In which I pretend this is a photo blog

I was in an artistic sort of mood this week, so I did a better job than usual of carrying (and remembering to use) my camera. Today the REU group went on a hike. Thus you get lots of pictures :-)

First off, my birthday was this week. My birthday cake:
I enjoy decorating birthday cakes, but when I get to choose the cake I go for a jam cake that does not lend itself to decoration. Go figure. It was delicious though :-)

Around campus:
My view from the bench where I sit to eat lunch every day.









A couple of pictures of different angles of the same flower. I'm not usually attracted to the red ones, but for some reason this one caught my eye.




I have no idea what these are, but they are really cool!








Squirrels!







And one from the construction site near the math building... The names that some companies decide to take never fail to amaze me.








Busiek State Forrest:
Not very far along the trail we came across a pretty butterfly. Shortly after that we reached a stream where the bridge was out...


And then we found the bridge...









The rest of the hike was very pretty, but not terribly eventful.











And of course we have flowers from along the trail:

Looks like Red, Tastes like Cake

This past Friday was Charlie's birthday (one of the other REU students). He wanted red velvet cake. Red Velvet cake is red simply because of ridiculous quantities of food coloring. As in, we dumped two one ounce bottles of red food coloring into the cake batter (well, first just into the buttermilk)

With the bright red batter mixed, we dumped it into the one casserole dish we have that is almost the right dimensions for a cake pan. You can see the two empty food coloring boxes behind it. In the process of baking, the red got darker - but no less red. We frosted it with cream cheese frosting and I used the few drops of red food coloring that did not make it into the cake itself to dye a little of the frosting red for decoration:

It is a tasty cake (delicious and moist with a hint of chocolate), but in the future I am going to be inclined to replace the food coloring with water or more buttermilk unless there is a good reason to need a red (or for that matter any brightly colored) cake. There will be more cake baking this week. This time it will not involve the dumping in of food coloring...

Friday night several of us met one of the professors for dinner at Hickok's Bar and Grill (their website is rather entertaining). It is themed after the old west and in particular after the shootout between Wild Bill Hickok and Davis Tutt. In fact, they were going to be doing a reenactment of the shootout later that night, but we decided we did not want to wait around that long. I was overall amused. Oh, and they had tasty fried okra :-) .

It was determined that the REU group ought to visit Branson at some point. One of the professors discovered that Yakov Smirnoff (a Russian comedian) has decided that he wants a talk show and so was giving out free tickets to the taping of a pilot episode. When we arrived at the theater one of the ushers that was catching everyone at the door realized that we were a group of mainly college students and as a result pulled us aside and took us in early to pre-seat us near the front.

It was an interesting experience. I can't say that it made me have any desire to watch the talk show if it actually becomes one, but then again I would be amazed if there were a talk show that I would consider worth watching. He had interesting ideas about the importance of laughter in relationships. I was amazed at how disorganized it all seemed to be. I would have thought that they would have figured out things like how they were going to get the right camera angle to film objects on the table ahead of time...

I finished both Hardy's Tess of the D'Urbervilles and Dickens's Oliver Twist this week. I am greatly enjoying reading both authors, but I think I have a slight preference for Dickens. Dickens likes to have a narrator that is not afraid of breaking away from the story for a moment to directly address the reader and for me this seems very natural (when I am relating a story to my friends I am very prone to going off on side tangents). Dickens also uses amazing analogies, for instance, in Oliver Twist the narrator is justifying jumping back to the countryside when Oliver has found himself in quite a mess in London and starts out his explanation with:

It is the custom on the stage, in all good murderous melodramas, to present the tragic and the comic scenes, in as regular alternation, as the layers of red and white in a side of streaky bacon.

