex), but sometimes we want to know the least number of colors that are needed to properly color a graph. This number is called the chromatic number of the graph.

Graphs are also used in the "stable marriage problem": given a set of people, each having a preference ranking of some of the other people, is there a way to pair off the people so that no two people both prefer the other over their partner? This problem is represented as a graph: each person is a vertex, and each pair of vertices has an edge if it is a possible partnership. Then each vertex has a ranking of the vertices that it's adjacent to. In the traditional scenario, where there are men and women that are to be married off, the answer is yes, there is a stable marriage. In this case, the algorithm for finding a stable marriage is incredibly amusing: it is best conceptualized in terms of one gender making proposals to the other, who accepts or rejects each proposal.

That's about it for giraffe theory. I'm sorry I haven't been updating my blog very often; I'll have to shape up soon, because my next entry will be about my geometry class. HA HA HA.

Learning mathematics.

Typically, a student in my program will take between three and five math classes. This semester, I am taking five math classes. This is what they are:

• Graph theory;
• Geometry;
• Combinatorics 2;
• Functional analysis;
• Mathematical logic.

I'll post about each one individually. For now, I'll tell you about my general experience with classes.

There are 26 math classes offered this semester (if I counted correctly). You can see the schedule and list of classes if you want. There are also a few other classes, such as: Hungarian language, some history class, and political philosophy—but of course I'm not bothering with being well-rounded.

Typically, each math class starts at n:15, goes until (n+1):00, has a fifteen-minute break from (n+1):00 to (n+1):15, and continues until (n+2):00, for n = 8, 10, or 12 depending on the class. There are a lot of cheap and good places to eat that are near the school, so sometimes during a fifteen-minutes break I go get a snack or lunch. On days when I have class from 8:00AM to 2:00PM, I've been in the habit of getting falafel to-go, from this restaurant right across the street.

This entry was supposed to be about math classes and now I'm talking about food yet again. Oops.

The classes are pretty much the same as college classes in America, with a few important exceptions. First, the chalkboards have two panels, and when one panel is lowered the other panel rises. You can write stuff on the first panel when it's in the lowered position, then lower the second panel—thus moving the stuff you already wrote out of the way—and then write more stuff on the second panel. It's really cool.

The professors aren't really available outside of class. They typically have only have one office hour each week, and it takes place during usual class time. It's not so much an office hour as it is a supplementary session of class; professors spend the hour answering questions about the class material or the homework, usually in front of the whole class. I haven't really needed one-on-one time with my professors (not yet, anyway), so this hasn't been such a big deal for me. It is nevertheless a sharp contrast to my college, where I can just walk into a professor's office and have a conversation pretty much whenever I want.

The professors are Hungarian, and they know varying amounts of English. Some professors know just enough English to give math lectures, but they can sometimes be unintelligible. Other professors have nearly perfect English, with just a slight accent. There is one professor who speaks acceptable English, but his words are obscured by his impressive moustache.

Course registration did not happen until three weeks into the semester. This is because, in the Hungarian system, students "shop around" at various classes for the first few weeks, and then they decide what to take based on actually going to the class. It's a really nice system that makes the decision a lot easier, but it also makes it extremely hectic during the first week or two when we're trying out like nine classes.

Next journal entry, I'll talk about graph theory. If you have never heard of graph theory, then you probably don't know what a graph is. But soon you will find out!

Miscellany.

I've started drinking milk and Nutella mixed together. The result, of course, is Nutella milk. It tastes delicious. I found a shop by my apartment that sells Nutella in 400-g (14-ounce) containers for just 550 forint. That's 2.60 dollars. I never knew happiness could be so inexpensive.

Since I can't really cook, my two roommates cook most of our meals, in exchange for which I do the dishes. Thus, I started a tradition in my apartment: while washing the dishes, I put on Brahms's Hungarian Dances at a high volume. It makes the chore feel much more dramatic.

Upon searching the web for Budapest delivery pizza, I discovered that there exist Pizza Hut locations in Budapest. Of course, I immediately went to the nearest Pizza Hut for dinner. It turns out that, in Budapest, Pizza Hut is a fairly classy sit-down restaurant. I ordered a garlic–chicken pizza. It obviously did not compare to Chicago deep-dish pizza, but it was nevertheless very good.

I'll write about my math classes in my next entry. For now, here's an interesting problem from my geometry class: give a bounded subset B of the Euclidean plane and an angle t such that the rotation of B by angle t is a proper subset of B. I couldn't think of an example on my own, but I later found one on the internet that was absolutely amazing.

