Dynamical systems and number theory
Take any positive integer. If is is 1, stop. Otherwise, if it is even, divide it by 2; if it is odd, multiply it by 3 and add 1. This gives you a new number. Repeat the procedure until you get 1. What do you think: will you ever come to 1 or not? For which initial numbers? (In fact, nobody knows yet). This simple rule is an example of a dynamical system. Another classical example is the motion of the solar system. Here one knows the gravitational forces between the bodies, and thus the accelerations. One can (more or less) easily write the corresponding system of differential equations. Now we want to predict the time-evolution of our system, for example, will Mars and Venus ever collide? Or will some planet eventually fly away? However, for this system it is very difficult, even impossible, to find an explicit formula for the solution. Instead, one has to pose more general questions: Are the solutions (in general) bounded for all times? How do they behave? Are they regular or chaotic? If unbounded solutions exist, how do the orbits escape to infinity? It is this kind of questions that the theory of Dynamical Systems deals with.
It appears that even ``easy-looking'' dynamical systems can exhibit a very rich spectrum of behaviours, from periodic to chaotic. The picture above illustrates the behaviour of the so-called Chirikov standard map: a simple-to-formulate dynamical system which exhibits chaotic behaviour, and whose complete description is still out of reach of the modern science.
Number theory is the study of integers and primes. Typical questions are: as X becomes large, how many primes less than X are there? Can all large odd numbers be written as a sum of three primes? Which integers can be written as sums of two (integer) squares? Can a sum of two cubes equal a cube?
The subject has connections with many branches of mathematics, e.g., algebraic geometry, combinatorics, harmonic analysis, and spectral theory. Somewhat surprisingly, there are also connections to mathematical physics, in particular quantum chaos: the zeroes of the Riemann zeta function (which encodes the distribution of prime numbers) very remarkably appears to be closely related to the spectrum of the quantization of a chaotic dynamical system (as predicted by random matrix theory). Quantum chaos is also concerned with models of "chaotic eigenfunctions"; M. Berry has proposed that such eigenfunctions in many ways should behave as random superpositions of randomly oriented waves (with the same frequency). One model for which rigorous results on the random wave model are known is the behaviour of Laplace eigenfunctions on the torus. Here it turns out that the length of nodal lines (i.e., the zero sets) of random eigenfunctions are intimately related to the ways an integer can be written as a sum of two squares.