Geometric Graph Theory
This class is making me glad I have the physics background I do. One of our proofs involved replacing some of the ideal rubber bands connecting the points of our graph with rigid bars so that we could make the "forces" at all of the vertices sum to zero while still forcing them not coincide. All of this is to prove that 3-connected planar graphs can be viewed as the skeleton of a convex polyhedron. I think this class is going to prove to be the most challenging, but it is fun :-). I'm definitely glad for my background with graph theory and knowing what sorts of techniques are useful for approaching graph theory problems. It made the first HW set much more manageable for me than it seemed to be for a lot of my classmates...
Topics in Geometry
It is a great deal of fun to do proofs that effectively involve staring until you understand the geometry of what is going on, translating it into Algebra so that you can actually explain what is going on, working through the Algebra until it gets stuck, translating back into the language of geometry to get past a few sticky parts and then going back to Algebra to finish the proof. I love that math is so wonderfully interconnected! It also highly amuses me that several times questions, such as "Is the empty set an affine subspace?" have gotten the response, "Well, some elementary text books say (that/it is), but I do not think that is a good idea."
Set Theory this week, aka Playing with Infinity
When playing with cardinals, we start out with all of the natural numbers. We say the number of natural numbers is aleph naught. If we add aleph naught copies of 1 together, we get aleph naught. If we add aleph naught copies of 2 together, we get 2*aleph naught, which is just aleph naught. In fact, if we take the sum of all of the natural numbers, then we get Aleph naught. However, if we multiply them all together, we get c (the size of the set of real numbers). Now we have c*c=c and c^(aleph naught)=c and for that matter c^c =c. However, 2^c>c and we call it c_1. In fact we can define countably many c_i's with the formula c_(n+1) = 2^c_n. If we add all of these c_i together, we get a cardinal that is strictly larger than any of the c_n, so we call it d. We can then follow the same procedure with d to create e and so on. Once we have countably many of these countable sets of levels of infinity, we can add all of them together to create an even larger infinity, and so on and so forth. It is exciting.
Conjecture and Proof
The number of different possible approaches to problems often amazes me. We had one this week about tiling a rectangle. Various solutions included counting different kinds of dominoes in certain parts of the board mod n, looking at what happens if you roll a sphere around in the rectangle, and two very different colorings of the board. Over all it continues to be a fun class full of really cool proofs.
Friday, September 28, 2007
Monday, September 24, 2007
Mellow is fun too :-)
Last Tuesday and Wednesday were not that terribly exciting. I made dinner with Christina (and Kyle on Wed) and did math. It is important to just relax sometimes :-). One day, and I don't remember which so I am just going to talk about it here, I ended up in another conversation with an older Hungarian lady on the bus. This one used to be a translator, so she spoke English. It is really quite interesting - she asked if I was Canadian. I don't know if I really don't come across as American or if people are afraid they will offend me if they assume I am and I happen to not be. Either way, it must not be obvious. I wonder what I do or don't do that makes it less obvious that I am American (several of my classmates have had people frequently identify them as American, sometimes before they say a word).
Thursday
The BSM program set up a showing of the documentary about Erdős Pál, "N is a Number," at the Renyi Institute and were highly encouraging all of us to go, so I went. Erdős was a very interesting person. I have to admit that the idea of not having anywhere in particular that you are expected to be and being able to do math and travel around the world to do math with other people on a whim is somewhat appealing. I don't think I would actually do it though, having some amount of stability is nice. It was also very interesting to see the footage that was filmed in Budapest. War scenes are much more striking when you recognize the buildings and bridges and statues as things you see on the way to school everyday...
Friday
Christina and I threw another dinner party at her apartment. We made a broccoli and pepper stirfry, lots of rice and carrot sticks for dinner. After dinner I made another batch of approximation cookies - this time with a spice cookie variation - the box of random spices had nutmeg and cinnamon and we had obtained fresh ginger for the stirfry. I was very, very pleased with the result :-)
While I was making the cookies, people decided that we should make an Apples to Apples game. Several of us had been wanting to play, but no one brought a set. The homemade game is likely going to be a work in progress for the whole semester. It is rather interesting version of the game: a large set of the cards are in (frequently mispelled) Hungarian, another large set are math terms, and a non-trivial set are inside jokes that have spawned in the time we have been here. Eventually we decided we had enough cards to try playing. It was hilarious, as Apples to Apples should be. I think my favorite play was when the adjective was "countably infinite" and the winning noun card was "Hungarian vowels".
Saturday
Christina, Chelsey and I planned to go to the Hummus Bar for lunch, but it was closed for Yom Kippur so we found lunch at their place instead. After lunch Christina and I went to track down a thrift store she had heard about to find a "classy dress" for the "classy sandwich party" two of our friends were throwing on Sunday. Shopping is not usually my thing, but it was fun :-). You can find interesting clothing in Hungarian thrift stores...
In the evening I went back to my apartment for dinner as one of my host mother's real daughters was coming over. We had fried eggplant with yogurt (which was surprisingly tasty), fried mushrooms, fried turkey, boiled potatoes and corn. Despite being mostly fried, the food was quite good. She also served wine with dinner. About half way through dinner she started refilling the wine glasses...I tried to tell her no thank you but she insisted on refilling it fuller than it was to start with *sigh*. About half a glass of wine had me slightly light headed. I was very glad I was did not have anything that needed to get done that evening...
Sunday
I actually watched my host mother make the Hungarian version of French toast! (usually she has it made before I wake up...) . First beat the egg (not excessively, but like you would for scrambled eggs) - just egg, do not add milk (you only add milk if the bread is dried out and not fresh). Heat (sunflower) oil in a frying pan while you soak the bread in the egg. Fry the egg soaked bread in the hot oil until it is golden brown. She always serves it with a mixture of powdered sugar and cinnamon. I think the biggest difference is using the wonderful fresh Hungarian bread rather than the sandwich loaves we get in the US...
