Almost periodic functions were introduced by Harald Bohr in 1920s as a result of his investigations into the Riemann's zeta function. The theory was further developed by Bochner and von Neumann in the 30s and 40s. An important result of the theory is Bohr-von Neumann approximation theorem stating that trigonometric polynomials are uniformly dense among almost periodic functions.

Bochner's characterization of almost periodicity has been used by Duncan and Ulger (1992) to study almost periodic functionals on Banach algebras. This provides a more general framework to study almost periodicity.

In this talk we discuss some connections between almost periodic functionals and representation theory. Perhaps the simplest connection is that every coordinate functional of a continuous finite-dimensional representation is almost periodic. In the reverse direction, we can show that if $A$ is an involutive Banach algebra, $\pi \colon A \longrightarrow \mathscr L(H)$ is an involutive representation, and $\xi, \eta \in H$ are algebraically cyclic vectors such that the associated coordinate functional $\pi_{\xi, \eta }$ is almost periodic, then $\dim H<\infty$. We also discuss a construction in which one can associate a residually finite-dimensional (RFD) Banach algebra $U(A)$ to a given Banach algebra $A$, and point out its similarities to almost periodic compactification of locally compact groups.

The results in this talk are joint work with M. Filali, from University of Oulu, Finland.

Speaker's homepage: http://web2.uwindsor.ca/math/monfared/Main.html

The goal of these two talks is to introduce models for mapping spaces in higher categories and show that they are all equivalent. In the process we will discuss straightening and unstraightening, infinity analogue of the Grothendieck construction. We will conclude with a discussion of cartesian fibrations.