|An algebraic geometer is called in when messages from an alien spacecraft appear to be asking questions about projective varieties. Though it may at first appear to be another "mathematics as a common language for first contact with aliens"-story, its purpose is actually to comment on one unexpected use of the internet. (See note below, but be warned that it contains "spoilers" that will give away important features of the story. You'll probably enjoy it more if you read the story first.)
Pop quiz is my favourite story [in the collection Reality Conditions]. It includes aliens! Mathematics is used as a universal language. It is common in Science fiction to use mathematics in order to communicate with aliens. The story is not as simple as it seems. There is Sarah, a mathematician, an expert for projective algebraic geometry. One day a satellite orbiting earth finds a foreign object (UFO) in Earth's orbit. Soon after that a message is sent to Earth. The signals send a mathematical question. They (I don't know who exactly) ask Sarah if she could help them. She gets three questions. The first one is easy for her. The second one is a bit harder. And the third one is a very hard one. She needs more time to solve it. In order to get the correct answer she has to make a new theory. Through all this questioning and answering she keeps questioning herself what will happen if her answer is correct or incorrect. At the end nothing really happens. Well, an alien visits Earth and apologises for the mistake that has been made. His son sent his homework to Earth so that they (we) would do it instead of him.
There are three things to discuss here. 1. Are there any aliens or are we alone?!? 2. The way we could communicate with aliens! 3. The level of our mathematical knowledge! My hobby is astronomy and I have to tell you that according to the huge number of galaxies, solar systems and planets in the universe it would be stupid of us to think that we are alone. The question is, how could we communicate with aliens. It is ironic, that in this story, Sarah went through so much trouble to solve those mathematical questions. She tried to figure out the message! In the end it turns out that Aliens speak English. It is possible that something like that could really happen. We always make things harder than they really are. After all, if a civilization was developed enough to build space ships and find a way to travel with the speed of light, it would be highly possible that they would be able to understand us or at least learn our language. The level of our mathematical knowledge is something that we rarely doubt about. We are convinced that we know a lot, that we are highly educated and that we know “everything”. This story shows us just how primitive we are. It would probably take us weeks or even months do to somebody's homework. I think that Egyptians and even mathematicians before them knew a lot more than we do now. Their knowledge got lost somewhere, somehow, … We call them primitive, but when we compare our and their mathematical knowledge, we are the primitive ones. We are only starting to learn things and we are certainly not the smartest beings in the whole universe. What shouldn't be left out is the role of women in this story. In the past women mathematicians had a hard time. Now things are different. In this story, a man (Franco) is supposed to solve those mathematical puzzles, but he sends for Sarah. He doesn't want to be the one who will make a mistake. He avoids responsibility! Sarah can handle that responsibility, she knows enough to find the correct answers and in the mean time, she even finds a new theory. I'm sorry that I can't tell you more about the mathematics that the story talks about. I know very little about projective algebraic geometry, Euclidean spaces, grassmannians, etc.
Of course, this story is completely fictional and some might even say that it is ‘far fetched.' But, it has more of a basis in reality than you might think! As a professor of mathematics, I quite frequently get e-mail from college, high school and middle school students from around the US and around the world, asking for help with their homework assignments. It is sometimes rather specific (e.g., “What is the formula for the 2-soliton solution to the KdV equation?”). One simply said: “I have to write a paper on surfaces. Can you give me information?” A local student here in Charleston wrote to the college and said that he had a report due in two weeks and had done nothing yet. He was hoping that for his report, one of the professors here would go to his class and make a presentation! The point is this: having been given access to the internet, a method for communicating almost instantaneously with experts in any part of the world, these students prefer to ask strangers for help rather than do the work themselves. And, with the huge number of people available to them on the internet, I wouldn't be surprised if they can generally find at least one person willing to do their assignment for them.
Some of the mathematics presented in this story is real. In particular, the stuff about homogeneous coordinates for projective space, Grassmannians and Grassmannian duality. (I left out a little detail about the grassmannian duality: we need a little bit of extra structure, like an inner-product on the underlying vector space, for the duality to exist.) However, the stuff Sarah figures out in answering the third question is completely made up, and probably makes no sense if you think about it carefully.