This is a practical workshop to discuss possibilities for extending facilities in Algebraic Geometry in existing computational algebra systems such as Magma. The intention is to develop ideas for applications in Algebraic Geometry by building upon or combining current tools in novel ways. We hope these discussions are concrete enough to influence development over the next two years.
Schedule
All talks will be in Room W0.02 in the Sir David Davies building on the Loughborough University campus. (Either type 'Davies' into the interactive map search, or download the campus map as pdf and find Building 8.)- 11.00-12.00 Gavin Brown (Loughborough)
- Working with minimal rational surfaces
- 2.00-3.00 Stephen Coughlan (Bayreuth)
- TBA
- 4.00-5.00 Michael Kerber (IST, Vienna)
- Geometric algorithms for algebraic curves and surfaces
- 9.30-10.30 Alexander Kasprzyk (Sydney)
- N or M? Computing in toric geometry
- 11.00-12.00 Niels Lubbes (RICAM, Linz)
- Lattice polygons and families of curves on surfaces
- Abstract: First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective surfaces.
- 1.30-2.30 Shabnam Kadir (Hannover)
- The arithmetic of Calabi--Yau manifolds, motives and modularity
- 3.00-4.00 Andreas-Stephan Elsenhans (Bayreuth)
- K3 surfaces and Picard groups
- 4.30-5.30 Josef Schicho (RICAM, Linz)
- Gallimaufries
- Abstract: Hironaka's resolution of singularities in char. 0 uses induction on the dimension: to resolve a singularity in dimension n, one introduces a resolution problem in dimension n - 1. This smaller problem depends on local choices of coordinates. A recent proof under construction (by Hauser and S.) of Hironaka's result is based on a new type of resolution problem, called gallimaufries, whose main feature is a lack of dependence on any local choices.
Wednesday 8 September
Thursday 9 September
Friday 10 September
Friday morning is for open workshop discussion for those who stay longer.Aims
- Implementations of algebraic geometry techniques on surfaces (and more-so 3-folds) is in a state comparable to that of curves 15 years ago: there are a limited number of particular calculations that can be applied in fairly precisely specified situations. Algebraic curves now have vast batteries of tools that can be applied in great generality. Most problems we would like to analyse in higher dimensions have clear theoretical solutions, but realising these as practical algorithms is slow, although some things are clear: Schicho's group progress with resolution, adjoints and related calculations; specialists in computational geometry progress on topological questions (a prerequisite for analytical methods); several groups have begun to exploit toric geometry more systematically; arithmetic geometers continue to set the challenges over nonclosed fields; and so the list goes on. We aim to survey the state of the art in higher dimensions, with special emphasis on identifying potential calculations that could be implementated as general tools at the next stage.
Organisers
- Gavin Brown (Loughborough)
- Alexander Kasprzyk (Sydney)