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

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)


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