Setmana Catalana

This past week was spring break, and not so coincidentally, it was also one of the most perfect weeks I’ve had in a long time. I headed out to Catalunya and visited with almost-relatives and old friends. I can’t even attempt to give a play-by-play of the week, but I’ll try and come up with a highlight reel. For the most part I was in Barcelona, staying with the parents of my Catalan aunt, Salvador and Vicenta. I knew I had made some good spring break plans when I showed up at their apartment on Sunday night, literally on the waterfront in the Barceloneta neighborhood. My first night there I spent sitting in their living room, listening to the waves. On Monday I went for a stroll, heading by the Plaça Catalunya and the Passeig De Gracies before finding a place in front of the Born market to have a coffee and read. That night I met my “pseudocousins” Julia and Claudia, who took me around the city a little bit. Tuesday morning I went for a fantastic run along the beach, and then I visited the Centro de Arte Santa Mónica, a smallish art museum at the foot of the Ramblas. The coolest thing I saw there was a collection of sculptures made as a project in an architecture school. What makes these so interesting is that they were all inspired by various microorganisms. What’s more, there is a good deal of mathematics that goes into their design; some were made by creating a triangular pattern, covering an icosahedron with the pattern, then projecting this onto a sphere to make the final sculpture. You can find some photos on Facebook.

Wednesday I traveled to Vic to visit with Ignasi, Goreti, their son Alan and his girlfriend Silvia, with whom I stayed in 2005. Ignasi and I caught up briefly but a Barça game against Arsenal was due to start soon so we rushed off to meet up with Alan and Silvia. I watched the game with him in a bar along with thirty or so of the most maniacal soccer fans I have ever met in my life. Barça ended up tying the game, but since they scored two goals in an away game, my primitive understanding of league soccer rules leads me to believe that this was a pretty good outcome. Thursday morning Ignasi and I drove over to Manlleu to visit with my old guitar teacher. We quickly established that I remembered absolutely nothing of what he taught me and that further, several months of being away from the guitar wreaked certain havoc with my basic skills, but all was forgiven and he even gave me a copy of his CD. I had a nice lunch with everybody before taking the bus back to Barcelona for a final night with Salvador and Vicenta. Salvador and I watched the Barcelona basketball team handle Madrid to advance to the European Final Four.

One constant feature of the trip was a sort of experimentation among my hosts to see how much food they could get into my stomach – it was here that I experienced my very first four-course breakfast. All of the food was of course fantastic and the quantity of jamon serrano that I consumed brilliantly reaffirms my belief that pig is in fact the tastiest animal. Pa amb tomaquet remains a perennial favorite (it means “bread with tomato” and is precisely that, plus a little olive oil and salt. Try it – it will rock your world). Other discoveries include fuac, a type of cured sausage, and the simple power of the truita frances - an omelette with potatoes and caramelized onions. It can also be safely said that Spanish wine is orders of magnitude better than its Hungarian competitors.

Quick Updates

I (barely) survived midterms week and with the chaos settling behind me I figured it was time to produce a brief “Yep, I’m still alive” type-post, especially given my travel plans, as I’ll be in Spain from Sunday. I haven’t gotten up to much in the past couple of weeks that would bear discussion here; mostly I’ve been settling in to the city, finding the places I like to go to regularly. I’ve found that Thursday night pool makes for a nice tradition; at the very least it reminds me of pool tournament night back at Reed. I gave up on the first pool hall I found here because it was far away and always very crowded. In its stead, I’ve started going to a place called “Amigo”, which is one of the stranger places I’ve found in Budapest in that it is a rock ‘n’ roll bar, with giant pictures of Elvis and Chuck Berry on the wall and a soundtrack that stops at about 1959. Apparently there is also some level of a rock ‘n’ roll scene still thriving in Budapest, as

Venice

A little over three hours elapsed on Friday afternoon in between my friend bringing cheap train tickets to Venice for the weekend to my attention and six of us pulling out of Keleti Palyudvar with a fourteen-hour, four-country train ride ahead of us. Rather last-minute, yes, but it made for a fantastic couple of days and a good way to spend a three-day weekend (Hungary is celebrating National Day today). You can check out the photos on Facebook here.

Yes, the train ride was fourteen hours long, and no, we didn’t spring for a sleeper car, which made our Friday night somewhat more uncomfortable than most. What little sleep we were able to get was interrupted by a slew of border crossings: Croatia (actually the Croatians boarded the train and stamped our passports at two different stops), Slovenia, and finally Italy in the wee hours of the morning. We arrived in Venice at around seven in the morning on Saturday. Our first objective was breakfast, which we found about forty feet from the train station. It was nothing special and cost something like eight euros, the first of many indications of just how touristy the city is. Luckily, our hotel was a fantastic find: it is still under construction, so we were able to book a room for six for twenty bucks each. We dropped our bags off and set off to talk a walk around the city.

This would be the part of the post where I’d gush about how beautiful the city is, how unique, how charming, etc. I’ll let the photos speak for themselves in this respect: there’s no denying that the city is every bit as beautiful as everybody says it is. What I hadn’t realized, and what eventually started to really bother me, is just how overrun the place is with tourists. This put me in a unique position: just like the 55,000 other people tromping noisily about the city, I was part of the problem. Just how big of a problem is it? This is a city that has fallen into a sort of Shoe Event Horizon: there are exactly five types of business in the city: hotels, restaurants, gelato stands, luxury-goods stores, and combination Murano-glass-and-Venetian-Mask shops. That’s it. There is nowhere to escape from the beaten path. There are no low-key, out-of-the-way spots that you can spend a couple of stress-free hours in, so crucial to a weekend trip in a city. It’s a nonstop barrage of timeless beauty and tourist traps; an exquisite Disneyland.

With that little rant out of the way, I suppose I should mention a few highlights, primarily food. We ate dinner Satuday night in a dime-a-dozen tourist restaurant: the food was delicious but again the place felt more like “Venice” than Venice. The real culinary highlight of the trip was lunch. Saturday we found a sandwich shop right by the Rialto Bridge which sold delicious little sandwiches for 1.50*, the (*) being because they conveniently forget to inform you that this is the take-away-price, and crank up the price by a euro for the privilege of sitting at a table. But even for 2.50, my sandwiches were fantastic: I had one with prosciutto and porcini mushrooms and another with brie and some kind of meat which reminded me of jamon serrano. I have seen bold new vistas in sandwichery open before my eyes. Gelato is ubiquitous and universally delicious (hard to imagine bad gelato) and I made the marvelous/dangerous discovery that hot chocolate goes really really well with Disaronno.

The other thing I really enjoyed was the Peggy Guggenheim Collection: this is housed in her private villa on the Grand Canal. There’s a sculpture garden outside which looks fantastic in the late afternoon light, and inside is her private art collection. The first thing you see when you walk in is a Calder mobile flanked by a couple of Picassos: straight ahead is a terrace which leads right out onto the waterfront. I really enjoyed the Yves Tanguy works: like Dalí, I like the fantastic landscapes that feel empty and unsettling, yet still manage to transfix. Magritte was another favorite: I saw Empire of Light and The Voice of Space. Umberto Boccione’s sculptures were very engaging: they seem to wrap through themselves, almost as if they are higher-dimensional objects pulled down into three-space.

We finished our Sunday strong, by finding the cheapest pizza shop we could. We took a final stroll around the city before heading to the train station for another fourteen-hour sojourn, this time accompanied by a cadre of incredibly talkative Italians who finally disembarked around three in the morning in the middle of Slovenia. All told, Venice was unspeakably beautiful but somehow sad: I feel like there isn’t really a city there any more, only a sort of city-sized museum gift-shop.

Buda Afternoon

I’ve just uploaded a bunch of photos I took yesterday afternoon in Buda to Facebook. I walked across the Chain Bridge into Buda, up into the hill by the castle, and down into the heart of Buda. It really was a fantastic little walk: even if it was cold, it was the kind of breezy cold that carries a hint of spring, and it helped that it was gloriously sunny out, especially because it looks like we’re going to get some snow tomorrow and Friday.  Check out the photos if you are interested: I added some captions so you should get a pretty good sense of what my walk was like.

Cafés

Last time I gave a quick run down on the kinds of food I’ve been eating here, Hungarian and otherwise. In keeping with what appears to be my new series, this time I’ll talk about good places to get good coffee.

I gave up on espresso a couple of years ago when it became clear to me that my demands far outstripped my equipment and my abilities: espresso takes a good machine and a good enough palate to be able to adjust the brewing process to taste, neither of which I really had. So I switched my focus to “American” style drip/French-press-brewed coffee, where the equipment is less expensive and more forgiving to novices without a really acute sense of taste. Plus it helped that I had the Stumptown Annex one beautiful bike ride away, which meant free cuppings every day (twice every day, actually!) and the world’s best cups of coffee. A couple of years and an unthinkable amount of pocket money later, I found myself in Europe where American-style brewed coffee is practically nonexistent – even where it does exist, the protocol in Europe is to roast the bejeezus out of it so that any interesting features in the beans disappear. So I started drinking espresso again. Apart from being able to tell what I like to drink and what I don’t, I haven’t really been able to distinguish much between different blends or roasts, which has gotten me curious – I’m sure there are tons of different things you can do with an espresso, just like different cups of brewed coffee can taste totally different. So I’m definitely in the exploratory phase, café-wise. That being said, there are places around town that I seem to be spending more and more of my life at these days, and they deserve a mention here.

First and foremost is Sirály Café on Király, where I often get my work done. For the price (280/400 Ft single/double espresso) the coffee can’t be beat, and there is a large and well-lit upstairs area with plenty of tables to spread out and spend an afternoon wrestling with Conjecture and Proof (which has become an official Monday tradition of mine). One website I came across described the atmosphere as “no-frills” but I disagree completely – it’s warm and inviting and feels exactly as a café should – it’s not flashy because it shouldn’t be. At least on the upper floor there is often more English than Hungarian being spoken; I’ve noticed a definite linguistic split between up- and downstairs. One of these days I’m going to stick around for the live music in the evenings: I’ve heard rumors of “Polish clarinet-driven jazz, French noise pop, and more”.

Closer to campus, on Rottenbiller, is Billy Rotten (get it?). A little sleeker (or at least with a more self-conscious attempt at decor), there are murals and designs painted on the walls which make for a nice point of focus while you are hashing out homework problems (I’m an especially big fan of the pattern with the cubes near the back). The coffee is more expensive (300/600) but very very good: the espresso is red-golden and creamy and tastes like everything at once. There is a little less room, it can get noisier (but who doesn’t like dubstep?) and there’s less light, but location and quality are two big draws to recommend this place.

If I ever do a post on bars, I will have to talk about Szimpla Kert at considerable length, but there is a second Szimpla on Kertesz just across the street from the Hummus Bar (a restaurant which is second only to Pita Inn in terms of its ability to make me perfectly content with everything in the world, incidentally) at which I had an absolutely fantastic espresso last Friday. Make no mistake; I will be back at Szimpla before the week is up, and I will bring a full report.

Food

I think I’m going to depart a little from my normal “Here’s something interesting I did this week” format and give a general rundown on one of my favorite parts of living in Hungary: the food. Before coming over, my only experience with Hungarian cooking was a vague memory of a “goulash” recipe from childhood and an extremely enthusiastic friend of mine (who happens to be a graduate of culinary school) exhorting me to eat as many pastries as possible. I took his suggestion to heart quickly, when I found two pastry shops in the same building as the language school. For the topologically-minded, I’m pretty sure it’s an actual theorem that pastry shops (as well as bars and cafés) are dense in Budapest: there exists one in any neighborhood. For under two hundred forints (that’s less than a dollar, people!) you can get all sorts of good things: the “taska” (literally “bag”) which holds a variety of delicious fillings (vanilla, apple, chocolate, etc.), the “retes” (strudel), even croissants filled with Nutella (the best shops also put a light orange glaze on top). Because of this profusion of delicious and unbelievably cheap baked goods (as well as the enormous quantities of pork products at similarly insane prices) I’m only half-joking when I say I might not survive the semester.

Having briefly mentioned pork, I feel I should say a little more. My friend is on the record as saying, “Pig is the tastiest animal” and I agree whole-heartedly. The Hungarians are a pig-crazy country – witness the Wooly Pig festival which I only missed because I was in Vienna, the mysteriously Spam-like sausages I was served for breakfast in Bükkszentkereszt, the virslis taska (“hot dog bag”, available wherever baked goods are sold, a.k.a. everywhere). They go so far as to not even stock beef in the supermarket, which I was disappointed to find out when I was attempting to make a stew, although the carnitas I ended up making were quite a good consolation prize. Incidentally, carnitas are a large factor in the “pig is the tastiest animal” calculation and I plan to experiment quite a bit more in the future.