Christina and I met up for church, but when we got to the Lutheran church just before 9, we noticed that the 11:00 service included a Bach Kantata. We glanced at each other and without hardly having to say anything concluded that we should go be productive and come back at 11. It turned out to be really cool, although the sheet they gave us with the German lyrics and Hungarian translation was not that terribly useful...
After church we had lunch and then went to Margit Sziget (the park island) to meet up with a couple of friends and do math outside. By the time we left the island, I had all but one of the problems due Monday done and written up. Christina and I went back to her place and got ready for the Classy Sandwich party. The party was fun, the sandwiches were tasty, and it was nice seeing most people dressed up. :-)
Sandwich assembly line:
Yesterday and today are math days. Tomorrow I am making dinner with Christina and we will have a math party. The theory is that if we can get on top of the HW during the week, then we can actually do bigger exciting things on the weekends. Hopefully this will work out well!
Oh, as a side note, apparently the Duracell bunny is much more adventurous that the Energizer bunny...
Thursday
The BSM program set up a showing of the documentary about Erdős Pál, "N is a Number," at the Renyi Institute and were highly encouraging all of us to go, so I went. Erdős was a very interesting person. I have to admit that the idea of not having anywhere in particular that you are expected to be and being able to do math and travel around the world to do math with other people on a whim is somewhat appealing. I don't think I would actually do it though, having some amount of stability is nice. It was also very interesting to see the footage that was filmed in Budapest. War scenes are much more striking when you recognize the buildings and bridges and statues as things you see on the way to school everyday...
Friday
Christina and I threw another dinner party at her apartment. We made a broccoli and pepper stirfry, lots of rice and carrot sticks for dinner. After dinner I made another batch of approximation cookies - this time with a spice cookie variation - the box of random spices had nutmeg and cinnamon and we had obtained fresh ginger for the stirfry. I was very, very pleased with the result :-)
While I was making the cookies, people decided that we should make an Apples to Apples game. Several of us had been wanting to play, but no one brought a set. The homemade game is likely going to be a work in progress for the whole semester. It is rather interesting version of the game: a large set of the cards are in (frequently mispelled) Hungarian, another large set are math terms, and a non-trivial set are inside jokes that have spawned in the time we have been here. Eventually we decided we had enough cards to try playing. It was hilarious, as Apples to Apples should be. I think my favorite play was when the adjective was "countably infinite" and the winning noun card was "Hungarian vowels".
Saturday
Christina, Chelsey and I planned to go to the Hummus Bar for lunch, but it was closed for Yom Kippur so we found lunch at their place instead. After lunch Christina and I went to track down a thrift store she had heard about to find a "classy dress" for the "classy sandwich party" two of our friends were throwing on Sunday. Shopping is not usually my thing, but it was fun :-). You can find interesting clothing in Hungarian thrift stores...
In the evening I went back to my apartment for dinner as one of my host mother's real daughters was coming over. We had fried eggplant with yogurt (which was surprisingly tasty), fried mushrooms, fried turkey, boiled potatoes and corn. Despite being mostly fried, the food was quite good. She also served wine with dinner. About half way through dinner she started refilling the wine glasses...I tried to tell her no thank you but she insisted on refilling it fuller than it was to start with *sigh*. About half a glass of wine had me slightly light headed. I was very glad I was did not have anything that needed to get done that evening...
Sunday
I actually watched my host mother make the Hungarian version of French toast! (usually she has it made before I wake up...) . First beat the egg (not excessively, but like you would for scrambled eggs) - just egg, do not add milk (you only add milk if the bread is dried out and not fresh). Heat (sunflower) oil in a frying pan while you soak the bread in the egg. Fry the egg soaked bread in the hot oil until it is golden brown. She always serves it with a mixture of powdered sugar and cinnamon. I think the biggest difference is using the wonderful fresh Hungarian bread rather than the sandwich loaves we get in the US...
Christina and I met up for church, but when we got to the Lutheran church just before 9, we noticed that the 11:00 service included a Bach Kantata. We glanced at each other and without hardly having to say anything concluded that we should go be productive and come back at 11. It turned out to be really cool, although the sheet they gave us with the German lyrics and Hungarian translation was not that terribly useful...
After church we had lunch and then went to Margit Sziget (the park island) to meet up with a couple of friends and do math outside. By the time we left the island, I had all but one of the problems due Monday done and written up. Christina and I went back to her place and got ready for the Classy Sandwich party. The party was fun, the sandwiches were tasty, and it was nice seeing most people dressed up. :-)
Sandwich assembly line:
Yesterday and today are math days. Tomorrow I am making dinner with Christina and we will have a math party. The theory is that if we can get on top of the HW during the week, then we can actually do bigger exciting things on the weekends. Hopefully this will work out well!
Oh, as a side note, apparently the Duracell bunny is much more adventurous that the Energizer bunny...
Friday, September 21, 2007
0^0 = 1
I have now been to two weeks worth of all of my classes, and remain excited about all of them, so this is a good sign. This is a math post. I have decided that bringing my laptop to school every day is a little ridiculous, and the keyboards on the computers here are British and thus strange and I don't want to type long things on them, so I should come up with some sort of schedule. My current plan is thus to have a non-mathy post on Monday/Tuesday and a mathy post on Thursday/Friday which of the two days in each case depending on things like what other plans/hw I have ;-).
