AnàlisiQuantitative rectifiability in metric spaces
Resum: The theory of quantitative rectifiability for Ahlfors regular subsets of Euclidean space was developed extensively by David and Semmes in the early 1990s, partly motivated by questions arising in harmonic analysis. They proved, among many other things, the equivalence of Uniform Rectifiability (UR) and the Bilateral Weak Geometric Lemma (BWGL). Roughly speaking, an Ahlfors regular set is UR if a large proportion of every surface ball coincides with some Lipschitz image of a Euclidean ball of the same radius; it satisfies the BWGL if most surface balls are locally wellapproximated by $n$dimensional affine planes in Hausdorff distance. In this talk we discuss the equivalence of UR and BWGL for Ahlfors regular metric spaces. While the above definition of UR makes sense in this context, BWGL does not. Instead, the BWGL condition is stated in terms of local GromovHausdorff approximations by $n$dimensional Banach spaces. This is joint work with David Bate and Raanan Schul.
Geometria TropicalAlguns aspectes aritmètics de $\mathbb{P}^1\times \mathbb{P}^1$.
Resum: Iniciarem amb aspectes generals de superfícies, restringintnos finalment al cas $\mathbb{P}^1\times \mathbb{P}^1$. Pel cas $\mathbb{P}^1\times \mathbb{P}^1$ començarem a estudiarne els seus twist i la seva relació amb cohomologia Galoisiana no abelina. Aquest és un ``work in progress" iniciat conjuntament amb E.Badr, F.B. i X.Xarles.
Teoria d'AnellsPackage Deal Theorems for Localizations over hlocal Domains
Resum: Let $R$ be a commutative ring with total ring of fractions $Q$, let $\Lambda$ be a (not necessarily commutative) $R$algebra, and let $M$ be a finitely generated right $\Lambda$module. For each maximal ideal $m$ of $R$, consider a (not necessarily finitely generated) $\Lambda_m$submodule $X(m)$ of $M_m$. For which such families is there a $\Lambda$submodule $N$ of $M$ such that $N_m=X(m)$? This question was answered by LevyOdenthal for $R$ a commutative Noetherian ring of Krull dimension $1$ under two consistency hypotheses:
