BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Herwig Hauser (Univ. Vienna) DTSTART:20230213T110000Z DTEND:20230213T123000Z DTSTAMP:20260423T010754Z UID:GPL/25 DESCRIPTION:Title: Mod uli of n points on the projective line\nby Herwig Hauser (Univ. Vienna ) as part of Geometry and Physics @ Lisbon\n\nLecture held in 6.2.33 (Semi nar Room\, Math\, FCUL).\n\nAbstract\nThis mini-course addresses Master an d Doctoral students as well as Postdoc and Senior researchers in mathemati cs.\nObjective: The problem of constructing normal forms and moduli spaces for various geometric objects goes back (at least\, and\namong many other s) to the Italian geometers (Enriques\, Chisini\, Severi\, ...). A highlig ht was reached in the 1960 and 70es when\nDeligne\, Mumford and Knudsen in vestigated and constructed the moduli space of stable curves of genus $g$. These spectacular\nworks had a huge impact\, though the techniques from a lgebraic geometry they applied were quite challenging. In the course\, we\ nwish to offer a gentle and hopefully fascinating introduction to these re sults\, restricting always to curves of genus zero\, that is\,\ntransversa l unions of projective lines ${\\mathbb P}^1$. This case is already a rich source of ideas and methods.\n\nContents: We start by discussing the conc ept of (coarse and fine) moduli spaces\, universal families and the philos ophical\nbackground thereof: why is it natural to study such questions\, a nd why the given axiomatic framework is the correct one? Once\nwe have bec ome familiar with these foundations (seeing many examples on the way)\, we will concentrate on n points in ${\\mathbb P}^1$ and\nthe action of $PGL_ 2$ on them by Mobius transformations. This is part of classical projective geometry and very beautiful. As long as the $n$ points are pairwise disti nct\, things are easy\, and a moduli space is easily constructed. Things b ecome tricky as the points start to move and thus become closer to each ot her until they collide and coalesce. What are the limiting configurations of the points one has to expect in this variation? This question has a lon g history - Grothendieck proposed in SGA1 a convincing answer: $n$-pointed stable curves.\n\nWe will take at the beginning a different approach by p roposing an alternative version of limit. Namely\, we embed the space of $ n$\ndistinct points in a large projective space and then take limits there in via the Zariski-closure. This opens the door to the theory of\nphylogen etic trees: they are certain finite graphs with leaves and inner vertices as a tree in a forest. Their geometric combinatorics\nwill become the guid ing principle to design many proofs for our moduli spaces. Working with ph ylogenetic trees can be a very\npleasing occupation\, we will draw\, glue\ , cut and compose these trees and thus get surprising constructions and in sights.\nAt that point a miracle happens: The stable curves of Grothendiec k\, Deligne\, Mumford\, Knudsen pop up on their own. We don’t\neven have to define them - they are just there. So the circle closes up\, and our j ourney is now able to reprove many of the classical\nresults in an easy go ing and appealing manner.\nThe course is based on a recent research cooper ation with Jiayue Qi and Josef Schicho from the University of Linz within the\nproject P34765 of the Austrian Science Fund FWF.\n\nHerwig Hauser\, F aculty of Mathematics\, University of Vienna herwig.hauser@univie.ac.at\n LOCATION:https://researchseminars.org/talk/GPL/25/ END:VEVENT END:VCALENDAR