Geometric Graph Theory
A few days ago we gave three proofs of the same theorem ( If G is a three connected planar graph, S is a nonempty subset of V(G) and f0 is a function that maps elements of S to the real plane, R, then there is a unique function f:V->R, such that f(i) = f0(i) for i in S and it is harmonic at all other vertices. ) The first proof of this involved random walks and probability theory. The second had us view the graph as an electrical circuit and used physics to obtain a proof. The third started out with ideal rubber bands that obey Hookes law from these we defined an energy function such that minimizing it gave the desired configuration. It was very interesting seeing these three very different approaches - and the resulting equivalences between statements about physics, math and probability theory. I really like math that plays in the areas between the defined subjects :-). Also, I want my shelf of textbooks that I have used in former classes. There are not that many on it yet, but it would be really nice to be able to look up the details of theorems I mostly remember...
Topics in Geometry
This class is also very good at getting into the area's between subjects! Proofs have a tendency to start out as a geometry problem, involve using algebra properties to define isometries, manipulating the isometries geometrically - playing with things like distance functions, and then drawing them back in to the group context to use the techniques that became so familiar in Abstract. It is really interesting seeing geometric interpretations of group theoretic ideas that previously did not seem like things to be visualized (granted, when the geometry gets into higher dimensions it also does not really get visualized.) Additionally, when we don't have questions during the "office hour" the prof does cool geometry stuff, like giving a geometric proof of the Cauchy-Schwarz inequality.
Conjecture and Proof
This class is definitely the three credit version of Putnam seminar. On Tuesdays the prof goes over cool proofs and points out various useful ideas and techniques. On Thursdays he collects the HW and then we go over it, having people present the different proofs they have come up with. It is really cool to see the sometimes drastically different approaches people take to problems. Problem solving is not exactly my favorite type of math, but the practice and being exposed to different techniques and ways of thinking is good for me and the proofs are definitely exciting.
Set Theory
We seem to be down to 4 people in Set Theory, which means that it will happen, but that it will be turned into a reading class where we are expected to teach more of it to ourselves and it will only meet once a week. *sigh* Ah well, it will still be fun. It will also teach me to pay very, very close attention to proofs in class. The prof has a tendency to go through a proof, look like he is ready to move on and then say "Oh, as a foot note, that was not a proof and the theorem is false as currently stated" and then we stare at the board for a minute or so until we figure out what the problem is. Today we had to figure out why a0<=a and b0<=b does not imply a0^(b0)<=a^b. In set theory, 0^0 = 1 and 0^a=0 for all a not equal to 1, so if b and b0 are both 0 it is false. We also looked at why the proof of the fact that the countable union of countable sets is countable requires the axiom of choice. It is a fun class :-)
Geometric Graph Theory
A few days ago we gave three proofs of the same theorem ( If G is a three connected planar graph, S is a nonempty subset of V(G) and f0 is a function that maps elements of S to the real plane, R, then there is a unique function f:V->R, such that f(i) = f0(i) for i in S and it is harmonic at all other vertices. ) The first proof of this involved random walks and probability theory. The second had us view the graph as an electrical circuit and used physics to obtain a proof. The third started out with ideal rubber bands that obey Hookes law from these we defined an energy function such that minimizing it gave the desired configuration. It was very interesting seeing these three very different approaches - and the resulting equivalences between statements about physics, math and probability theory. I really like math that plays in the areas between the defined subjects :-). Also, I want my shelf of textbooks that I have used in former classes. There are not that many on it yet, but it would be really nice to be able to look up the details of theorems I mostly remember...
Topics in Geometry
This class is also very good at getting into the area's between subjects! Proofs have a tendency to start out as a geometry problem, involve using algebra properties to define isometries, manipulating the isometries geometrically - playing with things like distance functions, and then drawing them back in to the group context to use the techniques that became so familiar in Abstract. It is really interesting seeing geometric interpretations of group theoretic ideas that previously did not seem like things to be visualized (granted, when the geometry gets into higher dimensions it also does not really get visualized.) Additionally, when we don't have questions during the "office hour" the prof does cool geometry stuff, like giving a geometric proof of the Cauchy-Schwarz inequality.
Conjecture and Proof
This class is definitely the three credit version of Putnam seminar. On Tuesdays the prof goes over cool proofs and points out various useful ideas and techniques. On Thursdays he collects the HW and then we go over it, having people present the different proofs they have come up with. It is really cool to see the sometimes drastically different approaches people take to problems. Problem solving is not exactly my favorite type of math, but the practice and being exposed to different techniques and ways of thinking is good for me and the proofs are definitely exciting.
Set Theory
We seem to be down to 4 people in Set Theory, which means that it will happen, but that it will be turned into a reading class where we are expected to teach more of it to ourselves and it will only meet once a week. *sigh* Ah well, it will still be fun. It will also teach me to pay very, very close attention to proofs in class. The prof has a tendency to go through a proof, look like he is ready to move on and then say "Oh, as a foot note, that was not a proof and the theorem is false as currently stated" and then we stare at the board for a minute or so until we figure out what the problem is. Today we had to figure out why a0<=a and b0<=b does not imply a0^(b0)<=a^b. In set theory, 0^0 = 1 and 0^a=0 for all a not equal to 1, so if b and b0 are both 0 it is false. We also looked at why the proof of the fact that the countable union of countable sets is countable requires the axiom of choice. It is a fun class :-)
Monday, September 17, 2007
A szep napot van!
