3:00pm - 5:00pmMassively parallel computations in algebraic geometry

Chair(s): Janko Böhm (TU Kaiserlautern, Germany), Anne Frühbis-Krüger (Leibniz Universität Hannover)

Massively parallel methods have been a success story in high performance numerical simulation, but so far have rarely been used in computational algebraic geometry. Recent developments like the combination of the parallelization framework GPI-Space with the computer algebra system Singular have made such approaches accessible to the mathematician without the need to deal with a multitude of technical details. The minisymposium aims at bringing together researchers pioneering this approach, discussing the current state of the art and possible future developments. We plan to address applications in classical algebraic geometry, tropical geometry, geometric invariant theory and links to high energy physics.

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

Tools for perturbative calculations from algebraic geometry

__Alessandro Georgoudis__

Uppsala University

In the last few years perturbative methods in scattering amplitudes have incorporated a lot of tools and methods from computational algebraic geometry. In this talk I will present some ideas that have been developed for reducing and calculating Feynman Integrals.

A massively parallel fan traversal with applications to geometric invariant theory

__Christian Reinbold__

TU Munich

The GIT fan of an algebraic group action on an algebraic variety describes all GIT quotients arising from Mumford’s construction in Geometric Invariant Theory. Its computation poses a challenging hurdle due to Buchberger's algorithm with double exponential worst case complexity. We present an optimized and scalable approach for computing the GIT fan by means of computational convex geometry. By factoring out symmetries and utilizing an ultra scale computing center, we were able to traverse the GIT fan of 6-pointed stable curves of genus 0 in approximately 13 minutes, yielding 249.604 chamber orbits.

Parallel algorithms for computing tropical varieties with symmetry

__Dominik Bendle__

TU Kaiserslautern

In this talk I will report on the massively parallel computation of tropical varieties, taking their symmetry into account. For this purpose we build upon a massively parallel fan traversal method implemented by Christian Reinbold using Singular in conjunction with GPI-Space. To pass between neighboring tropical cones of the Gröbner fan we use recently developed methods by Hofmann and Ren. This is a joint work with Janko Boehm, Mirko Rahn and Yue Ren.

Space sextics and their tritangents

__Yue Ren__

MPI Leipzig

In this talk, we will briefly review the well-known algebro-geometric oddity that complex space sextic curves admit exactly 120 tritangent planes.

We discuss recent works which shows that all tritangents can be totally real, as well as the current state on the question whether the 120 tritangents determine the curve uniquely. The latter context gives rise to computational algebro-combinatorial challenges in which parallelization would be of great help.

This is joint work with Turku Celik, Avinash Kulkarni, Mahsa Sayyary, and Bernd Sturmfels.