Oberseminar Algebra und Geometrie
Sommersemester 2017: Berkovich spaces
Organised by I. Halupczok, B. Klopsch, S. Schröer and M. Zibrowius.
All talks take place Fridays at 12:30 in 25.22.03.73.
Short description
A big problem in non-archimedean analytic geometry is that analytic functions (for example on $\mathbb{Z}_p$, the ring of $p$-adic integers) cannot simply be defined as functions that are given locally by a power series, due to the space being totally disconnected. There exist various remedies for this, one of the more recent ones being to replace the original space by the corresponding Berkovich space. This is a topological space, where, in some sense, the "missing points have been added" in such a way that spaces that "should be connected" are connected.Since the invention of these spaces, they found a whole range of applications, e.g.:
- they make it possible to do integration along paths in the $p$-adics
- they yield a $p$-adic version of the dessins d'enfants
- they are very useful for harmonic analysis of dynamical systems over the $p$-adics.
Talks and tentative dates
| 21.4.2016 | Talk 1: The Berkovich affine and projective line (Benno Kuckuck) |
| 28.4.2016 | Talk 2: Banach rings and Banach spectra (André Schell) |
| 5.5.2016 | Talk 3: More examples (Andrea Fanelli) |
| 12.5.2016 | Talk 4: The spectrum is non-empty, compact, Hausdorff (David Bradley-Williams) |
| 19.5.2016 | Talk 5: Strictly $k$-affinoid algebras (Johannes Fischer) |
| 26.5.2016 | Talk 6: Strictly $k$-affinoid domains (Benedikt Schilson) |
| 2.6.2016 | Talk 7: The structure sheaf and strictly $k$-affinoid spaces (Kevin Langlois) |
| 9.6.2016 | No talk. |
| 16.6.2016 | Talk 8: Strictly $k$-analytic spaces (Kevin Langlois) |
| 23.6.2016 | Talk 9: Analytification of algebraic varieties (Peter Arndt) |
| 30.6.2016 | Talk 10: Examples of curves (Stefan Schröer) |
| 7.7.2016 | Talk 11: Berkovich spaces vs. tropical geometry (Steffen Kionke) |
| 14.7.2016 | No talk; instead: Gemeinsames Seminar Algebra und Modelltheorie with Münster |
| Should somebody suddenly feel the urge to volunteer for one more talk, here are some options: | |
| ?.?.2016 | The Berkovich hyperbolic space (tba) |
| ?.?.2016 | Harmonic analysis on $\mathbb{P}^1_{\operatorname{Berk}}$ (tba) |
| ?.?.2016 | Realizations of buildings in Berkovich spaces (tba) |
Literature
Baker: An introduction to Berkovich analytic spaces and non-archimedean potential theory on curves
Ducros: Espaces analytiques $p$-adiques au sense de Berkovich
Nicaise: Introduction to Berkovich spaces
Payne: Analytification is the limit of all tropicalizations
Rémy, Thuillier, Werner: Bruhat--Tits theory from Berkovich's point of view. I -- Realizations and compactifications of buildings
Temkin: Introduction to Berkovich analytic spaces
More literature:
Baker: Non-Archimedean Geometry (lecture notes)
Baker, Payne, Rabinoff: On the structure of nonarchimedean analytic curves
Jonsson: Topics in Algebraic Geometry I --
Berkovich Spaces (lecture notes)
Payne: Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry