Mathematical Sciences: Classification of the Compact Extensions of a Multidimensional Bernoulli Shift

卡迈耶教授将研究多维伯努利位移的有限扩展。(这里的多维是指几个交换变换在概率空间上的作用，而不是单个变换的作用。基过程的有限扩展是它与有限置换群作用的斜积。特别是，她将试图确定这种延长本身是一个伯努利过程。她的猜想是，如果扩展是弱混合的，就会发生这种情况，否则可能性在数量上是有限的，并且是明确可识别的。所有这一切的一维情况都很容易理解，并且已经在微分几何中自然产生的流动中找到了应用。类似的应用被设想为多维的结果。卡迈耶教授将要进行的数学研究属于遍历理论的范畴。这个主题的基本关注点是，从长远来看，当一个空间的变换（比如一个具体的几何对象）被迭代多次时，平均会发生什么。这一领域的有用结果告诉我们，如果忽略底层空间的精细结构，并且如果转换完全混淆了各种情况，那么只会发生可管理数量的不同情况。正如卡迈耶教授的项目所示，现在有相当大的兴趣将这些结果扩展到同时和可交换地作用的几个转变的情况。该项目得到了妇女研究机会倡议的支持。

Professor Kammeyer will study finite extensions of multi- dimensional Bernoulli shifts. (Multidimensional here refers to the action of several commuting transformations on a probability space, as opposed to the action of a single transformation. A finite extension of the base process is a skew product of it with an action of a finite permutation group.) In particular, she will seek to determine when such an extension is itself a Bernoulli process. Her conjecture is that this occurs if the extension is weakly mixing, and that otherwise the possibilities are finite in number and explicitly identifiable. The one-dimensional case of all this is well-understood and has found applications to flows that arise naturally in differential geometry. Similar appli- cations are envisioned for multidimensional results. The mathematical research to be undertaken by Professor Kammeyer falls under the heading of ergodic theory. The basic concern of this subject is what happens on the average, over the long run, when a transformation of a space (say for concreteness a geometric object) is iterated many times. Useful results in this area tell one that if fine structure of the underlying space is ignored, and if the transformation mixes things up thoroughly, then only a manageable number of distinct situations can occur. There is now considerable interest, as exemplified in Professor Kammeyer's project, in extending such results to the case of several transformations acting simultaneously and interchange- ably. This project is supported under the Research Opportunities for Women initiative.