Down the River

This week's baking adventure was the result of having two overripe bananas but neither a loaf pan nor muffin tins (or, for that matter, my banana bread recipe). I creamed half a stick of butter with a handful of sugar and then mixed in an egg and the two mashed bananas. I then mixed in a pinch of salt, a little baking soda, some cinnamon and enough flour to make a reasonable batter (thicker than I would usually make banana bread batter but not as thick as I usually make cookie dough). I divided the resulting batter into six blobs on a cookie sheet and baked it at what my oven calls 325. This was the result :

They were a little spongier than I was expecting, but were deemed a tasty success. :-)

Two weeks ago a group of us decided to make pancakes for Sunday brunch. Last week we made pancakes, scrambled eggs and home fries. This week the group was larger and we had pancakes, scrambled eggs, home fries, bacon, and cinnamon rolls. I wonder if it will continue to grow next week...

While we were cooking this morning we got a call from one of our professors asking if we wanted to go canoing. The response was a resounding "yes!", so we arranged to meet the professors after we had our giant brunch. We went out to Lake Springfield Park and spent about an hour and half canoing on the James River. It was gorgeous, but I unfortunately don't have any pictures since I figured taking my camera on the canoe might not be the most brilliant idea in the world.


A couple of other interesting incidents from this last week:

Scene 1: My self and a girl and guy I don't know happen to arrive back at the dorm at the same time and get on the elevator. I'm wearing my blueberry pi shirt.

girl: *looks at my shirt* Is that a Hebrew letter?
me: *smiles* It's actually the Greek letter pi.
girl: *giggles and looks really embarrassed*
guy: *to girl* Well, you're on a college campus...
me: ...

At this point the elevator thankfully reached the 8th floor and I got off and I managed to get to my room before breaking down laughing. That was a new one.
___________________
Scene 2: I've gone to the library to pick up a couple of books on Gröbner bases so that I can understand a math article I am supposed to be reading and went ahead and picked up Oliver Twist and Hardy's Return of the Native while I was there.

kid running the circulation desk: What's a Gröbner base?
me: *recalls that I am not at Mudd and thus cannot assume that people know linear algebra* Um, do you know any linear algebra.
him: Not really, but I have a lin al book!
me: Ok... Well, its sort of like a generalization of the kind of basis you will encounter in linear algebra...you can sort of think of it giving us a way to represent a generalization of numbers...
him: So it has something to do with area?
me: ... not quite.
______________

Since then I have been trying to think how I would convey the idea of a Gröbner basis without using at least linear algebra. I think I've figured out how to get the general idea across without using anything more complicated than polynomials:

The degree of a polynomial is the highest power of the variable, say x , that appears in the polynomial. For example:

5x^2+3x+2 has degree 2.

Now, suppose we wanted to think about the collection of all polynomials that have degree at most two. Any such polynomial can be uniquely written as

ax^2+bx+c

where a, b and c are allowed to be any number, but can not have any x's. So a=42 is fine, a=6x is not allowed. As a mathematician I would then say that { x^2, x ,1} is a basis for the collection of polynomials of degree at most two.

However, it is also possible to write any degree at most two polynomial uniquely as

dx^2+e(x+1)+f

where I again allow d,e and f to be any number. This means that {x^2, x+1, 1} is also a basis for the collection of polynomials of degree at most two.

For a relatively simple collection like the set of polynomials of degree 2, any of the many possible bases (the plural of basis is bases) we choose is going to behave nicely. When we start getting into more complicated sets of polynomials (allowing more variables, allowing the coefficients themselves to contain variables, etc) it is possible to find a set that serves as a basis , but does not behave as nicely as we would like. A Gröbner basis is a basis that also behaves nicely in other ways. They are named after the Austrian mathematician Wolfgang Gröbner and were first introduced in 1965.

What does math research look like?
Lots and lots of paper:

Oh, and we had a couple of spectacular sunsets this week:

Well, That Was Different...

July 4

I made myself a really tasty breakfast by mixing some left over acorn squash with a handful of oatmeal, a little milk and some cinnamon and then curled up on the couch to read Great Expectations. Pip (the main character) managed to make me so mad that when one of our professors called suggesting that we should all go out to a movie, I actually decided to go. We went to the Moxie - a tiny independent cinema in downtown Springfield. The Fall is a really, really weird movie.

