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Projects
We will work on the following projects at this workshop:
Lead: Sasha Zotine
Leads: Giulia Gaggero and Oliver Clarke
Comprehensive Groebner Bases
Groebner bases are familiar objects that allow us to work practically with polynomial ideals. Similarly, Comprehensive Groebner Bases (CGBs) allow us to work with polynomial systems that contain parameter variables. They give a precise description of all Groebner bases that one can encounter by specialising those parameter values; grouping together Groebner bases with a similar shape.
In this project, we will implement a few algorithms for computing CGBs and testing them on small examples. In particular, we will implement the well-known Suzuki--Sato algorithm and, with time permitting, look into some optimisations.
Lead: Kisun Lee
This project aims to prototype certified numerical tools for approximating algebraic varieties inside Macaulay2. Starting from a polynomial system and one nonsingular point, we will construct local tangent-normal coordinates, use interval/Krawczyk certification to validate local boxes, and explore how these certified patches can be extended into curve and surface approximations. The package will also include visualization of certified algebraic varieties.
References: https://arxiv.org/abs/2502.05357, https://arxiv.org/abs/2602.07718
Lead: Diane Maclagan
We will work on improving the Tropical.m2 package, and updating gfanInterface.m2 for the new release of gfan.
Lead: Alejandro Vargas
In this project, we will work with split matroids and matroid valuative invariants. Split matroids are a large class of matroids, comprising paving matroids. From the perspective of the matroid base polytope, they can be thought of as the matroids obtained from the hypersimplex by cutting away vertices using splits, such that each vertex is eliminated by a unique split. This split structure makes computing valuative invariants really fast. We will implement an internal representation of matroids using splits, and several valuative invariants.
References: https://arxiv.org/abs/2208.04893