Kuramoto Oscillators: Dynamical Systems meet Computational Algebraic Geometry

When does a system of coupled oscillators synchronize? This central question in dynamical systems arises in applications ranging from power grids to neuroscience to biology: why do fireflies sometimes begin flashing in harmony? Perhaps the most studied model is due to Kuramoto (1975); we  analyze the Kuramoto model from the perspectives of algebra and topology. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks); our work also tackles more general situations.

We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. The talk will include a surprising (at least to us!) connection to Segre varieties, and close with examples of computations using the Macaulay2 software package "Oscillator". 

Joint work with Heather Harrington (Oxford/Dresden) and Mike Stillman (Cornell).

Frank Sottile

The infinitesimal rigidity of symmetric bar-joint framework 

A d-dimensional bar-joint framework is a realisation of a finite simple graph G in Euclidean d-space, where the bars are the realisations of the edges and the joints are the realisations of the vertices. A framework is said to be rigid if the only continuous motions that preserve bar-length are combinations of rotations and translations. Frameworks which present a non-trivial symmetry have been of interest to a number of rigidity theorists, due to interesting real-life applications. 

Nadir Fasola

Tetrahedron instantons via Donaldson-Thomas theory

One of the major problems in topological string theory is the study of instanton moduli spaces. D-brane systems have provided a very fruitful framework for approaching this problem systematically and a great stimulus for the development of new interesting mathematics. Indeed, the category of D-branes wrapping a submanifold of string spacetime can be broadly understood geometrically as the derived category of coherent sheaves on said submanifold. D-brane bound-state countings then have a natural interpretation in terms of sheaf counting problems in enumerative geometry, especially Donaldson-Thomas theory. Using the example of tetrahedron instantons, recently introduced in string theory by the work of Pomoni-Yan-Zhang, I will explain how the geometry of moduli spaces of instantons can be studied by looking at Quot schemes in the framework of Donaldson-Thomas theory, and how partition functions from physics are rigorously computed as generating functions of virtual invariants in a purely combinatorial way. This is based on joint work with S. Monavari.

Kaibo Hu