Many mathematical problems, although continuous in nature, are inherently combinatorial. Examples include influence processes on networks (synchronization, opinion dynamics, and related models) and continuous relaxations of combinatorial optimization problems (such as SDP and the Burer–Monteiro approach).

For continuous systems with combinatorial structure, can combinatorics truly explain their behavior? If so, problems that now seem difficult might collapse into simple combinatorial principles. This hope motivates my PhD research on global synchronizing graphs in the context of the Kuramoto model:

Previous approaches rely largely on algebraic arguments. For instance, for Erdős–Rényi graphs one can reparametrize the energy through a Burer–Monteiro–type low-rank factorized SDP formulation and prove that, w.h.p., every second-order stationary point is rank-one and hence synchronized.

Our approach is geometric. Specifically, we translate the problem of understanding an n-dimensional energy landscape into the much simpler task of analyzing planar geometric relations between vectors. Through this lens, we uncover several new globally synchronizing graph classes.

Beyond these new results, perhaps even more interesting is the mechanism that emerges. Algebraic methods typically rely on concentration phenomena and essentially argue that graphs “close to’’ complete graphs globally synchronize. In contrast, our work reveals a different mechanism: global synchronization can arise through a local-to-global propagation process along a combinatorial skeleton of the graph.

Remark 1. The graph classes we know so far represent only the tip of the iceberg. I have compiled, as comprehensively as I can, the currently known globally synchronizing graphs in the Gallery of Global Synchronizing Graphs.

Remark 2. Traced back to Huygens’ 17th-century observation of synchronized pendulum clocks, the global synchronization problem is not only mathematically intriguing; it also exhibits unexpected connections to several active directions in modern applied mathematics, including the low-rank Burer–Monteiro formulation for solving combinatorial optimization problem, and clustering phenomena in transformer dynamics, where the propagation of next-token predictions through network layers can be interpreted as a dynamical system. Empirically, this system often exhibits collapse to a single outcome or to a few distinct outcomes, a phenomenon known as rank collapse.

Global synchronization via modular decomposition (Hongjin Wu, Ulrik Brandes), in preparation.

Benign nonconvexity of synchronization landscape induced by graph skeletons (Hongjin Wu, Ulrik Brandes), Preprint.

Threshold graphs are globally synchronizing (Hongjin Wu, Ulrik Brandes), Preprint.

Global boundedness of the curl for a p-curl system in convex domains (Hongjin Wu, Xingfei Xiang), Mathematical Methods in the Applied Sciences, 2023.

Hongjin Wu, PhD Thesis: Global synchronization of inductively defined graph classes: a local-to-global perspective. Examiners: Ulrik Brandes, Afonso Bandeira, Claudio Tessone, Yinyu Ye.

Hongjin Wu, Master Thesis,  Analysis of semilinear systems involving the curl operator. Advisor: Baojun Bian.

Hongjin Wu, Bachelor Thesis, Three understandings on gradient flow, Advisor: Dajun Zhang.