In this talk I will discuss an algorithm that decomposes an arbitrary matroid polytope into some elementary pieces, called Schubert matroids. The algebraic structure in which the decomposition takes place is the valuative group of matroids, which received attention recently as the homology of permutohedral toric varieties.

We revisit Martin Hofmann's now-classical result on the relationship between extensional and intensional type theories by building on insights from homotopy type theory and leveraging the structure of left semi-model category on the category of models of type theory. Specifically, we demonstrate that extensional type theory (ETT) and intensional type theory (ITT) extended by the axiom of uniqueness of identity proofs (UIP) are equivalent in both a logical and a homotopy-theoretic sense. Whereas Hofmann's original proof was focused on initial, or syntactic, models, our approach generalizes to all cofibrant extensions of the base theories, encompassing types, terms, and propositional equalities. In doing so, this result unifies the verification of the analogue of Hofmann's result for all possible new extensions of intensional type theory at once.

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.