Mathematical Sciences: Topics in Ergodic Theory and Analysis

贝娄教授和琼斯教授将继续他们正在进行的研究几乎无处不在的收敛问题，这些问题与沿子序列和沿其他算子序列的逐点遍历定理有关。对某些等差数列的均值收敛将是一个特别感兴趣的焦点。Calderon-Zygmund型分解技术将用于研究这类问题。主要研究人员还将研究另一个极端的问题，涉及到强清扫性质，在这种情况下，a.e.在sup规范中的收敛性非常严重地失效。这个项目是遍历理论中的数学研究。该理论关注的是，当一个合适的底层空间的合适变换被迭代多次时，在长期运行中平均会发生什么。（通常，底层空间可能只是一个线段，转换可能会将线段切割成碎片，然后拉伸或收缩并重新排列它们。）一种观点是研究函数的实参反复变换时在空间上的行为。经常发生的情况是，函数迭代的平均值以一种有意义的方式收敛。这门学科最近的一个趋势是研究子序列的平均值，例如平方数或质数，而不是所有的整数指标。从这个意义上讲，贝娄教授和琼斯教授将对子序列进行遍历理论研究。

Professors Bellow and Jones will continue their ongoing investigation of almost-everywhere convergence questions arising in connection with the pointwise ergodic theorem along subsequences and along other sequences of operators. A.e. convergence in mean for certain arithmetic sequences will be an especial focus of interest. Calderon-Zygmund type decomposition techniques will be used to study such problems. The principal investigators will also pursue problems at the other extreme involving the strong sweeping out property, where a.e. convergence in the sup norm fails very badly. This project is mathematical research in ergodic theory. This theory is concerned with what happens on average over the long run when a suitable transformation of a suitable underlying space is iterated many times. (Typically, the underlying space might be just a line segment, and the transformation might cut the line segment up into pieces, then stretch or shrink and rearrange them.) One point of view is to study the behavior of functions on the space when their arguments are repeatedly subjected to the transformation. It frequently happens that the averages of the iterates of the functions converge in a meaningful fashion. A recent trend in the subject is to look at averages over subsequences, for instance the squares or the primes, instead of over all integer indices. Professors Bellow and Jones will pursue ergodic theory over subsequences in this sense.