Dynamics on Calabi--Yau and abelian varieties

这个探索性的项目将研究被称为代数簇的几何对象的对称性。这些形状是由代数方程定义的：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。对称是以可逆的方式将变数上的每个点发送到变数上的另一个点的函数或规则：例如，将一个圆围绕其中心旋转某个选定的角度。在我们的项目中，我们将研究更多维的代数簇，我们的目标是更好地了解这些簇上对称性的性质。特别是，我们想要了解对称性的熵。熵是对对称复杂性的衡量；越高的熵意味着对称越复杂，随着对称的反复应用，变化上邻近的点移动得更快。我们将研究两种不同类型的代数簇，称为Calabi-Yau簇和Abel簇。这类变种在几何、数论和理论物理中都很有趣。在这个背景下，有许多有趣的对称性可以研究，也有很多理论工具可以用来计算熵。通过在这一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.