The Dixmier-Moeglin equivalence and dynamical properties of automorphisms

我提议的研究涉及代数和动力学之间的相互作用。近年来，人们已经注意到，可以将代数对象与给定的几何对象以及某些几何数据相关联。这些物体被称为扭曲齐次坐标环。有趣的是，许多有趣的代数性质似乎与所谓的“动力学性质”有关。例如，当一个人有一个抽象空间和这个空间的变换时，一个人可以迭代这个变换，并询问随着我们重复迭代，空间中的某个给定点随着时间的推移发生了什么。这类问题是动力学的核心，理解其中的一些问题实际上会让我们深入了解相关的扭曲齐次坐标环。 代数所能拥有的最好的性质之一就是所谓的迪克斯米尔-莫格林等价成立。这种等价性使人们能够更好地理解代数的理想。目前，对于扭转齐次坐标环，DixmierMoegline等价是否成立还是个未知数，我建议的研究是证明这个等价确实成立。对于这些环，等价性可以从几何上理解，知道有关相关几何空间的某些动力学事实就足够了。 这项研究将结合许多不同领域的想法，因为它横跨几何和代数。

My proposed research deals with interactions between algebra and dynamics. In recent years, it has been noticed that one can associate an algebraic object to a given geometric object along with certain geometric data. These objects are called twisted homogeneous coordinate rings. What is interesting is that many interesting algebraic properties appear to be linked with what are called "dynamical properties". For example, when one has an abstract space and a transformation of this space, one can iterate the transformation and ask what happens to a given point in the space over time as we repeatedly iterate. These sorts of questions are at the heart of dynamics, and understanding some of these questions actually gives us insight into the associated twisted homogeneous coordinate rings. One of the nicest properties an algebra can have is for the so-called Dixmier-Moeglin equivalence to hold. This equivalence allows one to better understand the ideals of an algebra. At the moment, it is unknown whether or not the Dixmier-Moegline equivalence holds for twisted homogeneous coordinate rings and my proposed research is to show that this equivalence does indeed hold. For these rings, the equivalence can be understood geometrically and it is sufficient to know certain dynamical facts about the associated geometric spaces. This research will combine ideas from many different areas, as it straddles both geometry and algebra.