We had planned to go to the Thai restaurant near campus after the movie, but it turned out to be closed for the long weekend. The second Thai restaurant we tried was also closed. Happily, the third Thai place we tried was open. I was surprised to discover mango curry on the menu (I had never considered the idea of mango in curry) but it was delicious, if not deserving of the 5 star spiciness level they claimed...

After dinner, the professors pointed us in the direction of the fireworks show (they did not want to deal with the crowds). We were told that the fireworks were at the intersection of Division and 65. While driving down Division toward 65 we observed that there were a lot of people who were parked in various parking lots and set up there to watch the show. We parked in one of these lots and then decided to walk down Division to get closer. We briefly pondered how bad an idea it would be to walk across the overpass, but decided to go for it (the shoulders were really wide and there were plenty of other people crossing) and managed to get close enough that we could hear the music they were playing to go with the fireworks. Now, in my experience 4th of July fireworks displays are generally accompanied by a variety of patriotic music, but that was not quite what we found. For some reason the Springfield 4th of July fireworks show is organized by one of the local super churches and so we watched fireworks choreographed to the kind of music I would expect from a praise band at a "contemporary" church service. It was fine, but not quite what we were expecting. The fireworks were nice enough.


When the show was over we were rather amused that we were walking faster down Division than the traffic was moving until shortly before we made it back to where we had parked. Upon getting back to the dorm I had an amusing conversation with one of my suite mates:

Her: We should have ice cream!
Me: I'm more in the mood for cookies...
Her: (skeptically) Do we have cookies?
Me: We could.
I pause briefly to glance at the time, decide that 10:30 is not too late to make cookies, and pull a stick of butter out of the fridge.
Her: (excitedly and amazed) Are you going to make cookies?!
Me: Yes...

So we had oatmeal chocolate chip cookies - and managed not to let our oven burn them this time. I'm getting better at identifying how browned the bottoms of cookies are getting by smell...



July 5

The bank is right across the street from the mall and far enough from the dorm that we have not felt like walking. The suite mate who drives uses a different bank, but is a fan of the mall, so she took us to the bank and then we went to the mall. As a result I ended up spending multiple hours on the 4th of July weekend "going shopping" at the mall. Where by shopping I mean following the other girls from store to store and remembering why it is that it has been a while since I have bought anything other than pants at the mall... That said, I was somewhat impressed to discover that while it still contains nothing that I would wear, the junior department actually contains clothing that is not completely hideous. It was also nice to note that there is a pretty teal and purple combination that appears to be "in" this year.

We saw an interesting delivery truck as we were driving to the mall:



ps. I have discovered that my roommate's camera cord works with my camera, so you get pictures again :-)

Pie!

Last Thursday after dinner a couple of us were sitting and chatting and someone suggested that ice cream would be delicious. It was observed that the nearest ice cream place that is open later than 5pm is about a 40 min walk away. After considering this for a little while, we decided to go anyway :-). It was a very pleasant walk and since it was just getting dark the fireflies (incidentally, I have realized that I alternate randomly between calling them fireflies and lightning bugs... I wonder where I picked up which name...) were all out. I had forgotten how pretty they are! It is really too bad that we don't have them at Mudd... The ice cream was tasty (I had German chocolate with pecans and coconut) . We took a different route back to the dorms - I have to admit that I find residential neighborhoods rather more interesting than walking past used car lot after used car lot, but it was still a reasonably nice walk.

It was observed that yesterday (6/28) could be considered 2pi day and as such was determined that there needed to be pie. We made three pies - apple, strawberry rhubarb, and cherry. Making pie crust when it is extremely humid is quite the adventure (made even more so by the fact that the closest thing we had to a rolling pin was a coffee mug...), but we managed. It was relatively late in the day when we finally got around to baking and one of our professors (Jorge) had invited us all over for lunch today( Sunday) so we decided to save the pies to take with us to lunch.

Jorge and his wife are from Peru, so lunch included ceviche, yucca root and Peruvian corn (in addition to some of the sorts of food you would be more likely to expect at a barbecue). I had heard of ceviche, but never had it, and from what I had heard I was rather skeptical - fish "cooked" by leaving it in lime juice overnight rather than using heat just did not sound terribly appealing, especially as I am not generally a huge fan of seafood. However, I try to make a point out of trying everything at least once - and today reminded me why. It was absolutely delicious, especially with the yucca!

