Seminar on Banach and Metric Space Geometry

Date: March 21, 2024

Time: 10:00AM - 11:00AM

Location: BLOC 302

Speaker: Harisson Gaebler, University of North Texas

Title: Riemann integration and asymptotic structure of Banach spaces

Abstract: Let X be a Banach space. A bounded and Lebesgue almost-everywhere continuous function f:[0,1]\to X is Riemann-integrable. However, the converse statement is false in general. This motivates the following definition: X is said to have the Lebesgue property if every Riemann-integrable function f:[0,1]\to X is Lebesgue almost-everywhere continuous. In this talk, I will discuss my work during the last several years on the relationship between the Lebesgue property and asymptotic structures. I will begin by giving an overview of the Lebesgue property that includes relevant examples, older results, and a brief mention of my first paper on this topic which ultimately led to two more recent joint works. I will then spend the majority of the talk discussing the these two more recent papers whose results include 1) the characterization of the Lebesgue property in terms of a new asymptotic structure that sits strictly between the notions of a unique \ell_{1} spreading model and a unique \ell_{1} asymptotic model (this is a joint work with Bunyamin Sari) and 2) the complete separation of the Lebesgue property from a (uniformly) unique \ell_{1} spreading model (this is a joint work with Pavlos Motakis and Bunyamin Sari). Lastly, I will mention two relevant open problems.