Research Project Best Practices
The final project is an original contribution — not a survey, not a tutorial replication. You are expected to produce something that did not exist before: a proof, an implementation with novel results, or a precise critical analysis that advances understanding of where a major ML system approximates or fails a categorical ideal. Choose one of the three tracks below.
Track A: Mathematical
Formalize an open connection between category theory and machine learning precisely enough that the result could appear in a mathematics or theoretical ML venue. The contribution is a theorem with a proof, or a precise counterexample.
Example Projects
- Prove that mean-field variational inference is (or is not) the pointwise left Kan extension of the variational family inclusion functor.
- Characterize the conditions under which the enriched Kan extension over Meas coincides with variational inference.
- Prove that the RG fixed-point condition implies a specific form of representation invariance in a formal model of a neural network.
- Show that nonzero sheaf cohomology of the loss landscape implies the existence of spurious local minima.
Track B: Computational
Implement a categorical structure and demonstrate it on real domain data, with results that are meaningfully compared to a non-categorical baseline. The contribution is working code, reproducible experiments, and an analysis of what the structure buys.
Example Projects
- Persistent homology pipeline on protein backbone data as preprocessing for fold classification; compare to a baseline without topological features.
- Functorial data migration between two heterogeneous scientific datasets using Catlab.jl; demonstrate that the migration preserves a specific invariant the baseline ETL pipeline does not.
- Discrete Kan extension engine applied to a crystallographic defect energy database; compare generalization accuracy to a standard regression baseline on held-out configurations.
- Sheaf-based contradiction detector for a pair of heterogeneous physical models; quantify the H¹ obstruction on real data and show it correlates with known modeling errors.
Track C: Critical
Write a paper that precisely identifies where a major ML system approximates a categorical ideal and what breaks at the seams. The contribution is a precise diagnosis — not a vague critique — with a specific proposal for what the correct categorical structure would be.
Example Projects
- A precise analysis of where transformer attention is and is not a categorical construction, with an identification of the morphism structure it implicitly encodes.
- JEPA vs. RG: a formal comparison of what JEPA's encoder discards versus what an RG transformation discards, with a specific experiment that distinguishes them.
- A categorical reading of diffusion models: identify the category, the functor, and the sense in which the reverse process is or is not a Kan extension.
- An analysis of graph neural networks as approximate functors: when is the GNN message-passing operation functorial, and what breaks when it is not?
Project Timeline
| Week | Milestone |
|---|---|
| 4 | Track selection and one-paragraph project proposal |
| 7 | Literature review and problem statement (2 pages) |
| 10 | Progress report: main result stated, partial proof or preliminary results |
| 13 | Final presentation (15 minutes) and written report (8–12 pages) |
What Makes a Strong Project
- Precision over breadth: A narrow result proved rigorously is worth more than a broad claim supported by intuition.
- The categorical lens question: Every strong project answers clearly: what is the categorical ideal, and what is the gap?
- Falsifiable claims: If you cannot state what would count as evidence against your claim, the claim is not precise enough.
- Honest failure analysis: A result that doesn't work as expected, analyzed carefully, is more valuable than a result that is cherry-picked to look good.