Yesterday morning as I was sitting and waiting for the bus an elderly Hungarian lady came and sat down on the bench next to me. She noted that the weather was beautiful and I smiled and agreed. Then she asked me something that I could not quite figure out... I explained that I only know a little Hungarian. She smiled, nodded and then talked to me using very simple Hungarian. Sometimes she would have to reword things, but there was only one question that I never managed to figure out! Even better, I knew enough Hungarian to answer her questions - my grammar was far from perfect and I did not always use complete sentences, but I could make myself understood. They were simple things, like where I was from and where in Budapest my classes are, but I understood the questions and new how to answer them! Also, I apparently do not have a strong American accent in my broken Hungarian - she asked me if I was German and was genuinely surprised when I explained that I was American. Her stop was before mine and as she got off I wished her "szep napot" - have a beautiful day - and she smiled and waved from the stop as the bus pulled away. I think it made both of our days :-)
Sunday Christina and I tracked down a small Anglican congregation whose service is in English. Their sanctuary looks like it was adapted from the same sort of wine cellar as the bar we went to to listen to live Hungarian music. Their priest was in Austria at some sort of meeting, so they were planning to have a basic morning prayer service but there was a priest who just happened to be visiting from England, so they ended up having a normal service after all. There were only about 20 people there and the 'sanctuary' could not have held more than twice that, but everyone was very friendly and it was really nice to actually understand everything. I also really enjoyed the number of different accents :-) Everyone spoke English, but we were still from all over the world. We will probably go back there frequently, though it is also nice to visit the gorgeous churches that are all over Budapest.
In Budapest there is a phenomenon known as a tanchaz - as best as I can figure out, a tanchaz (literally "dance house") is like a contra dance but with traditional Hungarian Dancing. This past Saturday was the tanchaz ball where all of the different teachers and bands and dancers came together starting at 8pm and there was supposed to be dancing until 5 am. Christina, Chelsey and I decided to check it out. We got there around 8 and discovered that there were 3 different rooms with different kinds of dances in each room and the dancing in any given room changed every half hour or so as teachers familiar with the dances from different regions switched out. We tried dancing some, but my feet and ankles are not cut out for that kind of intense foot work (I realized that part of why I like contra is that the emphasis is not on footwork). Chelsey and Christina were much better. I really enjoyed watching and listening to the live music though. The boot slapping dances that the men do are especially impressive. We left around 10:30 so that we could catch the metro home, but it was definitely a lot of fun :-)
Erzebet Tér has been invaded by cows. Here are a few of the more interesting ones:
And this one even had flies
Last Thursday Christina, Kyle and I climbed Gellert Hill - a hill taller than the one the castle is on and that houses the citadel. It has gorgeous views of the entire city. It also has a pretty garden and a simple playground consisting of swings and concrete dinosaurs. Other than that words really are not useful, so here are pictures
The castle in Buda
The Danube
Pest
Sunday Christina and I tracked down a small Anglican congregation whose service is in English. Their sanctuary looks like it was adapted from the same sort of wine cellar as the bar we went to to listen to live Hungarian music. Their priest was in Austria at some sort of meeting, so they were planning to have a basic morning prayer service but there was a priest who just happened to be visiting from England, so they ended up having a normal service after all. There were only about 20 people there and the 'sanctuary' could not have held more than twice that, but everyone was very friendly and it was really nice to actually understand everything. I also really enjoyed the number of different accents :-) Everyone spoke English, but we were still from all over the world. We will probably go back there frequently, though it is also nice to visit the gorgeous churches that are all over Budapest.
In Budapest there is a phenomenon known as a tanchaz - as best as I can figure out, a tanchaz (literally "dance house") is like a contra dance but with traditional Hungarian Dancing. This past Saturday was the tanchaz ball where all of the different teachers and bands and dancers came together starting at 8pm and there was supposed to be dancing until 5 am. Christina, Chelsey and I decided to check it out. We got there around 8 and discovered that there were 3 different rooms with different kinds of dances in each room and the dancing in any given room changed every half hour or so as teachers familiar with the dances from different regions switched out. We tried dancing some, but my feet and ankles are not cut out for that kind of intense foot work (I realized that part of why I like contra is that the emphasis is not on footwork). Chelsey and Christina were much better. I really enjoyed watching and listening to the live music though. The boot slapping dances that the men do are especially impressive. We left around 10:30 so that we could catch the metro home, but it was definitely a lot of fun :-)
Erzebet Tér has been invaded by cows. Here are a few of the more interesting ones:
And this one even had flies
Last Thursday Christina, Kyle and I climbed Gellert Hill - a hill taller than the one the castle is on and that houses the citadel. It has gorgeous views of the entire city. It also has a pretty garden and a simple playground consisting of swings and concrete dinosaurs. Other than that words really are not useful, so here are pictures
The castle in Buda
The Danube
Pest
Thursday, September 13, 2007
Math!
I have now been to all of the classes I think I am planning to take at least once, so this post is going to be my first impressions. For those of you who are not as interested in the math, at the bottom there are also a few pictures ;-)
Geometric Graph Theory
Last spring I took Mudd's Graph Theory course. Throughout the class, we made a point out of the fact that the specific drawing, embedding, of a graph was not important. We focused on properties inherent to the graph regardless of drawings. Drawing were useful for seeing what was going on and obtaining intuition, but did not tend to play a crucial role in the proofs. I get the feeling that this semester the most important part of most of the proofs is going to be finding the correct embedding of a graph (sometimes in 2-space, sometimes in 3-space). It is going to involve things like observing that every three connected planar graph can be drawn as the skeleton of a convex polytope. In turn, given the skeleton of a convex polytope we can prove that it is a 3-connected planar graph.
