Dynamics on Calabi--Yau and abelian varieties

这个探索性项目将研究被称为代数变异的几何对象的对称性。这些是由代数方程定义的形状：最简单的例子是半径为1的圆，它是方程x^2+y^2=1的解的集合。对称是一种函数或规则，它以可逆的方式将品种上的每个点传送到品种上的另一个点上：例如，以选定的角度围绕其中心旋转一个圆。在我们的项目中，我们将研究具有更多维度的代数变种，我们的目标是更好地理解这些变种上的对称性。特别是，我们想要了解对称性的熵。熵是对称复杂性的量度；更高的熵意味着对称性更复杂，随着这种对称性的一次又一次地应用，变体上附近的点移动得更快。我们将研究这一问题的两种不同的代数变种，称为Calabi- Yau变种和abelian变种。这些变体在几何、数论和理论物理中都很有趣。在这种情况下，我们可以研究许多有趣的对称性，也可以使用许多理论工具来计算熵。通过在这种背景下研究我们的问题，我们希望能更好地理解对称熵的可能性。

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.