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.