The other two key components in the Hungarian national cuisine are sour cream and Erõs Pista (which translates as “Strong Steve”), a spicy pepper paste which goes well on everything (Erõs Pista and sour cream is particularly problematic, given how delicious it is in relation to how likely it is to one day kill you). I was initially distraught to realize that there is not a drop of Cholula in this country (I’m still pining for the half-gallon we managed to import from Mexico to Portland…), although as soon as I discovered Erõs Pista all of my worries evaporated (and now I have an excuse to look for a Hungarian grocery store in the United States).

As far as actual Hungarian cooking goes, I’ve had a few dishes, all of which more than lived up to my expectations. One of my first nights here I found a café that serves Hungarian cooking, and I had a beef stew in a paprika-sour cream sauce with a kind of fresh-made Hungarian noodle/dumpling, which was absolutely delicious – I’ve heard you can also make it with veal. I’ve also had a couple of encounters with dishes of the following format: “_ breast stuffed with _ “, with {turkey, chicken} in the first component and {dried plums and Camembert, peppers and Roquefort} in the second. These are also amazing, especially when you deep-fry it, which was the case with the turkey breast. I tried my hand at an approximation to chicken paprika, which was good enough that I think I’m going to make it again this week. It’s very simple: take some chicken (it can be anything: I used chicken legs), sear it for a little while with some onions and garlic, then transfer to a big pot and add some canned tomatoes and plenty of paprika. Let this stew until the chicken has finished cooking, then remove the chicken, stir in sour cream, and add the chicken back in. I ate this with egg noodles.

Although this isn’t strictly Hungarian, it does involve pork and I’m cooking it in Hungary so I feel compelled to mention it all the same; I’ve been working on a recipe for pork chops. Coat the chops in salt, oregano, and crushed garlic; then lightly fry in olive oil. While they are cooking, squeeze the juice of a lemon over them. Serve with white rice. It’s fast and flavorful. Okay, at this point I’m starving and I haven’t eaten dinner yet so I’m going to end things here.

MC Escher and the Hyperbolic Plane

I found this article by the absolutely singular Tom Wieting in the Reed Magazine this month and felt compelled to pass it on – one of the best articles on mathematics for the general audience I’ve read in a while. I feel like I should pay more attention in Tom’s classes from now on…

Adventures In Rural Hungary

I got out of the city this weekend and had a couple of days up in the mountains that were absolutely perfect. I visited the town of Bükkszentkereszt by way of Miskolc. My friend from Reed who is also doing BSM and I met up at Keleti train station on Saturday morning, where we got on a train to Miskolc in the northwest corner of the country. It’s about three hours by train which for us meant three hours stuffed in a six-seater cubicle with four other people and a heater stuck on high – I was distinctly reminded of my experiences in the sauna at Szechenyi baths a few weeks back. The train ride, it would turn out, would be the biggest hitch in an otherwise totally seamless trip.

We found ourselves in Miskolc at ten thirty in the very foggy morning; after some asking around we got on the street car and headed into the city center. We stopped for coffee and headed back out, walking further into the city, when we suddenly found ourselves in the middle of a festival – the Kocsonya (ice soup?) Fesztival, to be precise (we never did really figure out what kocsonya was). Most of the festival seemed to consist of various stands selling various foods and drink: we were never more than ten feet from forralt bor and sausages. Things got decidedly stranger when we spotted a man on stilts in a frog costume, and, in a moment I can only characterize as (David) Lynchian, we came across a man in some sort of hellish-looking furry outfit with a horned mask making an incredible racket on a giant wooden version of those noisemakers that you spin around.

Our original destination was a tourist office the whereabouts of which we had a vague idea of via the Miskolc metro map. A little bit more wandering and we found it; the woman inside showed us how to get to the bus station from which we could get out of Miskolc and up towards Bükkszentkereszt. We found the place with no problem and half an hour later we were on our way. The road to Bükkszentkereszt is gorgeous, winding through the hills outside of Miskolc. This time of year, the trees are all bare but the hillsides are covered in snow. We arrived at the edge of town after an hour’s bus ride: it was extremely foggy.

