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沿着与某些几何数据相关联。 这些物体被称为扭曲的齐次坐标环。 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一个变换时，他可以对这个变换进行重复，并问当我们重复时，空间中的一个给定点随着时间的推移会发生什么。 这类问题是动力学的核心，理解其中一些问题实际上让我们深入了解了相关的扭曲齐次坐标环。 代数所能具有的最好的性质之一就是所谓的Dixmier-Moeglin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，这是未知的Dixmier-Moegline等价是否为扭曲的齐次坐标环，我提出的研究是要表明，这种等价确实持有。 对于这些环，等价性可以从几何上理解，并且知道关于相关几何空间的某些动力学事实就足够了。 这项研究将结合联合收割机的想法，从许多不同的领域，因为它跨越几何和代数。

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.