Please note the different time.We will discuss some recent developments in dimensional-free bounds for the Hardy--Littlewood averaging operators defined over convex symmetric bodies in $\mathbb R^d$. Specifically we will 
be interested in $r$-variational bounds. nWe also prove the dimension-free bounds on $\ell^p(\mathbb Z^d)$ with $p>3/2$ for the discrete  maximal function associated with cubes in $\mathbb Z^d$. Using similar methods we also give a new simplified proof for the dimension-free bounds on $L^p(\mathbb R^d)$ with $p>3/2$ for maximal functions corresponding to symmetric convex bodies in $\mathbb R^d.$ If the time permits we will discuss problems for discrete Euclidean balls in $\mathbb Z^d$ as well.
This is joint project with J. Bourgain, E.M. Stein and B. Wr\'obel.