Overview

This course examines the intersection of Abelian Sandpile Theory and Tropical Geometry—two frameworks revealing how discrete local rules generate global geometric order.

The sandpile model is a cellular automaton whose dynamics exhibit self-organized criticality, a mechanism of emergent complexity observed in physics, computation, and biology.

Tropical geometry, the piecewise-linear limit of algebraic geometry, provides a natural language for describing these emergent structures in combinatorial and metric terms.

Together, they form a bridge between algebraic geometry, dynamical systems, and complexity theory. Their shared ideas resonate with the mathematics underlying modern data analysis and machine-learning architectures, where global coherence emerges from distributed local interactions.

Offered by the Topological and Geometric Complexity Group at ICMS Sofia, the course welcomes doctoral students and researchers interested in the geometric foundations of complex systems.

Course Topics

• Spanning trees and the sandpile group
• Abstract tropical curves, divisors, and Jacobians
• Extended sandpiles and canonical morphisms
• Tropical series and sandpile solitons
• Tropical sandpile model and scaling limits

Learning Outcomes

Participants will:

• Relate discrete sandpile dynamics to algebraic and geometric invariants
• Understand tropical Riemann–Roch and Abel–Jacobi theorems
• Apply combinatorial and homological tools to model emergent order
• Recognize how concepts of criticality and relaxation connect to ideas in complex-systems science and learning dynamics

References

1. Corry & Perkinson, Divisors and Sandpiles (AMS 2018)
2. Kalinin et al., Self-organized criticality and pattern emergence through the lens of tropical geometry, PNAS 115 (2018)
3. Kalinin & Shkolnikov, Sandpile solitons via smoothing of superharmonic functions, Commun. Math. Phys. 378 (2020)
4. Mikhalkin & Zharkov, Tropical curves, Jacobians and theta functions, Contemp. Math. 465 (2008)
5. Shkolnikov, Algebraic Limits of Sandpiles, IMRN 2025