Algorithms to compute and compare toric degenerations

Develop algorithmic methods to compute toric degenerations of a given algebraic variety and to compare distinct toric degenerations arising from different theoretical frameworks such as algebraic geometry, representation theory, cluster algebras, and tropical geometry.

Background

Toric degenerations provide a powerful bridge between general algebraic varieties and toric varieties, allowing invariants of the original variety to be studied via the combinatorial geometry of polytopes. Numerous constructions exist across several areas (algebraic geometry, representation theory, cluster algebras, tropical geometry), each yielding toric degenerations under different paradigms.

Despite this progress, there is a notable algorithmic gap: there is no general-purpose procedure to compute a toric degeneration for an arbitrary variety, nor to compare degenerations obtained by different approaches. Addressing this gap would systematize computations and facilitate unification across frameworks.

References

However, despite the progress made in these domains, the challenge remains: there are currently no known algorithms tailored for computing toric degenerations of a given variety or for facilitating comparisons between disparate cases from different theoretical frameworks.

— A Family of Monomial Gröbner Degenerations of Determinantal Ideals with Minimal Cellular Resolutions  (2403.17736 - Mohammadi, 2024) in Section 1 (Introduction)