My name is Hongjin Wu. I recently defended my PhD thesis at ETH Zürich on 3 October 2025 under the supervision of Prof. Ulrik Brandes.
Hongjin Wu, PhD Thesis: Global synchronization of inductively defined graph classes: a local-to-global perspective. Defense Slides. Examiners: Ulrik Brandes, Afonso Bandeira, Claudio Tessone, Yinyu Ye.
My thesis develops new methods and provide new insights into global synchronization phenomena on graphs. This is a problem sitting at the intersection of combinatorics, dynamical systems, and nonconvex optimization. If you have ever wondered why graph, spring, egg-box, these three completely different objects, actually talk to each other mathematically, you can take a look at What is... a global synchronizing graph?.
Before my PhD, I wrote a master’s thesis in mathematics on the regularity of solutions to a PDE system (the p-curl system) arising from electromagnetism, supervised by Prof. Baojun Bian at Tongji University. Prior to that, I completed my bachelor’s degree in mathematics at Shanghai University.
Research statement of PhD study
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:
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:
Which graphs guarantee that all nodes on the unit circle converge to a single point when linked by attractive springs, no matter where they start? Or which graphs induce benign nonconvexity in the corresponding nonconvex optimization landscape? Here benign means that every local minimum coincides with the global minimum.
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.
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.
Publication
Hongjin Wu, PhD Thesis: Global synchronization of inductively defined graph classes: a local-to-global perspective. Defense Slides. Examiners: Ulrik Brandes, Afonso Bandeira, Claudio Tessone, Yinyu Ye.
Hongjin Wu, Ulrik Brandes, Global synchronization via modular decomposition, to appear, 2025.
Hongjin Wu, Ulrik Brandes, Skeleton of global synchronizing graphs and benign nonconvexity, to appear, 2025.
Hongjin Wu, Ulrik Brandes, Threshold graphs are globally synchronizing, 2025.
Hongjin Wu, Ulrik Brandes, Skeleton of global synchronizing graphs and benign nonconvexity, to appear, 2025.
Hongjin Wu, Ulrik Brandes, Threshold graphs are globally synchronizing, 2025.
Hongjin Wu, Benign nonconvexity and global synchronization with nonlinear interactions, Under review, 2024.
Hongjin Wu, Xingfei Xiang, Global boundedness of the curl for a p-curl system in convex domains, Mathematical Methods in the Applied Sciences, 2023.
Hongjin Wu, Master Thesis, Analysis of semilinear systems involving the curl operator, Tongji University, 2020.
Hongjin Wu, Bachelor Thesis, Three understandings on gradient flow, Shanghai University.
Hongjin Wu, Xingfei Xiang, Global boundedness of the curl for a p-curl system in convex domains, Mathematical Methods in the Applied Sciences, 2023.
Hongjin Wu, Master Thesis, Analysis of semilinear systems involving the curl operator, Tongji University, 2020.
Hongjin Wu, Bachelor Thesis, Three understandings on gradient flow, Shanghai University.
Activity
Invited talk, Skeleton of Global Synchronizing Graphs, at Mittagseminar, Institute of Theoretical Computer Science, ETH Zürich, 13 Nov 2025. Slides.
Poster presentation, Benign nonconvexity and global synchronization with nonlinear interactions, at Workshop Algorithmic Optimization: Tools for AI and Data Science (50 years Yurri Nesterov's research), UCLouvain, Belgium, 27-30 August 2024. See Abstract and Poster.
Thematic school, Optimization & algorithms for high-dimensional machine learning and inference, Institut de Mathématiques de Toulouse (IMT), CNRS, 7-11 October 2024.
Thematic school: Models & methods for high-dimensional machine learning and inference, Institut de Mathématiques de Toulouse (IMT), CNRS, 14-18 October 2024.
Summer school, Network Systems in Science and Technology, S.I.D.R.A., the Italian Control Systems Society, Bertinoro University, Italy, 9 July 2022.
Poster presentation, Benign nonconvexity and global synchronization with nonlinear interactions, at Workshop Algorithmic Optimization: Tools for AI and Data Science (50 years Yurri Nesterov's research), UCLouvain, Belgium, 27-30 August 2024. See Abstract and Poster.
Thematic school, Optimization & algorithms for high-dimensional machine learning and inference, Institut de Mathématiques de Toulouse (IMT), CNRS, 7-11 October 2024.
Thematic school: Models & methods for high-dimensional machine learning and inference, Institut de Mathématiques de Toulouse (IMT), CNRS, 14-18 October 2024.
Summer school, Network Systems in Science and Technology, S.I.D.R.A., the Italian Control Systems Society, Bertinoro University, Italy, 9 July 2022.
Teaching
[Repeatedly, 2021-2025] I enjoy mentoring students in research projects on Applied Network Science Seminar, Prof. Ulrik Brandes, ETH Zürich, Zürich.
Spring 2019, Analysis (bachelor level), Prof. Peipei Shang, Tongji University, Shanghai.
Spring 2019, Convex Analysis (master level), Prof. Ge Xiong, Tongji University, Shanghai.
Fall 2018, Introduction to PDEs (master level), Prof. Xingfei Xiang, Tongji University, Shanghai.
Fall 2018, Advanced Mathematical Statistics (bachelor level), Prof. Yin Zhuang, Tongji University, Shanghai.
Spring 2019, Analysis (bachelor level), Prof. Peipei Shang, Tongji University, Shanghai.
Spring 2019, Convex Analysis (master level), Prof. Ge Xiong, Tongji University, Shanghai.
Fall 2018, Introduction to PDEs (master level), Prof. Xingfei Xiang, Tongji University, Shanghai.
Fall 2018, Advanced Mathematical Statistics (bachelor level), Prof. Yin Zhuang, Tongji University, Shanghai.
Education
PhD advised by Prof. Ulrik Brandes and Prof. Afonso Bandeira (second supervisor), Social Networks Group, D-GESS, ETH Zürich, Zürich.
Master in Mathematics with focus on PDE, Advisor: Prof. Baojun Bian, School of Mathematical Sciences, Tongji University, Shanghai.
Bachelor in Mathematics, Department of Mathematics, Shanghai University, Shanghai.
Master in Mathematics with focus on PDE, Advisor: Prof. Baojun Bian, School of Mathematical Sciences, Tongji University, Shanghai.
Bachelor in Mathematics, Department of Mathematics, Shanghai University, Shanghai.