Quantitative rectifiability in metric spaces
Matthew Hyde (University of Warwick)
Dia: 9 / 2 / 2023
Hora: 14:00
Lloc: Dpt Matematiques, Seminari C3B (C3B/158)

Web: Grup d'Análisi matemàtica i Harmònica

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 Bi-lateral 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 well-approximated 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 Gromov-Hausdorff approximations by $n$-dimensional Banach spaces. This is joint work with David Bate and Raanan Schul.

Geometria Tropical

Alguns aspectes aritmètics de $\mathbb{P}^1\times \mathbb{P}^1$.
Francesc Bars (U.A.B.)
Dia: 7 / 2 / 2023
Hora: 13:00
Lloc: Dpt Matematiques, Seminari C3B (C3B/158)

Resum: Iniciarem amb aspectes generals de superfícies, restringint-nos finalment al cas $\mathbb{P}^1\times \mathbb{P}^1$. Pel cas $\mathbb{P}^1\times \mathbb{P}^1$ començarem a estudiar-ne 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'Anells

Package Deal Theorems for Localizations over h-local Domains
Román Álvarez (UAB)
Dia: 6 / 2 / 2023
Hora: 15:00
Lloc: Dpt Matematiques, Seminari C3B (C3B/158)

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 Levy-Odenthal for $R$ a commutative Noetherian ring of Krull dimension $1$ under two consistency hypotheses:

1. $X(m)=M_m$ for almost all maximal ideals $m$ of $R$;

2.$ X(m)\otimes Q=M\otimesQ$ for all maximal ideals $m$ of $R$.

They called these kinds of results Package Deal Theorems.

In this talk, I will give a version of this result for a larger class of domains, namely $h$-local domains, which were introduced by Matlis in the 1960s. $H$-local domains are commutative domains with the property that each non-zero element is contained in only finitely many maximal ideals of the ring and each non-zero prime ideal is contained in a unique maximal ideal of the ring. From results of previous work of Herbera-Příhoda and the original techniques of Levy-Odenthal, I will conclude with a Package Deal Theorem for traces of projective modules over $h$-local domains.