Abstract: Let $X$ be a smooth toric variety with dense torus $T$. It is known that the syzygy order of equivariant cohomology $H_T^∗(X)$ as an $H^∗(BT)$-module can be computed from the combinatorics of the underlying fan. Suppose $S\subset T$ is a subtorus, we will compute the syzygy order of equivariant cohomology $H_S^∗(X)$ with respect to $S$ as an $H^∗(BS)$-module in some cases.

Abstract: In this presentation we intend to illustrate in a brief way the study of optimal constants in some inequalities in $L^p$ norm of operators involving the Hardy operator defined as $Hf (x) = (1/x) \int_{0}^x f(t) dt$. In 1925, Hardy proves that $H$ is bounded from $L^p$ to $L^p$, if $1 < p ≤ ∞$, with norm exactly equal to $p′$. This result has given rise to a new field of study of great interest in Mathematical Analysis and Operator Theory. The aim of this talk is to go into this line of research by exposing the results that have been obtained recently, the open problems in this field, such as the problem formulated in 1996 by G. Bennett and, finally, to present some of our contributions giving solutions to several questions within this subject.