Tutorial Speakers

Titles and Abstracts

Janko Böhm
Universität Kaiserslautern, Germany

and
Anne Frühbis-Krüger
Carl von Ossietzky Universität, Germany

Massively parallel computations in algebraic geometry

Abstract:

Introducing parallelism and exploring its use is still a fundamental goal for the computer algebra community. In high performance numerical simulation, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. Recently, this approach has turned out to be also successful in computer algebra, for example, for handling complicated parallel structures arising in geometry, which are based on local to global principles for schemes and sheaves. Exploiting parallelism of this kind so far was a significant challenge due to the unpredictability of Buchberger's algorithm.

In this tutorial, we introduce the audience to the use of the Petri net based workflow management system GPI-Space in conjunction with the computer algebra system Singular as frontend and the computational backend. We start out by illustrating how to use the Singular/GPI-Space framework for realizing basic programming constructs. We will then discuss various applications with an in depth discussion of one of them, the verification of smoothness of an algebraic scheme based on a criterion of Hironaka. In this application parallelism arises through a tree of charts built by the algorithm.


Marc Moreno Maza
University of Western Ontario, Canada

Design and implementation of multi-threaded algorithms in polynomial algebra


Pierre Vanhove
CEA, France

Differential equations for Feynman integrals

Abstract:

Feynman integrals enter the evaluation of many physical observables in particle physics, gravitational physics, statistical physics and solid-state physics. They are multidimensional integrals which cannot be evaluated with elementary methods. It is an important question to understand what kind of special functions they are of their physical parameters.

In this tutorial we will present algebraic geometrical approaches for determining the Feynman integral D-modules. We will first introduce the parametric representations of a generic Feynman integral, and explain that Feynman integrals are holonomic functions, and that their ideal of annihilators generates a holonomic D-module. Then we will present a derivation of the holonomic D-module using the creative telescoping algorithm. We will then analyse the relation between this D-module and the generalized Gel'fand-Kapranov-Zelevinsky D-module obtained from a toric geometrical approach to the Feynman integrals. This tutorial will be illustrated by several examples.