Title: Riemann Integration and Asymptotic Structure of Banach Spaces
Abstract: I will introduce the topic of Riemann integration in Banach spaces in this talk and show briefly the curious property that a Banach-valued Riemann-integrable function need not be Lebesgue a.e. continuous. A Banach space X such that every X-valued Riemann-integrable function is Lebesgue a.e. continuous is said to have the Property of Lebesgue (PL). It is an open problem to characterize the PL in terms of asymptotic structure. I will detail what is currently known with respect to this problem and where the research is now headed. In addition, I will discuss at the end some related questions such as non-linear stability of the PL and whether or not a Banach space with the PL has the Radon-Nikodym property.