The Department of Mathematics at the Faculty of Science and Technology, University of Macau (UM), has achieved a remarkable milestone: two major research papers have been simultaneously accepted by the Journal of the European Mathematical Society (JEMS), one of the world’s most prestigious mathematics journals. It is exceptionally rare for a single academic unit to have two independent works accepted concurrently by such a top-tier journal—an accomplishment that powerfully demonstrates UM’s outstanding research strength and deep scholarly foundation in both pure and applied mathematics.

JEMS is widely recognized as one of the leading mathematical journals globally, standing alongside Communications on Pure and Applied Mathematics (CPAM), Duke Mathematical Journal, and Geometric and Functional Analysis (GAFA). Often regarded as the foremost journal following the “big four” in mathematics, JEMS maintains exceptionally rigorous selection standards. The simultaneous acceptance of two papers from UM is a resounding endorsement of the high caliber of its mathematical research.

The first paper, titled “The Applications of the Algebraic Structure on the Toda Systems Associated with Dₙ and F₄ Types,” was co-authored by Prof. Changfeng Gui and Associate Professor Wen Yang from the Department of Mathematics, together with Prof. Chang-Shou Lin from National Taiwan University, Associate Professor Aleks Jevnikar from the University of Udine (Italy), and Dr. Leilei Cui from Central China Normal University.

Toda systems are classical yet profoundly significant models of nonlinear partial differential equations, originally introduced in the 1960s by Japanese physicist Morikazu Toda to describe nonlinear wave propagation in lattices. Over time, they have become central objects linking Lie algebra root systems, Weyl group symmetries, quantum field theory, random matrix theory, and geometric analysis—playing crucial roles in instanton solutions in gauge theory and Yang–Mills equations on Riemann surfaces. A key analytical challenge lies in understanding the behavior of blow-up solutions, particularly the classification of local masses at singular points.

For Toda systems of types Aₙ, Bₙ, Cₙ, and G₂, prior work successfully expressed local masses via double summations using permutation groups and monodromy theory of complex ordinary differential equations, enabling a priori estimates and existence results. However, Dₙ and F₄ systems present far greater algebraic complexity: Dₙ systems exhibit fundamentally different structures depending on whether the rank is odd or even, while F₄ systems lack a Fuchsian ODE structure, making explicit permutation representations of their Weyl groups extremely difficult. These obstacles have long hindered progress.

This groundbreaking study achieves—for the first time—the complete classification of blow-up local masses for both Dₙ and F₄ Toda systems. For Dₙ, the team derived distinct local mass formulas tailored to odd and even ranks. For F₄, researchers identified patterns among 120 identities and constructed a triple-sum expression for local masses using four generating permutations—a significantly more intricate framework than those for Aₙ，Bₙ，Cₙ types. Building on this classification, the authors determined critical parameters for both systems and, by using Morse theory, proved the existence of solutions on manifold where the Euler characteristic is non-positive. This work not only opens new avenues for analyzing other Toda systems (e.g., affine types) but also offers valuable insights for blow-up studies in other critical exponent equations.

The second paper, “A Theory of First Order Mean Field Type Control Problems and their Equations,” was co-authored by UM Research Assistant Professor Hongwei Yuan, Prof. Alain Bensoussan (University of Texas at Dallas), Associate Professor Deguang Wang (Shenzhen University), and Prof. Shangzhi Ren (The Chinese University of Hong Kong).

Mean field type control theory extends classical control theory to large-scale multi-agent systems—such as fleets of autonomous vehicles, financial traders, or social media users—where each agent’s decision depends on both its own state and the collective distribution (“mean field”) of all agents, which in turn evolves under the influence of individual actions. This creates a complex, infinite-dimensional coupled system. The Bellman equation is the cornerstone for describing such systems, yet its strong nonlinearity and nonlocality have left fundamental questions—existence, uniqueness, and regularity of solutions—even in seemingly simple cases like “linear dynamics plus nonlinear perturbations”—largely unresolved. Existing approaches rely heavily on restrictive assumptions such as linear-quadratic structure, separable Hamiltonians, or globally bounded second derivatives of the Hamiltonian, limiting real-world applicability.

This paper establishes, for the first time, a complete and rigorous theory for general first-order mean field type control problems. It proves the global existence and uniqueness of classical solutions to both the Bellman equation and the master equation. It introduces an innovative “cone condition” and develops novel a priori estimates based on forward–backward ODE systems, with a key insight centered on the positive definiteness of the Schur complement of the Hessian of the Lagrangian. It validates the theory by solving nontrivial nonlinear quadratic examples previously intractable by conventional methods, and further applies the framework to analyze deep residual networks with batch normalization, forging a new bridge between control theory and deep learning.

Though rooted in distinct branches—geometric analysis and control theory—both studies tackle long-standing, fundamental challenges with original ideas and sophisticated analytical tools. Together, they underscore the UM’s growing prominence and international impact in advancing the frontiers of mathematical science. Adding to this momentum, Dr. Sicheng Liu, a postdoctoral fellow in the Department of Mathematics, has recently been awarded the prestigious “Overseas Excellent Young Scientists Fund” (Overseas Youth Talent Program), injecting fresh talent into the research team. These concurrent breakthroughs in both research output and human capital vividly reflect UM Mathematics’ strategic commitment to deep foundational inquiry, interdisciplinary innovation, and balanced talent development—laying a solid foundation for sustained, high-quality growth in the years ahead.