The pies went over well. It is always interesting to note which pies get finished first... Today the strawberry rhubarb was finished and the apple was down to one piece before anyone took the cherry. I had never had strawberry rhubarb before but friends have been telling me for some time that I needed to learn to make it, and I had happened to notice rhubarb at the grocery store, so I figured I might as well try. I was somewhat surprised at how excited people got at the prospect of rhubarb pie - prior to going to college I had never even encountered the idea (or at least had not taken note of the idea)- but it really is quite good.

We then spent the afternoon playing Scattergories and chatting with the professors and their wives (or at least I did, some people watched the soccer Euro 2008 final game and some movie). Overall, it was a very pleasant afternoon. :-)

Math is going well. I'm working on planning out a series of "higher level math for the non-mathematician" posts that will work toward giving you an idea of what it is I am working on this summer.

I have finished Bleak House and will be starting in on Great Expectations (also by Dickens) next. It is really interesting to be reading several of Dickens's novels in succession. I am noticing recurring character types and situations between the novels that I almost certainly would not have noticed if the other novels we not so current in my mind.

"There are dark shadows on the earth, but its lights are stronger in the contrast."

The above quote from the Pickwick Papers seems particularly apt given the rather dark clouds that have been coming and going all week (and that sent us to the basement Thursday with a tornado warning). I still appreciate actually having weather though :-)

Earlier this week I decided that we needed cookies. Since discovering in Budapest that I could make fine cookies without the bother of carefully measuring things out, I have almost exclusively been making "variations on approximation cookies." Now, given this style of baking I am never quite sure how long the cookies are going to take to bake and I am well aware of the fact that different ovens have different cooking times - but they usually take ~10-12 minutes. Less than 5 minutes after putting the first tray of cookies in the oven the apartment began to smell of popcorn (why the burning bottoms of cookies smells like popcorn I have yet to figure out)... They were definitely still edible, but their bottoms were rather blackened. I turned the oven down from "350" to "300" and gave it time to readjust the temperature before putting in the second tray - 12 minutes later the cookies were still slightly gooey. I took them out anyway and they /mostly/ set up nicely as they cooled and ended up being really soft. I think I need to experiment more with baking cookies longer at a lower temperature. I also need to remember to be really careful when dealing with the oven here...

Friday night a group of us from the REU walked into downtown Springfield (roughly the same distance that it is from Mudd to the Village) to find dinner. After wandering around for a while (which reminded me rather strongly of wandering up and down Raday in Budapest) we finally decided to eat at Trolleys and sat outside so that we had a view of the square. My chicken sandwich was good, though not spectacular, but it was served with sweet potato fries. I was slightly skeptical of the fries when I saw them on the menu (the time Mudd's dining hall tried to make them they seasoned them with salt and pepper, it really didn't work) but decided to try them anyway. I'm really glad I did - they were fantastic. I was through about half of them before I finally figured out that they had sprinkled them with cinnamon :-)

While we were eating dinner we discovered that there were local bands giving a free concert on the square later in the evening, so we ended up hanging out on the square in downtown Springfield listening to the music and people watching.

• an old bearded guy sitting right up by the stage who was /really/ into all of the music regardless of the style
• a group of three middle aged cowboys complete with cowboy hats and holsters with varied combinations of knives and guns
• plenty of little kids climbing on the big geometric sculpture and playing in the fountain
• a guy handing out glowsticks to any kid brave enough to go ask for one
• a group of students who didn't seem to know how to get clothes that fit and liked baseball caps with spikes
• and the list could go on and on

In short there was a rather large spectrum of people all gathered on the square.