Our first encounter with the native wildlife was with a little dog that we found wandering the streets. Shortly thereafter we spotted a chicken sitting on someone’s front porch. We headed through the town and walked up to a ridge overlooking the valley in which Bükkszentkereszt sits, where we saw two horses tied up in a field. We had a very slushy walk back down, at which point we started looking for lodging, but not before finding a deer sitting in someone’s driveway. This is when I started to wonder if maybe we were going to run into a bear next. We found a promising-looking inn, but even our extremely limited Hungarian skills were enough to tell us that the place was full up. Luckily, at that moment an English-speaking guest was passing by, who directed us to another nearby inn. Here, we were able to get a room for 3500 forint apiece, and for another 2000 we got a full Hungarian dinner and breakfast. We unpacked in the room and went over to the adjoining restaurant, where we did a little work and then had dinner: schnitzel with rice and french fries. We went down the street to the Nimród Sörözõ with the intention of playing pool, where we learned that the pool table was “nem jó” and so we had a drink and then went to bed.

The next morning we woke up to fresh snow and a brilliant new day; we had toast and what appeared to be spam sausages for breakfast. We walked a little ways out of town and headed up a hill, hoping to get a good view from the top. Although the walk up was very pretty with all the fresh snow, we were only able to get a small glimpse of the area through the tree cover. This was the second time I wondered if we were going to run into a bear. Back in the town, we were still without a bear encounter, although we did see in short succession a horse-drawn sleigh and a guy out for a ride on one of the horses we had seen tied up in town the day before. We warmed up and had some coffee in a café and then headed through town to the same ridge we had visited the day before. This time, we headed all the way up the (bare) hill, and did manage to get some spectacular views, both of the town and the surrounding area. We walked back into town and found the bus stop; ten minutes later we were on our way back to Miskolc.

The rest of the trip was fairly uneventful; we made it back to the train station and eventually to the train, which this time was at a safe temperature for human occupancy. All in all, it was a perfect weekend: relaxing and rustic. Photos are available on my facebook page.

Week 2; An Apology

With classes getting more or less into full swing, I’m finding less and less time to go do the sorts of things that are actually interesting to write up and post on the internet. This is the apology alluded to in the title. I have had a very good week, however, and I will do my best to put up some of the highlights, mathematical and otherwise.

The most interesting thing I did this week was go to Buda, (actually I did this twice in one day). On Tuesday I found myself with an afternoon to kill, and on the advice of a friend I took the 4/6 tram all the way across the river and out into Buda. This part of town being very hilly, when I got off the tram I had a simple algorithm: always go up. This worked perfectly: I soon found myself at the gates of a little walled “village” on top of one of Buda’s hills. It’s very pretty up there, very quaint (if rather touristy: I saw restaurants advertising meals for nine euros, which brings out two separate red flags: price and currency). A bit of wandering found me at a sort of lookout/fortification structure from which I had a fantastic view of the city. At this point I was acutely aware of my decision that morning when packing up my bag to leave my camera on my desk, a mistake I will not make again. It was a perfectly clear day, not too cold, and I could see across the river to Parliament and out across the entirety of Pest: the occasional steeple rising up in the middle distance, the high rises melting into the haze on the horizon. Perfect. I headed down the hill through some side streets and explored a couple of the main streets in Buda, before heading back across the river to meet up with a friend.

We went out for a walk and soon found ourselves back at the river by the Szechenyi Chain Bridge. We walked across and, in Buda once again, we headed up the hill towards the Buda Castle. At this point the sun was going down and as we headed up the hill the lights were coming on all over the city. The Szechenyi bridge in particular was gorgeous with a string of lights running across the width of the river. We got to the top and watched the city at night. Afterwards, we took a quick walk around the Buda castle grounds. Having a dinner party to go to, we were a bit pressed for time, so the next time I go back I should be able to give a fuller report, and I’ll be sure to bring a camera.