Chocolate Festival and the Opera.

I did two big things today. First, I went to the Chocolate Festival. Then, I went to the Opera.

The Budapest Chocolate Festival is exactly what it sounds like. There were about a hundred stands outside, each one selling different kinds of chocolate. There were caramel chocolates, hazelnut chocolates, marzipan chocolates, chocolate candy bars, chocolate gelato, fruit-filled chocolates, wine-filled chocolates, free samples of chocolate, chocolate biscuits, big chocolates, small chocolates, more chocolate than my body had room for. Metaphorically, anyway. Probably the best thing I tried was a chocolate filled with marzipan and nutella in layers. I had a breathtakingly delicious afternoon.

The Opera was good, too. I went with a few friends to Budapest's beautiful and famous opera house. We saw Don Pasquale, by Donizetti. It was in Italian with Hungarian surtitles, so I doubly didn't understand it. Fortunately, the program had a summary in English, so I vaguely understood what was supposed to be happening. The music was good, and so were the performers. This was the first time I had ever been to an opera, and hopefully not the last.

And now, the moment I know many of you have been awaiting: my next journal entry will be about my first week of math classes!

Not the right question.

I was riding the metro the other night when I overheard a couple speaking American English. They were trying to figure out when their stop was; knowing the metro pretty well, I asked them where they needed to get off, and it turned out to be the next stop.

They asked me, "Do you live here?"
I do live here for the semester (I got my residence permit a week ago!), so I said yes.
They then said, "Your English is very good."
Rather embarrassed at the misunderstanding, I said, "Well actually I'm American."

It turned out that they were here on a cycling tour (I assume they meant bicycling, because do they even make tricycles for adults?) and had just been in Bratislava. It also turned out that one member of the couple grew up in the same state that I'm from in America—and here we are, meeting in Budapest. What a huge small world.

Also, people need to pay attention to the difference between "living" and "being from".

THE ICE-CREAM EXCHANGE RATE.

I mentioned in a previous post that the meals here are significantly cheaper than in America. Thus, while a ten-dollar meal is cheap-to-medium by American standards, it is medium-to-expensive in Budapest. Thus, the official exchange rate of 200 forints to 1 U.S. dollar doesn't really give a good sense of how expensive a meal is, in the context of other Budapest meals.

I have remedied this situation by choosing a relatively stable food item and defining it to be the same price in both locations. Ice cream seemed like the obvious choice. I estimate that a nice cheap scoop of ice cream costs 2 dollars in America and 150 forint in Budapest. Thus, by the ice-cream exchange rate, we define

2 dollars = 150 forint

(this would look so much better in Latex). Thus, there are 75 forint to the dollar.

Let's see how this works in practice. Suppose I buy a meal for 2100 forint. By the official (but misguided) rate, this is 10.5 dollars—pretty reasonable, right? However, by the ice-cream rate, my meal is 28 dollars. Golly!

Well, I admit that ice cream could be cheaper in Budapest by a significantly greater degree than most other food is. Maybe. More research has yet to be done on how much of a difference this makes. Until then, I'm going to use this as an excuse to talk about ice cream and be a cheapskate.

Prague post.

We had a few free days between the end of the language program and the beginning of the math classes. Thus, some new friends and I went to Prague for three days. (The photographs in this Prague post were taken by me!) We took an overnight train and arrived in Prague at 4:30 in the morning. We were really tired and stressed out at first, but our trip got Pragressively better. We Czeched out the famous Charles Bridge, toured the Old Town, went inside a stunning cathedral, and walked the grounds of the royal palace. We also paid our respects at the John Lennon wall, a huge wall of Lennon- and Beatles-themed graffiti:

 One of the best parts of my trip was the food. We had a lot of traditional Czech cuisine, including several varieties of meat, with delicious creamy gravy. And the dumplings. Oh, the heavenly Bohemian dumplings.

We also toured the old Jewish quarter. There were lots of interesting museums and synagogues, with historical Czech-Jewish artifacts. Many of the old ritual objects were ones that I recognized from growing up in a Jewish family; I felt like I was connecting with my Eastern-European–Jewish heritage.

We came across a shop selling absinthe ice cream. I did not partake, but some of my friends did. The ice cream looked creepily green. My friends said it didn't taste like anything—it just burned. So that was a fun experience.

One last thing, Prague over the Danube at twilight:

Food.

The teacher of my Hungarian class told us about Túró Rudi, a candy unique to Hungary. It is a chocolate bar filled with cheese curds. Intrigued, I bought some, and enjoyed it immensely. I would bring some with me when I come home in December, but I'm pretty sure they have to be refrigerated.