When being less social and not doing math I have been doing a lot of reading. Having finished the Pickwick Papers (incidentally, Springfield has consecutive streets named Pickwick and Weller...), I started and have finished a collection of Dickens's Christmas writings and started Bleak House. I'm finding that I really enjoy Dickens. He manages to draw me into the story while maintaining the feeling that I am sitting with the narrator listening to him/her tell me the story. This latter impression is built by the small digressions and comments clearly directed at the reader that do not typically appear in novels. I was especially amused by Dickens's digression on idioms at the beginning of A Christmas Carol:

Old Marley was as dead as a door-nail.
Mind! I don't mean to say that I know, of my own knowledge, what there is particularly dead about a door-nail. I might have been inclined, myself, to regard a coffin-nail as the deadest piece of ironmongery in the trade. But the wisdom of our ancestors is in the simile; and my unhallowed hands shall not disturb it, or the Country's done for. You will therefore permit me to repeat, emphatically, that Marley was dead as a door-nail.

I have also been entertained by all of the allusions to various nursery rhymes in Bleak House and slightly surprised at how many of them I recognize before having them pointed out in the notes (and that I have found at least one that is not in the notes at all).

Now, it has been requested that I explain what exactly one does when doing "Math research." The simplest possible answer I can give is "math", but that is probably not a very satisfying answer. There are a few different kinds of broad things that count as math research (and this list is probably not exhaustive)

• Start with a problem that someone else has come up with but that no one knows the answer to and try to find the answer.
  
• Alternately, it is sometimes good to take a problem that we know the answer to and find a new way to get to the solution. This sort of thing is useful in mathematics since it can often give us a new way of looking at mathematical structures and lead to new insights.
  
• There are some areas of math that have been around since ancient times that are fairly well explored and understood. There are also areas of math and mathematical objects where things are named after mathematicians who are still alive today. In these areas it is quite possible to find interesting questions that no one has thought to ask and to then try to answer them.

My particular research this summer falls into some combination of the first and last of the above categories. I spent a good bit of the first week here learning about the theory of Local Rings and looking at the kinds of questions that my adviser for the summer has been interested in. From this we have come up with a vague question for me to attack - and working toward that is producing smaller more manageable problems to understand and solve. In particular, I am trying to understand the relationship between the zero divisor graph of finite local rings and the zero divisor graph of their corresponding graded ring.

I have been thinking about how to explain the general idea of what I am doing without using such technical terms, but I'm not quite there yet. I shall continue working on that :-)

Greetings from Springfield, MO

My parents and grandparents have complained that they knew what I was up to better when I was on the other side of the world than when I am just across the country and have thus requested that I revive this blog. I must admit that Springfield is somewhat less exciting a city than Budapest and I am not convinced that my life here is going to be quite as interesting to anyone still reading this as it was last fall, but I suppose I may as well give it a shot.

After a spring semester involving rather less math and rather more non-technical courses than I have grown accustomed to (but which was nevertheless enjoyable) I have taken off to Missouri to spend my summer doing math research. For those of you who understand the language of math, I am exploring the zero divisor graphs of local rings and in particular am analyzing the relationship between the zero divisor graph of a local ring and the corresponding graph of its associated graded ring. Other than math I have been doing quite a bit of reading and have finished the Pickwick Papers - next semester I am taking a class on Dickens and Hardy, so my summer reading is going to be largely centered around these two authors.

I'm not exactly sure what else to talk about, so I am going to turn to the weather. I remembered from back when we lived in Cape Girardeau that Missourians are fond of the saying "Don't like the weather? Wait a couple hours." but I had forgotten just how true it is. As an example, Sunday morning dawned bright and sunny. By noon we had light fluffy clouds and by mid afternoon we were in the midst of a thunderstorm that decided briefly to hail. Before sunset it had cleared up again...so that it could cloud over by morning and sprinkle on us as we walked from our dorm (Sunvilla, I'm on the 8th floor) to the math building. Since the forecast (hah) was predicting thunderstorms I packed a lunch. At noon I sat on a bench outside in the shade and watched the fluffy white clouds move across the bright blue sky. By 5pm it was completely overcast and threatening to rain again. The variability seems to be bothering some people, but I consider it a welcome change from the, shall we say, more stable climate of southern California.

(As a side note, I think spending a semester reading Shakespeare and the last week reading Dickens may have slightly altered my writing style...)