We first providing a method of projection that gives us a straight line embedding in the plane with no crossings - pick any face of the polytope and choose a point "close" to that face, where close means that if we extend the plane of any other face of the polytope that point and the polytope are in the same half space, and project onto the plane of the original face via that point. To prove that it is three connected we need to show that removing any two arbitrary points a and b leaves the graph connected. Select any 4 points a,b,c,d in the polytope. Either these 4 points are in a plane or they form a tetrahedron. If they form a tetrahedron, then we can find some plane that will separate a and b from c and d. Consider the portion of the polytope in the half space containing c and d. If c and d are adjacent, then we are done, otherwise follow a path from c and a path from d that always moves away from the dividing plane. Eventually, we will not be able to move farther away (since we are on a convex polytope). At this point, the paths will intersect, the final verticies of the paths will be the endpoints of an edge of the polytope and we can connect paths entirely in that "half" of the polytope, or the final vertices of the paths will be two of the vertices bounding a face of the polytope that is parallel to the dividing plane - in which case they are part of a cycle and we can connect the path. In all of these cases we have shown that c and d must be connected in the "half" of the polytope that does not contain a and b and so removing a and b does not disconnect the graph. If all 4 points are coplanar, we use that plane to divide the polytope and then restrict ourselves to stay on one side of the plane (it does not matter which unless they are all part of the same face, in which case we obviously take the side of the plane that contains the rest of the polytope) and use the same argument.
I find it very exciting that we can use phrases like "always moving away from the specified plane" when proving things about graphs! I am going to learn a completely new way of thinking about graph problems :-). The class is being cotaught. One of the professors is amazing, the other is slightly harder to follow, but the material is exciting enough that it is definitely going to be worth the work.
Topics in Geometry
The first day was mostly just outlining what we are going to be covering and the approach we are going to take, but I am still excitied. We are going to start with Euclidean Geometry, but instead of using the axiomatic approach (which the professor deemed "stupid") we are going to study geometry via transformations. In particular, we are going to spend a lot of time studying the isometry group of R^n. (Anyone know of any nice way to write math in a blog?) I really like seeing the connections between subjects, so studying Geometry via Algebra should be fun :-). After Euclidean, we are going to take a similar approach to studying the spherical, projective, and hyperbolic geometries. It will be nice to formally study them for more than a week long class at Mathcamp.
Conjecture and Proof
The best description I have come up with so far is that that this class looks like it is going to be the equivalent of turning the Putnam Seminar into a 3 credit class (and then forbidding collaboration on most of the problems...). It is full of very shiny proofs - the sort that are wonderfully simple and elegant and make you wonder how on earth anyone every found them. One of the things I really like about it is the same thing I really like about the Putnam Seminar - it is going to make me stop and think about all sorts of different areas of mathematics. It is far too easy to learn material one semester and then let it drift to the back of my brain and get rusty from disuse. I suspect C&P will make me use most of the math I know at some point or other :-). The fact that collaboration is mostly forbidden makes me sad though. Mudd has me very used to sitting with friends and bouncing ideas around...
Set Theory
Everything is a set. For instance, the ordered pair (a,b) can be thought of as the set {{a},{a,b}}. The basic axioms of set theory allow us to easily prove that {{a},{a,b}} = {{c},{c,d}} iff a=c and b=d, so it does indeed act like an ordered pair. Set Theory looks like it is going to be all about looking at familiar things in an unfamiliar way. The professor has a good, though subtle, sense of humor. Overall it looks like it will be a good class. The only problem is that there were exactly 6 of us that went yesterday, at least 2 of which I suspect are not going to actually take the class. If a class drops under 6, they turn it into a reading class rather than a normal class (as long as there are still at least 3 of us, which I think there will be). I would rather have it as a normal class, but a reading class will be fine too if it comes to that.
And now the non-math.
Some of the flowers that are around the building most of our classes are in:
Near the giant hour glass that I posted about earlier, there is a large parking lot. Currently, this parking lot appears to be some sort of advertisement show - it has billboards set up all over it. In addition to the billboards, there are animals made out of trash and oddly decorated phone booths.
Last night several of us hung out at Christina and Chelsey's place and had görögdinnye and tea. It is important to note that this was not organized by either of the mathcampers that are here this semester (though we were both there).
Geometric Graph Theory
Last spring I took Mudd's Graph Theory course. Throughout the class, we made a point out of the fact that the specific drawing, embedding, of a graph was not important. We focused on properties inherent to the graph regardless of drawings. Drawing were useful for seeing what was going on and obtaining intuition, but did not tend to play a crucial role in the proofs. I get the feeling that this semester the most important part of most of the proofs is going to be finding the correct embedding of a graph (sometimes in 2-space, sometimes in 3-space). It is going to involve things like observing that every three connected planar graph can be drawn as the skeleton of a convex polytope. In turn, given the skeleton of a convex polytope we can prove that it is a 3-connected planar graph.
We first providing a method of projection that gives us a straight line embedding in the plane with no crossings - pick any face of the polytope and choose a point "close" to that face, where close means that if we extend the plane of any other face of the polytope that point and the polytope are in the same half space, and project onto the plane of the original face via that point. To prove that it is three connected we need to show that removing any two arbitrary points a and b leaves the graph connected. Select any 4 points a,b,c,d in the polytope. Either these 4 points are in a plane or they form a tetrahedron. If they form a tetrahedron, then we can find some plane that will separate a and b from c and d. Consider the portion of the polytope in the half space containing c and d. If c and d are adjacent, then we are done, otherwise follow a path from c and a path from d that always moves away from the dividing plane. Eventually, we will not be able to move farther away (since we are on a convex polytope). At this point, the paths will intersect, the final verticies of the paths will be the endpoints of an edge of the polytope and we can connect paths entirely in that "half" of the polytope, or the final vertices of the paths will be two of the vertices bounding a face of the polytope that is parallel to the dividing plane - in which case they are part of a cycle and we can connect the path. In all of these cases we have shown that c and d must be connected in the "half" of the polytope that does not contain a and b and so removing a and b does not disconnect the graph. If all 4 points are coplanar, we use that plane to divide the polytope and then restrict ourselves to stay on one side of the plane (it does not matter which unless they are all part of the same face, in which case we obviously take the side of the plane that contains the rest of the polytope) and use the same argument.
I find it very exciting that we can use phrases like "always moving away from the specified plane" when proving things about graphs! I am going to learn a completely new way of thinking about graph problems :-). The class is being cotaught. One of the professors is amazing, the other is slightly harder to follow, but the material is exciting enough that it is definitely going to be worth the work.
