Tropical Moduli Project

This research/coding project emerged from the University of Washington's Tropical Geometry Seminar.

This repository is dedicated to computing the cells/graphs within the moduli space of tropical curves $\mathcal{M}_{g,n}^\text{trop}$.

Project Members: Dhruv Bhatia, Zawad Chowdhury, Dora Kassabova, Patrick O'Melveny, Andrew Tawfeek

The initial/foundation sources that motivated this project is

Chan, Melody. "Combinatorics of the tropical Torelli map." Algebra & Number Theory 6.6 (2012): 1133-1169. (Slides)

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Current Goals:

• Compute the $f$-vector of the poset being output by the current implementation of $\mathcal{M}_{g,n}^\text{trop}$ and check against Theorem 2.12 of Melody's paper¹
• Can we implement a dynamic-approach to the computation, using the observation that $\mathcal{M}_{g,n}^\text{trop}$ admits a stratification by lower moduli $g$ and $n$? (Also, can we find an explicit source for this as a mathematical statement? Maybe in this²?)
• Implement top-down approach to computation (e.g. by generating all maximal first, then contracting).
• Write-in a SageMath to/from Python translation (particularly cleanly between networkx and graphs for tropical curves) to expand compatibility between platforms (and to assist with later work, e.g. matroids...).
• Look into UW mathematics department/general usage computing cluster usage for large-scale computation of poset to store for future-use/offer publicly.
• Continue research into what aspects of the package are worthwhile developing based on most-recent publications (e.g. cohomology direction of cone complexes? etc. -- find and compile sources).

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Future Questions/Directions:

• Using Sage's convex rational polyhedral cone package (paired with toric plotter) we could potentially display $\mathcal{M}_{g,n}^\text{trop}$ as a generalized cone complex and have this interplay nicely with the poset of dual grpahs.
• Can we extend the code to interact well with the matroid perspective³ of tropical geometry, perhaps with the aid of Sage's already well-developed matroid package?
• Following up on the cone complexes: we should then be able to compute the link $\Delta_{g,n}$ of the cone complex -- potentially paving the way towards cohomology⁴...
• (Hard) What can we do towards extending our work to compactifications of other moduli spaces, such as K3 surfaces?⁵⁶

Relevant Documentation Links:

• graph theory
• convex rational polyhedral cones
• toric plotter
• matroid theory

Various helpful links:

• Tropical Geometry Seminar
• Coding Project Homepage
• Google Drive (UW-email required)
• Zulip Chat (UW-email required)

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Footnotes

1. Chan, Melody. "Combinatorics of the tropical Torelli map." Algebra & Number Theory 6.6 (2012): 1133-1169. ↩

2. Arbarello, Enrico, Maurizio Cornalba, and Phillip A. Griffiths. Geometry of algebraic curves: volume II with a contribution by Joseph Daniel Harris. Springer Berlin Heidelberg, 2011. ↩

3. Ardila, Federico. "The geometry of matroids." Notices of the AMS 65.8 (2018). ↩

4. Chan, Melody, Søren Galatius, and Sam Payne. "Tropical curves, graph complexes, and top weight cohomology of $\mathcal{M}_g$." Journal of the American Mathematical Society 34.2 (2021): 565-594. ↩

5. Ranganathan, Dhruv. "Tropical Geometry: Forwards and Backwards." Notices of the AMS 70.7 (2023). ↩

6. Alexeev, Valery, and Philip Engel. "Compact moduli of K3 surfaces." Annals of Mathematics 198.2 (2023): 727-789. ↩