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

教授Kammeyer将研究有限扩展的多- 维Bernoulli移位（这里的多维是指 几个交换变换对概率的作用 空间，而不是一个单一的转换的行动。一 基过程的有限扩张是它的斜积， 一个有限置换群的作用）。特别是，她将 试图确定这样的扩展本身何时是伯努利 过程她的猜想是，如果扩展是 弱混合，否则可能性是有限的， 数字和明确识别。一维情形 所有这一切都很容易理解，并已应用于流 在微分几何中自然产生的。类似应用- 阳离子被设想用于多维结果。 教授将要进行的数学研究 卡梅耶福尔斯属于遍历理论的范畴。基本 这个问题的关注点是平均而言， 从长远来看，当一个空间的转换（比如说具体化） 几何对象）被迭代多次。有用的结果在 这个区域告诉我们，如果底层空间的精细结构 被忽略，如果转换彻底混淆了事物， 则只能发生可管理数量的不同情况。 现在有相当大的兴趣，如在教授 Kammeyer的项目，在扩展这样的结果的情况下， 几种变换同时作用并相互交换- 巧妙地。该项目由研究机会资助。 妇女倡议。

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.