Topics in Geometry
The first day was mostly just outlining what we are going to be covering and the approach we are going to take, but I am still excitied. We are going to start with Euclidean Geometry, but instead of using the axiomatic approach (which the professor deemed "stupid") we are going to study geometry via transformations. In particular, we are going to spend a lot of time studying the isometry group of R^n. (Anyone know of any nice way to write math in a blog?) I really like seeing the connections between subjects, so studying Geometry via Algebra should be fun :-). After Euclidean, we are going to take a similar approach to studying the spherical, projective, and hyperbolic geometries. It will be nice to formally study them for more than a week long class at Mathcamp.
Conjecture and Proof
The best description I have come up with so far is that that this class looks like it is going to be the equivalent of turning the Putnam Seminar into a 3 credit class (and then forbidding collaboration on most of the problems...). It is full of very shiny proofs - the sort that are wonderfully simple and elegant and make you wonder how on earth anyone every found them. One of the things I really like about it is the same thing I really like about the Putnam Seminar - it is going to make me stop and think about all sorts of different areas of mathematics. It is far too easy to learn material one semester and then let it drift to the back of my brain and get rusty from disuse. I suspect C&P will make me use most of the math I know at some point or other :-). The fact that collaboration is mostly forbidden makes me sad though. Mudd has me very used to sitting with friends and bouncing ideas around...
Set Theory
Everything is a set. For instance, the ordered pair (a,b) can be thought of as the set {{a},{a,b}}. The basic axioms of set theory allow us to easily prove that {{a},{a,b}} = {{c},{c,d}} iff a=c and b=d, so it does indeed act like an ordered pair. Set Theory looks like it is going to be all about looking at familiar things in an unfamiliar way. The professor has a good, though subtle, sense of humor. Overall it looks like it will be a good class. The only problem is that there were exactly 6 of us that went yesterday, at least 2 of which I suspect are not going to actually take the class. If a class drops under 6, they turn it into a reading class rather than a normal class (as long as there are still at least 3 of us, which I think there will be). I would rather have it as a normal class, but a reading class will be fine too if it comes to that.
And now the non-math.
Some of the flowers that are around the building most of our classes are in:
Near the giant hour glass that I posted about earlier, there is a large parking lot. Currently, this parking lot appears to be some sort of advertisement show - it has billboards set up all over it. In addition to the billboards, there are animals made out of trash and oddly decorated phone booths.
Last night several of us hung out at Christina and Chelsey's place and had görögdinnye and tea. It is important to note that this was not organized by either of the mathcampers that are here this semester (though we were both there).
Wednesday, September 12, 2007
Hey look, the Danube is flooding
I went to intermediate Hungarian yesterday. The teacher informed us that the class always ends up horrible, but that he has a new crazy plan for this semester. We are going to spend about 3 weeks on grammar and then study Hungarian by translating the song lyrics from a hip hop band that did an album with songs about Hungarian History. My immediate thought was that this would be like learning English by studying School House Rock or that Animaiacs song about the Presidents... It should be interesting.
I will write about the math classes once I have been to all of them. :-)
Monday, September 10, 2007
The lull before the Math
Thursday
Chelsey and I made dinner for a large group of people again. This time it was Mexican themed. However, tortillas proved difficult to locate here, so Chelsey made tortillas while I concocted something that approximated a vegetarian chili. It came out delicious, but there does not seem to be anything actually called chili powder here. Instead we used something labeled paprika that smelled about right. Thus we ended up with "chili", rice, chewy (but amazing) homemade tortillas and various vegetables, cheese, and sour cream to pile on the tortillas. I love cooking :-)
Sunday
Christina and I met for breakfast and then went to the Lutheran Church again. It was exciting because we managed to understand enough to recognize one of the readings (the fruits of the spirit) despite the fact that it was in Hungarian! I also caught that part of the sermon was about students (by the words for student, school, studying, etc...), though I do not know exactly what about students.
In the afternoon we decided that it was too nice a day to stay inside and that we should go on an adventure. We took the metro to Moskva Tér and then rode the 56 tram out to its terminus in the hills. After wandering around we found a marked trail and went for a nice walk through the woods :-)
A pretty tram stop and a nice view of Buda
Christina and I went to the National Gallery – the big art museum full of work by Hungarian artists from the past several centuries. The percentage of art that depicts war and war heroes is indicative of just how war torn
A view of Buda from near the Museum
After the museum we decided to make dinner and invited several friends over. Dinner parties are much more our style than going to bars ;-). We were very excited to find brokoli at the Match (local grocery store chain) so we made an alfredo sauce with trappista cheese (the best description I have heard is that it is like a really nice white cheddar), pasta and steamed broccoli as the main dish. The paradiscom (tomatoes) here are very cheap and amazing, so we made a tomato and mozzarella salad and garlic bread to go with it. I was very pleased with the outcome. On top of it being delicious, the ingredients for the whole thing came out to less than $15 and we had left over ingredients. I really like the cheap produce here. After dinner we decided we wanted cookies, so I raided the supply of half bags of ingredients again while a couple of people went to get a chocolate bar to break up into make shift chocolate chips and we concocted a batch of oatmeal chocolate chunk cookies. I probably shouldn’t be, but I am still surprised at how well the cookies come out when I am just adding ingredients until it “looks right” and not measuring anything.