That’s the major bit of tourism I accomplished this week; otherwise it’s been pretty exclusively mathematical. We’re studying arithmetical functions in number theory which is some of my favorite material: I love the Möbius function in particular. An arithmetical function is just a function from the integers to the complex numbers.  Many important features of the structure of the integers can be studied through arithmetical functions: properties of an integer such as how many divisors it possesses (a question obviously connected to the distribution of the prime numbers) can be naturally phrased in terms of arithmetic functions – simply construct the function which maps an integer to the number of positive divisors such that . Whereas before I’ve approached arithmetical functions from an algebraic point of view, we’re getting into the analytic side of things in this class. One major question is what the “average” value of an arithmetic function is: for an arithmetical function we’re computing the sum ; if we divide this by we have a sensible notion of “average”. One result we showed in class this week is that for the number of divisors function , the average approaches , which is to say that on average, a large integer has about as many divisors as it does digits (more precisely, digits “in base e”).

If all goes well, I should be on a train tomorrow morning to Miskolc; from there I’m going to head out on a bus to a little town called Bükkszentkereszt where I’m hoping to have a relaxing weekend in the woods. I’m going to make a point of bringing my camera so I’ll be sure to report back.

The First Week (Part 2)

I’ll continue with my rundown of the first week of my classes. Next up is Commutative Algebra, which is the study of commutative rings; recall that a ring is a set of objects with an addition and a multiplication defined, and a commutative ring is one with commutative multiplication: for all . By contrast, a noncommutative ring is one where there are some elements with ; an example would be a ring of matrices with matrix multiplication as the operation. Commutative rings can be thought of as “enlargements” of the familiar ring of integers . The set of all polynomials in one or more variables with coefficients drawn from a commutative ring comprise a large subclass of commutative rings. One important question to be able to answer about a ring is what its ideals are. An ideal of a ring is a subset that is closed under addition and strongly closed under multiplication: we require for and also for all , a stronger condition than merely requiring . In the integers, the ideals are just all multiples of a given number: proving that any ideal has this form requires the Euclidean Algorithm which is basically just how we do long division. The structure of ideals in other rings is one thing that one studies in Commutative Algebra. There’s actually a lot of geometry lurking in the background, which I mentioned in an early post. The connection is that some commutative rings can be understood as rings of polynomials defined on geometric objects: a simple example is the ring of polynomials with complex coefficients, which is the ring of polynomials on the complex plane . Then there is a natural identification of a certain class of ideals of (the maximal ideals which are not contained in any other proper ideals) and the points of the underlying space . Unfortunately, not all commutative rings arise as rings of polynomial functions defined on a geometric object, but all rings have maximal ideals, so you can actually study the “geometry” of any commutative ring by thinking backwards and assigning a geometry to your ring by treating maximal ideals as “points”. Thus there is this two-way traffic between algebra and geometry where you can investigate algebraic objects geometrically and vice-versa, and this forms the starting point for the study of algebraic geometry.

I’ve only had one Complex Analysis course so far but it has been pretty satisfying. We’ve started out by studying some of the geometry of the complex plane. Recall that a point in the complex plane can be specified by giving a length and an angle above the real axis. The way multiplication is defined in the complex plane, multiplication of some complex by another complex number with length and angle has a geometric understanding as scaling by and then rotating counter-clockwise by . So, for example, the complex number has length and angle , from which has length and angle . But the complex number with length and angle is just , as we expected. This actually gives rise to a good deal of geometry on , and our first project in the class will be to discuss the so-called fractional linear transformations on . It would take a little while to explain in full detail what these are, but they are a class of transformations of the complex plane with certain nice geometric properties: they preserve angles, and they map lines and circles to lines and circles (as in, a line can be mapped to either of a line or circle, and vice versa). They also have a very natural understanding as rotations and translations of the Riemann Sphere, an object which can be identified with the complex plane once you add a “point at infinity”. If this sounds interesting, let me know; I’d like to try my hand at writing this up in more detail.

I had a pretty good weekend, all told. On Friday I ate at a Hungarian restaurant, where I had a turkey breast stuffed with dried plums and Roquefort, deep fried. One of my friends got a chicken breast stuffed with apples and Camembert, deep-fried. I sense a pattern. Yesterday I bought rather a large quantity of a mystery cut of pork by accident; I was looking for stew beef. I’m going to try marinating it in olive oil, garlic, and paprika, and then roasting it (thanks, Dad!). Math and cooking: pretty much an ideal Sunday.