The project has three main strands, which are the speed of mixing of random walks, self-similar fractals and arithmetic properties of random polynomials. These seemingly unrelated problems are, in fact, intimately connected by common underlying mathematical phenomena.
Generating a random object with a specified distribution is an important problem in mathematics with manifold applications in the sciences. One commonly used approach to this problem is a random walk, which starts with a non-random starting object, which is then modified step by step in a random manner gradually increasing its randomness. Perhaps the most familiar instance of this is card shuffling. The order of the cards in the deck is modified in a random manner, making it more and more random after each shuffle. After sufficiently many steps of shuffling, the deck becomes truly random, each possible ordering of the cards becomes (almost) equally likely. It is an important question to understand how many shuffle does it take to make the deck random, and more generally how many steps of a random walk is needed to be taken to reach the desired target distribution. The mathematical notion, which makes this question precise is called the mixing time. One aim of the project is to understand the mixing times of certain types of random walks, which are endowed with an underlying algebraic structure. The interest in this problem is twofold. First understanding it will shed light on the underlying algebraic structure and on related problems in pure mathematics. Second, these random walks serve as "toy models" for more complex problems.
Self-similar fractals are arguably among the most natural families of fractals. They are characterized by the property that they contain exact scaled copies of themselves at all scales. The dimension of a fractal is a mathematical notion designed to measure its "size". It is one of the most basic problems in fractal geometry to determine the dimensions of fractals, yet this is still far from properly understood even for self-similar fractals. One of the aims of the project is to address this problem.
The third and last strand of the project aims to understand number theoretic properties of random polynomials. One such property is irreducibility, that is whether or not a polynomial can be written as the product of two or more polynomials. This is analogous to the notion of prime numbers. Such questions are very interesting in their own right, but the primary reason for including them in the project is their intimate relation to the other two strands.
