The idea for this post came from a question I saw in a math help forum about improper integrals. While this problem has a very simple solution using basic tools in integral calculus, I want to show a more geometric approach which can be generalized much further (and for the Hebrew speakers among you, I made a video explaining this as well here ).
Imagine going to a new amusement park for the first time. Once you get there, you go to the first ride that you see, and when you finish it, you randomly choose one of the roads leaving it and follow it until you reach another ride. You continue like this for a long time – every time you go on a ride, then choose a random road (might even go back on the same road that you came) and walk on it until you reach your next ride.
After doing this for a long enough time, on which ride would you go on the most? Go on the least? And how are these questions connected to eigenvectors and eigenvalues?
The radar is a detection system that was developed before and during World War II for military uses, though by today it has many other applications including, for example, astronomical and geological research. The name radar is an acronym for RAdio Detection And Ranging, and as its name says, it uses radio waves to detect targets. In this post we will try to understand one of the basic ideas behind the operation of the radar, which while it seems quite simple at first glance, when we start to try to understand the details, we encounter an interesting problem. In order to solve it we will need to use a mathematical theorem first formulated by the Chinese mathematician Sunzi from the 3rd century AD.
Mathematical research is something that most people don’t really understand. They can imagine someone in a lab mixing chemicals or doing experiments with some scientific machinery, but mathematical research?
The goal of this post is to share with you a small part of this process – how we start with a simple problem, what happens when we complicate this problem, and how eventually we get what mathematicians call a research problem. For those interested, I made a video about this subject which you can see here and a Hebrew version here.
The square billiard table
This problem’s origin is in the well known game of billiard. While in a general billiard game there are many balls, we will simplify it and have a square billiard table and a single ball. We then hit the ball and assume that it moves in a constant speed (so it doesn’t lose any momentum), and when it hit the edges of the table, it bounces back they we expect it to.
In this post we consider an interesting mathematical process which can be easily simulated by a computer, and generates interesting pictures. A video version of this post can be seen in here (for now in Hebrew).
Let and be the vertices of a triangle and choose a random point in the triangle. Next, choose one of the triangle vertices uniformly at random and move half of the way towards it – we denote this point by . We now repeat this step infinitely many times – given the point we choose uniformly at random a vertex from , move half way towards that vertex and denote this point by .
The “limit” of this process produces a very interesting picture, as can be seen below.
This process is called a random walk, since at each step we choose one of the vertices at random and continue according to this choice.
Which text gives us more information – the full body of work of Shakespeare, or 884,647 random words written by 1000 monkeys?
To answer this, we talk with our good friend Alice, and ask her to send both of these texts to her close friend Bob. It should seem reasonable that the more information “dense” a text is, the harder it is to send it. Since the time of Shakespeare, we invented the internet over which we can only send messages over the simple 0\1 alphabet so Alice’s job is to translate the text to this 0\1 alphabet. If she manages to find a good translation with fewer 0\1 bits, the easier her job is. If the original text contained a lot of redundancy, namely many parts of the text don’t contain “new” information, then we expect Alice could shorten the message a lot. Conversely, the more information the message has, the harder it is to compress it.
So now, Alice needs to chose for each English letter a word over and then the message is coded to be . This leaves us with the question of what is the best that we can hope to gain this way?
We came to the point where we have already seen how lattices appear naturally in problems arising from number theory. In this post we construct a nice space which parametrize the set of all lattices of a certain dimension, with particular emphasis on 2-dimensional lattices.
In the post about number theory and lattices, we tried to determine when is the Euclidean distance in is actually a Euclidean norm and we were led to study the embeddings of rings such as as lattices in . As mentioned there, not all rings can be nicely embedded in as lattices, and in this post we will try to find the “right” way to view them as lattices, and what other algebraic problems can be translated to this setting.
In the post about Diophantine approximation, we saw that in order to find “good” rational approximations to a real number, it is enough to prove that given a lattice and a “big enough” box around the origin, the box must contain a nonzero lattice point. This very intuitive result is due to Minkowski which started what is today known as the geometry of numbers. After several examples in order to get some intuition, we shall prove this result and see several of its applications.
Number theory can mean a lot of thing to a lot of people. This is a very big part of mathematics, and it contains many areas starting with the elementary number theory (“simple” congruence like arguments), algebraic number theory (e.g. field extensions and algebraic integers), analytic number theory (Riemann zeta function) and many more. I will mainly concentrate on the algebraic part, though this post will be very basic. I recommend reading my post about lattices before reading this post.
The main goal of this post is to describe a method that gives geometric interpretation to many algebraic questions. In this post I will only mention one such question, but in later posts I will mention several others.