Software

Overview

I work on open-source software on github, where I have added about 500.000 lines of codes and modified about another 500.000 lines of code. My coding experience includes gap, python, julia and C++.

ToricVarieties_project

Overview

This software is written in GAP-4 and C++, is part of the homalg_project of Mohamed Barakat and uses the CAP_project. A visualisation of the ToricVarieties_project can be found here.

Documentation (last update 01/01/2022)

• AdditionsForToricVarieties,
• cohomCalgInterface,
• H0Approximator,
• QSMExplorer,
• SheafCohomologiesOnToricVarieties,
• SpasmInterface,
• ToolsForFPGradedModules,
• TopcomInterface,
• ToricVarieties,
• TruncationsForFPGradedModules.

For the latest package documentation, visit github.

Installation

I provide an installation script here for Debian9 and Ubuntu18.04 (last updated on January 2, 2022). Once complete, navigate to gap4.11.1/local/pkg/ToricVarieties_project:

• Build documentation: make doc.
• Execute tests: make test (latest daily tests on github).

Toric Varieties

• ToricVarieties (together with Sebastian Gutsche): General functionality for toric varieties.
• TopcomInterface: Interface to Topcom, so that toric varieties can be constructed from triangulations.

Coherent sheaves

Coherent sheaves on toric varieties can be modelled by f.p. graded S-modules (S being the Cox ring of the toric variety) (c.f. Gabriel morphisms and the computability of Serre quotients with applications to coherent sheaves):

• FreydCategoriesForCAP: F.p. graded S-modules.
• ToolsForFPGradedModules: Resolutions.
• TruncationsOfFPGradedModules: Truncations, which are heavily used in the computation of sheaf cohomologies.
• SheafCohomologyOnToricVarieties: Various algorithms for sheaf cohomologies, including those described in cohomologies of coherent sheaves and massless spectra in F-theory.

We provide interfaces to cohomCalg and spasm:

• cohomCalgInterface: Line bundle cohomologies via the famous cohomCalg algorithm.
• AdditionsForToricVarieties: Vanishing sets via cohomCalg.
• SpasmInterface: Faster sheaf cohomology algorithm modulo a high prime number. This can be used as approximation.

H0Approximator

On a hypersurface curve in a del-Pezzo 3 surface, we consider the pullback of a line bundle from the dP3. This package approximates the allowed values of global sections of this line bundle on the moduli space of all deformations of this curve. This implementation is based on Machine Learning and Algebraic Approaches towards Complete Matter Spectra in 4d F-theory.

Quadrillion F-theory Standard Models

The largest currently known class of F-theory Standard Models without chiral exotics and gauge coupling unifications was described in Quadrillion F-theory Standard Models. For short, we term these solutions QSMs (QSMs). We have investigated details of these geometries:

• Martin Bies, Mirjam Cvetič, Ron Donagi, Muyang Liu, Marielle Ong Root Bundles and Towards Exact Matter Spectra of F-theory MSSMs
• Martin Bies, Mirjam Cvetič, Muyang Liu Statistics of Root Bundles Relevant for Exact Matter Spectra of F-theory MSSMs

The QSMExplorer – written with Muyang Liu – reflects these efforts and allows to list data of these vacua very easily. Each vacuum comes with canonical nodal curves, which we use to learn more about these F-theory vacua. Most importantly, we provide algorithms to count limit root bundles (as introduced in Moduli of roots of line bundles on curves).

Freyd Categories (as part of the CAP_project)

The first implementations of a PresentationCategory category in the language of the CAP_project were available via

• CAPCategoryOfProjectiveGradedModules – documentation
• CAPPresentationCategory – documentation
• PresentationsByProjectiveGradedModules – documentation

These packages are by now deprecated because the concept has been much better understood in A constructive approach to Freyd categories and Tensor products of finitely presented functors. Together with Sebastian Posur, I have remodelled these packages in the package FreydCategoriesForCAP.

OSCAR Computer algebra

Currently, the ToricVarieties_project is being integrated into the OSCAR Computer Algebra System, which uses the modern programming language Julia. The documentation of the OSCAR Computer Algebra System is available here. Details on the toric varieties functionality can be found here.