PLEASE NOTE ROOM CHANGE FROM LAST TERM:  SEMINAR WILL NOW BE HELD IN FINE 110.    In recent times - particularly the last two decades - discrete analogues in harmonic analysis have gone through a period of considerable changes and developments. This is due in part to Bourgain's pointwise ergodic theorem for the squares on $L^p(X, \mu)$ for any $p>1$. The main aim of this talk is to discuss recent developments in discrete harmonic analysis. We will be mainly concerned with $\ell^p(\mathbf Z^d)$ estimates $(p>1)$ of $r$-variations $(r>2)$ for discrete averaging operators and singular integral operators along polynomial mappings. All the results are subjects of the ongoing projects with Elias M. Stein and Bartosz Trojan.