One day, the ice-cream shop by my apartment was selling Túro-Rúdi–flavored ice cream. I love ice cream, and I love Túro Rúdi, so surely I would love this flavor. But I didn't. I am glad that I tried it though: it was an interesting taste experience, and now I know that I shouldn't put cheese items in my ice cream.

I really like the ice-cream shop by my apartment. They always have the basic flavors, like chocolate, but I'm pretty sure they switch up the other flavors that they offer. There were a few days when they had a Tiramisu ice cream; I got that once and it was amazing. And the ice cream is pretty cheap, too—only 140 forint per scoop.

At this point I should explain Hungary's money system. The currency is called the forint. The exchange rate is approximately 200 forint to 1 dollar. As a result, I'm a millionaire in forints. There are coins worth 5, 10, 20, 50, 100 and 200 forint (there are no 1-forint coins). There are bills worth 500, 1000, 2000, 5000, 10000 and 20000 forint. A medium-price meal in Budapest tends to be about 1000 forint. However, note that this is only about five dollars! Also note that the ice-cream cone costs the equivalent of 0.70 dollar, which is ridiculously cheap in the United States. This is my motivation for the ice-cream exchange rate—but more on that another day.

The chicken here is really good. This is fortunate, since chicken is my favorite animal to eat. One time I got chicken with a paprika sauce. Another time I got chicken with a cheese sauce. Both were excellent. The cheesy chicken in particular gave me that distinct mix of feelings that arises from eating something incredibly delicious but also really unhealthy. It was a satisfied contentedness, mixed with guilt and regret, mixed with astonishment and wonder. If "feeling the fat clogging one's arteries" is an emotion, then add that to the mix as well. I'm not sure if this unique emotional state has a name. Let's call it foojitty.

I also felt quite foojitious when I got a palacsinta (crêpe, basically) filled with cheese and topped with cheese and sour cream. I got it at Nagyi Palacsintázója (English: Granny's Pancakes), which I mentioned in my first post. The banana–nutella palacsinta that I got there was the best breakfast food I've ever had. It's really cheap, too: depending on the kind of palacsinta, it costs 200 to 400 forint apiece.

Budapest has a lot of gyro restaurants. I get the feeling that gyros in Hungary are the cultural equivalent of Mexican food in America. Just like America, Budapest has a lot of Chinese-food places, including fast-food ones. In fact, my first meal in Budapest was at a Chinese buffet. The other day, I went to a Thai-food restaurant and ordered something that ended up being pasta with turkey and garlic, Italian-style. So I've eaten Italian food from a Thai restaurant in Hungary. I'm so multicultural.

Magyar, the language of suffixes.

English and Spanish, the two languages I know, are both Indo-European languages, along with hundreds of European and Indian languages. On the other hand, Hungarian (magyar) shares a language family with just two other major languages: Estonian and Finnish. That is, Hungarian is a really weird language.

I've been taking an intensive two-week introductory course on Hungarian which is supposed to be equivalent to a semester-long course. It's different though: with the two-week course, I don't have a whole semester to let everything sink in. I've pretty much been submerged in Hungarian for days. Fortunately, I find the language really interesting. I will now talk about it, for the benefit of my linguistics friends.

The alphabet is an extended Latin alphabet: in addition to our 26 letters, there are also ö, ü, gy, ty, ly, ny, cs, sz, zs, dz, and dzs. Most of these extra letters are pairs of characters, but in Hungarian they count as letters since they make their own distinct sounds. Also, the seven vowels all come in short and long forms; the long form is denoted by putting an accent mark over it. The hardest letters to pronounce are gy and ty, which make sounds that are not found in most other languages. Fortunately, knowing how a Hungarian word is spelled is enough to know how to pronounce it. That is, the spelling determines the pronunciation, unlike in English (cough dough rough through). A Hungarian word's stress is always on the first syllable, no exceptions.

Everything in Hungarian is a suffix. You want your noun to be a direct object? Add -t to the end. Indirect object? Add -nak or -nek to the end. You want your noun to belong to me, you, him/her? Add -m, -d, -ja respectively to the end. You want your verb to denote an action that I do to you? Add -lek to the end. A lot of nouns can be made into verbs by adding the ending -zik: for example, "piano" is "zongora", and "plays piano" is "zongorázik". A lot of prepositions are made using suffixes, too. For example, to say "in [noun]", just add the suffix -ban or -ben to the noun; to say "with [noun]", add the suffix -val or -vel to the noun. There are more suffixes that I learned about, and even more I'm sure that I haven't learned about.

