BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alejandro Vargas (U. Frankfurt) DTSTART:20230519T140000Z DTEND:20230519T150000Z DTSTAMP:20260423T021056Z UID:LAGARTOS/59 DESCRIPTION:Title: Valuated matroids\, tropicalized linear spaces and the affine building o f PGL_{r+1}(K)\nby Alejandro Vargas (U. Frankfurt) as part of (LAGARTO S) Latin American Real and Tropical Geometry Seminar\n\n\nAbstract\nValuat ed matroids were introduced by Dress and Wenzel in the 90s to\ncombinatori ally study metric spaces that arise naturally in $p$-adic\ngeometry and in phylogenetics.\nIn tropical geometry\, they encode the information of the tropicalization\nof a linear space.\nAffine buildings were introduced by Bruhat and Tits in the 70s as highly\nsymmetric simplicial complexes to ex tract the combinatorics of algebraic\ngroups.\nThe affine building associa ted to the projective linear group\n$PGL_{r+1}(K)$ admits a description vi a norms\, and by work of Werner a\ncompactification via semi-norms.\nInspi red by Payne's result that the Berkovich analytification is the\nlimit of all tropicalizations\, we show that the space of seminorms on\n$(K^{r+1})^ *$ is the limit of all tropicalized \\emph{linear} embeddings\n$\\iota : \ \mathbb{P}^r\\hookrightarrow\\mathbb{P}^n$ and prove a faithful\ntropical ization result for compactified linear spaces.\nThus\, under a suitable hy pothesis on the non-Archimedean field $K$\, the\npunchline is that the ran k-$(r+1)$ $K$-realizable valuated matroids\napproximate the compactificati on of the affine building of\n$PGL_{r+1}(K)$ in a precise manner\, and thi s can be regarded as the\ntropical linear space associated to a universal $K$-realizable valuated\nmatroid.\n LOCATION:https://researchseminars.org/talk/LAGARTOS/59/ END:VEVENT END:VCALENDAR