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Diane Maclagan edited this page Jun 27, 2026 · 7 revisions

We will work on the following projects at this workshop:

Toric Vector Bundles

Lead: Sasha Zotine

The purpose of this group will be to polish/improve/overhaul the ToricVectorBundles package. Our goals, in vaguely descending order, are:

  1. Reverse the order of the filtrations used in the Klyachko data so as to match the literature. This will require changes to compatibility with other packages such as PositivityToricBundles.
  2. Implement Weil decorations as another data type for interacting with toric vector bundles; see https://arxiv.org/abs/2412.03476
  3. Possibly merge with PositivityToricBundles.
  4. Polish and improve the code/documentation whereever possible. Knowledge about vector bundles is not necessary, but some familiarity with toric varieties is expected.

Computing comprehensive Gröbner systems

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.

Certified approximation of algebraic varieties

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

Tropical Geometry

Lead: Diane Maclagan

We will work on improving the Tropical.m2 package, and updating gfanInterface.m2 for the new release of gfan.

Split Matroids and matroid valuative invariants

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

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