This research will focus on two topics: (1) the almost everywhere convergence of weighted averages, and (2) spectral type of rank one transformations. The first problem operators on probability spaces and the study of sequences of measures for which the corresponding operators fail to converge. In the second study, rank one transformations has been a source of numerous and important examples in ergodic theory; however, their spectral type has not been studied in depth. This research is in the general area of modern analysis where researchers examine the theory of operators and transformations on various spaces of finite and infinite dimension.