Hungarian is a beautiful language, partly because of the principle of vowel harmony: each word's vowels are either all low vowels (a, o, u) or all high vowels (e, i, ö, ü). All of the suffixes come in high-vowel forms and low-vowel forms, and the one you use depends on the harmony class of the word. For instance, "in the bag" is "táskában", whereas "in the cabinet" is "szekrényben". This principle gives words a nice flow and makes Hungarian elegant and interesting.

One more thing: Hungarian does not have gendered nouns, like many languages do. In fact, Hungarian doesn't even have gendered pronouns (he or she). Take that, English.

Budapest Semesters in Mathematics.

I apologize for the length of my first entry. This entry will be shorter, I'm sure.

I am spending a semester in Budapest, Hungary, for the purpose of studying mathematics. I am one of about 70 undergrad math students from America and Canada on the Budapest Semesters in Mathematics (BSM) program. This program takes us to Budapest because many of the most important twentieth-century mathematicians are from Hungary—most famously, Paul Erdős. The program is hosted by a university in Budapest. If you're interested, you can read all about BSM on their website.

From what I've heard, the combinatorics and graph theory here are among the best in the world. The BSM professors are Hungarian, but they will teach in English. We don't choose classes until two weeks into the semester; before that point, we just try out different classes and see what we like the most. For those of you that care, I am thinking of taking: combinatorics, functional analysis, geometry, graph theory, and mathematical logic.

There is also a class called Conjecture and Proof, in which we just get miscellaneous, insanely difficult math problems to work on. I've heard lots of good things about it, but I've also heard lots of scary things. Apparently a lot of people drop the class before halfway, because it gets too hard. I may end up taking this class, but it's more likely that I'll just sit in on it and do whatever problems I feel like doing. Mostly I just don't want it to take away from my enjoyment of other classes or of life.

All this stuff is really exciting to me. I can't wait to start the math. However, before then, I am taking a two-week intensive Hungarian-language course. In my next entry, I will talk about the Hungarian language and my experiences with learning about it.

How I sat in a flying metal box for a day and ended up in Europe, and what ensued.

I find air travel literally incredible. I look out the window when I'm on an airplane, in sheer disbelief that this thing can actually fly—with people inside, even. I tried reading the Wikipedia page on lift, and it just made me more confused. (By the way: the thing about air having to take the same amount of time to pass over both top and bottom of the wing—that's incorrect.) It amazes me that I can start off in Chicago, and just eight hours later end up in Frankfurt, Germany, without even having to walk. (My connecting flight was in Frankfurt.)

The Frankfurt connection happened without a hitch. In the airport there, I had only to have my passport stamped by some German guy. I'm not sure whether that counted as customs. I guess I didn't need the three hours layover after all, but one can never be too careful. From there, it was a two-hour flight to Budapest, Hungary, where I would be living for the next four months. (Customs at Budapest's airport, if it existed at all, was nearly undetectable; I went up to some officer who just waved me through without even checking my passport.)

Budapest is famous for its paprika. For dinner on the first day of our visit, my two roommates and I found a restaurant called "Paprika", about a third of a block from our apartment. I had chicken with paprika, and it was really good. Earlier that day, we had had lunch at a Chinese fast-food place, which was not memorable. It is amusing, though, that my first meal in Budapest was Chinese food.

It was hard, not knowing where stuff is. Our apartment wasn't destined to have internet until three days after we arrived, and we needed internet in order to contact our parents and check for important emails from Budapest people. So we had to wander around the city and look for internet at cafés or bars (or internet cafés, which are apparently not just cafés that have internet). This was really nerve-wracking to have to do while jet-lagged on my first day in a country where I didn't speak the language and I didn't know how anything works or where stuff is. We walked into many a locale, only to have unintelligible conversations with Hungarians and walk out in shame.

We also got lost more often than things that happen several times a day happen. After my first day in Budapest, I was really nervous and scared and uncomfortable. Fortunately, things quickly got better, as I became acclimated to the city. It turns out that getting lost is a good way to learn one's way around.

From reading a Budapest guidebook, I found a restaurant that serves stunningly delicious pancakes at dirt-cheap prices. Although they are called "pancakes" on the restaurant's English-language menu, they were actually crêpes. I got one with bananas and honey. It was like heaven, but easy to get to (just take the metro to the other side of the Danube). By the way, the Hungarian word for pancake/crêpe is "palacsinta", pronounced "PA-la-chin-ta".

Coming up next: the study-abroad program that I'm in, Hungary's money system, and the Hungarian language.