Dynamics on Calabi--Yau and abelian varieties

点击翻译按钮获取中文摘要

This exploratory project will study symmetries of geometric objects called algebraic varieties. These are shapes defined by algebraic equations: the simplest example is the circle of radius 1, which is the set of solutions of the equation x^2+y^2=1. A symmetry is a function or rule that sends each point on the variety to another point on the variety, in a reversible way: for example, rotating a circle around its centre by some chosen angle. In our project, we will study algebraic varieties with more dimensions, and our goal is to get a better understanding the properties of symmetries on these varieties. In particular, we would like to understand the entropy of the symmetries. Entropy is a measure of the complexity of a symmetry; higher entropy means that the symmetry is more complicated, with nearby points on the variety moving apart more quickly as the symmetry is applied again and again. We will study this problem for two different kinds of algebraic varieties, called Calabi--Yau varieties and abelian varieties. These kinds of varieties are of interest in geometry, number theory and theoretical physics. In this context there are many interesting symmetries that we can study and numerous theoretical tools that we can use to calculate entropy. By studying our problem in this context, we hope to gain a better understanding of the possibilities for entropy of symmetries.