Sunday, July 13, 2008

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:






2 comments:

Karen said...

Hey!
-Suggestion for future canoeing or other boating trips where you would like to take pictures: buy one of those old disposable cameras;)! That way, hopefully you can get lots of great pics, but should it happen to get too wet, you haven't ruined your good camera :)
-Also I really enjoy your dialogues! They crack me up, lol!!!
-And congrats! You have successfully given me at least a basic understanding of a Grobner Basis! It makes me feel smart :)However, I think I will let you continue learning all those patterns and generalizations of numbers and how they go together, and I will stick with memorizing all the pieces and parts of the body and how they all work together =D

Love you!!!
Karen :)

Sarah said...

Obtaining one of those disposable cameras would require me to have more advance notice of what we are doing, but I'll keep that in mind :-)