Friday
We had the orientation session for the main portion of the program. Some of the bad scheduling conflicts present in the initial version of the schedule have been worked out, so my current plan is to start with Geometric Graph Theory, Set Theory, Topics in Geometry (which is going to be an algebraic geometry flavored class), Conjecture and Proof and Intermediate Hungarian. If I find that any of the math classes are not as exciting as I expect or that I am spending to much time working I will drop or switch to auditing one of them. I wanted to take the Hungarian Culture class, but it conflicts badly with some of the math, so I am just going to make sure I explore the culture outside of class. I am really excited for math to start! After the orientation they had a reception where we could meet some of the professors. It was kind of funny, I talked with the geometry prof and one of the geometric graph theory profs (it is being cotaught) and the first thing either of them asked me about was my background and whether I really had the prereqs – including things like linear algebra. I had to stop and remember that at places other than Mudd you can’t just assume that by junior year everyone knows how to work with eigenvalues and eigenvectors…
In the evening Christina and I decided to explore the outer boundary of the castle. We were reminded of when we were little and would go to playgrounds and climb around the play castles. Now, as big kids we find ourselves in
Chelsey and I made dinner for a large group of people again. This time it was Mexican themed. However, tortillas proved difficult to locate here, so Chelsey made tortillas while I concocted something that approximated a vegetarian chili. It came out delicious, but there does not seem to be anything actually called chili powder here. Instead we used something labeled paprika that smelled about right. Thus we ended up with "chili", rice, chewy (but amazing) homemade tortillas and various vegetables, cheese, and sour cream to pile on the tortillas. I love cooking :-)
Sunday
Christina and I met for breakfast and then went to the Lutheran Church again. It was exciting because we managed to understand enough to recognize one of the readings (the fruits of the spirit) despite the fact that it was in Hungarian! I also caught that part of the sermon was about students (by the words for student, school, studying, etc...), though I do not know exactly what about students.
In the afternoon we decided that it was too nice a day to stay inside and that we should go on an adventure. We took the metro to Moskva Tér and then rode the 56 tram out to its terminus in the hills. After wandering around we found a marked trail and went for a nice walk through the woods :-)
A pretty tram stop and a nice view of Buda
Thursday, September 6, 2007
Este az duna szép.
Before I start, a short note:
I have noticed that these posts can get rather long. Is this a good thing? Should I be more selective about what I post about? Are there things I post about that are not necessary? Is there anything you want to here about that I am not talking about? Please feel free to give me comments, either on this blog or via email. :-)
Weather exists here! The last couple of days have been on the rainy coldish side. They are not good for taking pictures, but the existance of actual weather makes me happy.
I have noticed that these posts can get rather long. Is this a good thing? Should I be more selective about what I post about? Are there things I post about that are not necessary? Is there anything you want to here about that I am not talking about? Please feel free to give me comments, either on this blog or via email. :-)
Weather exists here! The last couple of days have been on the rainy coldish side. They are not good for taking pictures, but the existance of actual weather makes me happy.
Szombat (Saturday)
While we were at school on Friday, Zoli went to the allatkert (zoo), so he was excited to talk about animals. I learned that there are az majom és az kis majom (a monkey and a baby monkey) amongst other allatok (animals). For dinner I went over to Christina’s and she made spaghetti for dinner. After dinner I took the random baking supplies (all labled in Hungarian) that had been left in her apartment by the girls who were here last spring and made something that approximated cookies. We had no measuring implements (well, other than 2 measuring funnels…) so I just did everything by eye and did not use a recipe. Given that I think the venture was a success.
Cookies!
Vasárnap (Sunday)
Christina and I met up with Kyle and went to Mass at Szt. Istvan’s Basillica. The building is very ornate, breathtakingly beautiful and completely distracting. I had a very difficult time focusing on the sermon (in Hungarian of course, it is good practice – I can pick out words and phrases sometimes). After church we went over to Margit sziget and wandered around in the gardens. There is an exciting fountain that is choreographed to music, lots of pretty flowers and the ruins of an abbey.
Here is a cool sculpture we passed on our way from church to the island. It has 3 parts, the one closest to the front is like a 3-D version of a möbius strip :-)
Pictures I took at Margit Sziget
Turtles!
Also, this is a picture of the Hungarian equivalent of those machines you put 50 cents into in the US and get some cheap toy out.
Yes, that is a Rubik cube.
Hétfő (Monday)
We all met up with another study abroad program for a 2 hour cruise on the Duna (Danube) after class. It was very pleasant and Christina I got some good pictures.
Kedd (Tuesday)
I met up with Mandi for dinner at a little hummus bar, which is not Hungarian food but was delicious and cheap. We then met up with several other friends at a bar that you would not find if you did not know it was there. From the street you can only see an unlabeled doorway that does not look as though it leads anywhere in particular. If you walk back through it a ways you get to a courtyard with tables set up and on the far side of the courtyard is the building that contains the bar itself. The inside seating area is in what looks as though it may have at one time been a wine cellar, but it has great acoustics (and is non-smoking!) and they had a live band playing traditional Hungarian music (which is why we were there). It was very cool.
Szerda (Wednesday)
Today was the last day of the language school. We spent a little bit of time reviewing, but the vast majority of class was dedicated to writing and performing a skit in Hungarian about the “Adventures of an American Student in
After class Christina and I set off to find the concert schedule at the palace of the arts. This turned out to be more of an adventure than expected as the 2-ös tram line currently has a gap in the middle caused by construction, so we got to take a bus and then find the tram again. Overall, Budapest’s public transportation system is amazing though. It is slightly confusing at first as there is the underground metro, an above ground tram system, busses and trolleys, but once you get an idea where the buses run you can easily (and fairly quickly) get anywhere in the city. The biggest problem is that the whole system stops running at about 11 pm... One of these days I will figure out the night bus system though, and then that will be fine too ;-)
Szia!Saturday, September 1, 2007
Jon a tigris!
Sziasztok!
If you don't want to read through all of my rambling, there are pictures at the bottom ;-).
