Beginners introduction to polymake. Overview over functionality, polymake's rule system and data types.
Class Field Theory and Applications
Class Field theory deals with the classiﬁcation of abelian extensions (ie. ﬁeld extensions with an abelian Galois group). Based on the type of the ﬁeld we have global CFT (for number ﬁeld and plane curves over ﬁnite ﬁelds) as well as local CFT (for p-adic ﬁelds and Laurent series over ﬁnite ﬁelds). Given an extension of number ﬁelds K∕k a norm equation is trying to ﬁnd α ∈ K s.th. N(α) = 𝜃 for a given 𝜃 ∈ k. Classically norm equations are linked to for exmple sums-of-squares: 𝜃 ∈ ℤ is a sum of two squares iﬀ 𝜃 is a norm for ℚ(i). Norm equations, apart from being classical objects have many applications in algebra. Classically, the solvability of norm equations is of course investigated locally: if there is a solution, there will ba one modulo every prime. CFT now classﬁes abelian extensions through suitable norm groups. This can and is used algorithmically, an obstacle being that local solubility is neccessary, but not suﬃcient, in general. In this talk I will present some of the core ideas of CFT and their links to norm equations. The corresponding algorithms are practical and available (mostly) through Hecke (hence in Oscar) and also in Magma.
TU Berlin / MPI MiS Leipzig
We will also show a first few use cases of the new computer algebra system OSCAR, which combines the features of ANTIC, GAP, Singular, polymake and other ingredients in Julia.
TU Berlin / Adam Mickiewicz University in Poznań
, Benjamin Lorenz TU Berlin
Building Polymake (C++, Toolchain) and polymake.jl
Details on how to compile polymake from source and overview of the functionality of polymake.jl
Recognizing Spaces with TOPAZ
One of the most important task in Topology is the recognition of a given space. In this talk we will explore various algorithms and heuristics implemented in polymake that will help us in this regard. We will look at classical methods like homology computations and less known ones like random approaches to discrete morse theory, bistellar flips and simple homotopy theory.
Polymake's JSON File Format and the polyDB Database
Introduction to polymake's new data format in JSON and the polyDB database for objects in discrete geometry and related areas.
, Lars Kastner
Patchworking and Tropical Compactifications
We present an implementation of Viro's patchworking using tropical hypersurfaces, as well as an efficient way to compute the Z_2 homology of the resulting algebraic hypersurface. We will describe how to construct cellular sheaves in polymake, in order to compute tropical homology.
MPI for Mathematics in the Sciences
Administrative ContactSaskia Gutzschebauch
MPI für Mathematik in den Naturwissenschaften