On Friday morning as I was riding the bus to school I realized that I was starting to feel much more comfortable with the general flow of life in Budapest. For instance, riding the 16 bus to school feels natural, I can quickly navigate the mess of a metro station (everything meets there) that is Deák Tér, Astoria (the square the language school is located on) has begun to feel very familiar and I am able to read (or at least discern the meaning of) more and more signs. Friday afternoon, Christina and I decided to go find a park that we had noticed on the map, so we took the blue line farther south than we had previously been in the city. The metro station there had a completely different feel, than the ones we use frequently. Once out of the metro station, we realized that that part of the city also had a rather different feel to it, and we quickly concluded that, while a very small portion of the city has started to feel like home, we are definitely still in strange, new city.
The Hungarian language is absolutely fascinating.
Possession uses a very different construction. If you want to say Marsha's dog, you add a possessive ending to the word dog, not the word Marsha. The Hungarian word for dog is kutya, so you would say "Marsha kutyaja".
It also has no word that is the equivalent of "to have". If you want to say "Marsha has a dog" instead of "Marsha's dog" you simply add "van" (which means "he/she is") to obtain "Marsha kutyája van".
This week has consisted largely of going to language class and studying the language. As interesting as I find it, I have to admit that I agree with the general consensus that we will be glad when math classes start... On Szerda (Wednesday) several of us met for dinner at the palacsinta place and then went over to Christina's to hang out and play set. On Csücsörtök este (Thursday evening) Éva's grandson Zoli came over. He started out trying to just talk to me, but quickly realized that I did not understand him at all, so he started quizzing me on the colors and counting everything in the room :-) The color sárga seems to incorporate everything orange and yellow. He stayed with us until this morning and has been patiently teaching me various Hungarian words. For instance, yesterday while I was at school he went to the állatkert (zoo) and saw az anya majom és a kis majom (a mother and baby monkey). As a mentioned before, Christina and I went to explore a park on Péntek (Friday). It was really pretty and had lots of trees. We were happy to find that it felt much less urban than most of the other parks we have been to here. After walking around the park for a while we went to the Pink Cadillac - a pizza place that had been recommended to her by several people. It turned out to be a really cute place with absolutely delicious pizza. We split a pizza that had ham, onions, spinach and garlic. After the pizza we stopped at one of the little ice cream shops - I had citrom, lemon. As we were walking back towards the metro station we looked down a side street and noticed and amazing sunset, so we set off towards the river and walked along the Duna (Danube) for a while. It was quite pleasant. Tonight I am planning to meet up with Christina to do something - we have not decided what yet, but it will undoubtedly be fun.
And now, pictures:
Buildings on the top of the hill I live on in Buda:
I tend to ride the 16 busz to and from Buda
At the park we found this building. It was fenced off and we think it was some sort of electrical building, but it looked cool and I got a nice picture.
Swinging at the park
A street in Pest
Sunset over one of the bridges
If you don't want to read through all of my rambling, there are pictures at the bottom ;-).
On Friday morning as I was riding the bus to school I realized that I was starting to feel much more comfortable with the general flow of life in Budapest. For instance, riding the 16 bus to school feels natural, I can quickly navigate the mess of a metro station (everything meets there) that is Deák Tér, Astoria (the square the language school is located on) has begun to feel very familiar and I am able to read (or at least discern the meaning of) more and more signs. Friday afternoon, Christina and I decided to go find a park that we had noticed on the map, so we took the blue line farther south than we had previously been in the city. The metro station there had a completely different feel, than the ones we use frequently. Once out of the metro station, we realized that that part of the city also had a rather different feel to it, and we quickly concluded that, while a very small portion of the city has started to feel like home, we are definitely still in strange, new city.
The Hungarian language is absolutely fascinating.
Possession uses a very different construction. If you want to say Marsha's dog, you add a possessive ending to the word dog, not the word Marsha. The Hungarian word for dog is kutya, so you would say "Marsha kutyaja".
It also has no word that is the equivalent of "to have". If you want to say "Marsha has a dog" instead of "Marsha's dog" you simply add "van" (which means "he/she is") to obtain "Marsha kutyája van".
This week has consisted largely of going to language class and studying the language. As interesting as I find it, I have to admit that I agree with the general consensus that we will be glad when math classes start... On Szerda (Wednesday) several of us met for dinner at the palacsinta place and then went over to Christina's to hang out and play set. On Csücsörtök este (Thursday evening) Éva's grandson Zoli came over. He started out trying to just talk to me, but quickly realized that I did not understand him at all, so he started quizzing me on the colors and counting everything in the room :-) The color sárga seems to incorporate everything orange and yellow. He stayed with us until this morning and has been patiently teaching me various Hungarian words. For instance, yesterday while I was at school he went to the állatkert (zoo) and saw az anya majom és a kis majom (a mother and baby monkey). As a mentioned before, Christina and I went to explore a park on Péntek (Friday). It was really pretty and had lots of trees. We were happy to find that it felt much less urban than most of the other parks we have been to here. After walking around the park for a while we went to the Pink Cadillac - a pizza place that had been recommended to her by several people. It turned out to be a really cute place with absolutely delicious pizza. We split a pizza that had ham, onions, spinach and garlic. After the pizza we stopped at one of the little ice cream shops - I had citrom, lemon. As we were walking back towards the metro station we looked down a side street and noticed and amazing sunset, so we set off towards the river and walked along the Duna (Danube) for a while. It was quite pleasant. Tonight I am planning to meet up with Christina to do something - we have not decided what yet, but it will undoubtedly be fun.
And now, pictures:
Buildings on the top of the hill I live on in Buda:
I tend to ride the 16 busz to and from Buda
At the park we found this building. It was fenced off and we think it was some sort of electrical building, but it looked cool and I got a nice picture.
Swinging at the park
A street in Pest
Sunset over one of the bridges
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