On John Mather's Work

Abstract: John Mather is a great scholar who was dedicated to mathematics in his whole life. His works in mathematics can be characterized as original and foundational. He laid out the foundation of singularity theory while he was a graduate student. He also laid out the foundation of modern Hamiltonian dynamical systems. Those fields became main stream in mathematics and it attracts many talents to pursue. His other works on characteristic classes, foliations, celestial mechanics, prime ends of conformal mappings are of the same quality with great influence in mathematics.

• The paper surveys John Mather’s groundbreaking methods in singularity theory, establishing rigorous criteria for stability and finite determinacy of differentiable map-germs.
• The paper details innovative applications of variational methods in Hamiltonian dynamics, including the formulation of Aubry-Mather sets and mechanisms underlying Arnold diffusion.
• The paper examines Mather’s extension of characteristic classes to singular varieties, connecting Nash blow-up techniques with modern algebraic and differential geometry.

Critical Review of "On John Mather's Work" (2511.09913)

Introduction

"On John Mather's Work" presents a comprehensive survey of John Mather's seminal contributions to mathematics, covering singularity theory, Hamiltonian dynamics, characteristic classes for singular varieties, and foliation theory. The exposition emphasizes the originality, technical depth, and theoretical cohesion of Mather's research, articulating both foundational results and their enduring influence across geometric analysis, dynamical systems, and topology.

Mather's Foundational Work in Singularity Theory

Mather's development of stability theory for differentiable mappings fundamentally advanced singularity theory. Building on the notions of stable and generic maps introduced by Thom and Whitney, Mather formalized the equivalence relations for map-germs using the $C^k$-topology and $k$-jets. Through a sequence of pivotal papers, Mather established rigorous criteria for the stability and finite determinacy of smooth map-germs, integrating algebraic and differential perspectives.

A crucial milestone is Mather's Division Theorem, which strengthened Malgrange's preparation theorem for $C^\infty$ mappings. This result provided the necessary regularity and decomposition for handling function germs, enabling the extension of finite determinacy arguments from the holomorphic to the differentiable setting.

Mather also proved the equivalence between infinitesimal stability and stability for proper $C^{\infty}$ maps, demonstrating that control of linearized deformations is sufficient for global stability under small perturbations.

For the algebraic classification of stable germs, Mather showed that the isomorphism class of the local algebra $Q(f)$ (modulo the ideal pulled back from the target) classifies the germ up to differentiable equivalence, reducing the classification of isolated singularities to the study of finite-dimensional local algebras.

He further established the role of transversality: stability of smooth maps is generically equivalent to the transversality of $k$-jet prolongations to their orbits under diffeomorphism group actions, completed by the multijet transversality theorems.

Another major contribution is the proof that topologically stable maps are dense in the space of smooth maps, laying the groundwork for stratification theory (Whitney stratifications) and the characterization of singular varieties.

Mather's techniques and results were further extended by the Mather-Yau theorem, relating the moduli algebra of complex hypersurface singularities to their classification up to biholomorphic equivalence, establishing a bridge between smooth and holomorphic singularity theory.

The classification of simple (ADE) singularities and their connection to monodromy, deformation theory, and Dynkin diagrams underpin a large part of modern singularity theory, with ramifications in algebraic geometry, representation theory, and mathematical physics.

Variational Methods and Hamiltonian Dynamics

From the late 1970s, Mather focused on Hamiltonian dynamical systems, leveraging variational methods for the qualitative study of area-preserving maps and Lagrangian systems.

He constructed the Aubry-Mather sets as action-minimizing invariant sets for area-preserving twist maps, generalizing Birkhoff's work on geodesics and providing new mechanisms for orbit selection in non-integrable systems. The variational framework used minimal configurations and Peierls barriers, yielding precise characterizations of invariant circles and the structure of quasi-integrable dynamics.

Mather addressed the problem of connecting orbits (heteroclinic and homoclinic connections) in Birkhoff regions of instability, both for two-dimensional maps and, crucially, for higher-dimensional systems. His approach generalized to the study of minimal measures for Tonelli Lagrangians, defining rotation vectors and action-minimizing measures using convex duality ($\alpha$ and $\beta$ functions).

These ideas enabled the formulation of generalizations of Aubry-Mather theory, including the existence and structure of minimal invariant measures for arbitrary degrees of freedom, the definition of generalized Peierls barriers, and the precise categorization of connecting orbits between Aubry sets, ultimately leading to the concept of Mather's mechanism for interregion connections in phase space.

A particularly influential application is the resolution of Arnold diffusion in near-integrable Hamiltonian systems. Through variational constructions and normal forms, Mather and successors proved the generic existence of orbits that traverse pre-assigned regions in action space for systems with at least two and a half degrees of freedom. The formal statement of Mather's last theorem rigorously establishes the density of such diffusive trajectories and quantifies the typicality of Arnold diffusion in the parameter space of perturbations, with explicit conditions on regularity and non-degeneracy.

Characteristic Classes of Singular Varieties

The construction of Chern classes for singular varieties was historically challenging due to the degeneration of the tangent bundle. Mather addressed this via the Nash blow-up and the associated Nash bundle, defining the Mather class as the push-forward of Chern classes of the Nash bundle on the Nash transform of the variety.

This approach was instrumental in MacPherson's proof of the Deligne-Grothendieck conjecture, resulting in a unique, functorial extension of Chern classes (MacPherson classes) to singular algebraic varieties, using the Mather class and local Euler obstruction as core ingredients.

Further research revealed the equivalence between Wu classes and Mather classes, and between Schwartz classes and MacPherson classes, bridging constructions rooted in algebraic geometry, topology, and singularity theory. This line of work connects to the theory of polar varieties, establishing explicit intersection-theoretic formulas for characteristic classes of singular projective varieties.

Foliation Theory and Classifying Spaces

Mather, in collaboration with Thurston, made significant contributions to the topology of foliations. By analyzing the classifying spaces and associated cohomology (Haefliger structures), they connected the (co)homology of classifying spaces for homeomorphism and diffeomorphism groups to the topology of foliated manifolds. These results had profound consequences for obstruction theory and the construction of foliations with prescribed normal bundles.

Implications and Prospects

Mather's work has deep, enduring implications across several domains:

• In singularity theory, the criteria for stability, determinacy, and classification of map-germs form the backbone of the field, with extensions to symplectic, contact, and algebraic settings.
• Variational methods developed for Hamiltonian dynamics enable both fine-grained analysis of orbits and global statements about orbit complexity and instability, with direct applications to celestial mechanics, statistical mechanics, and ergodic theory.
• The construction of characteristic classes for singular spaces is critical for modern algebraic and differential geometry, impacting enumerative geometry, string theory, and intersection theory.
• The topological analysis of foliations continues to influence higher-codimension foliation theory, dynamical systems, and geometric topology.

Future directions include quantitative refinements of Arnold diffusion, higher-dimensional generalizations of Aubry-Mather mechanisms, extensions to random and nonautonomous systems, and further interactions between singularity theory and algebraic geometry (e.g., moduli spaces, perverse sheaves, and Hodge theory).

Conclusion

"On John Mather's Work" provides a scholarly and exhaustive account of Mather's contributions to modern mathematics. The synthesis of singularity theory, variational calculus, Hamiltonian dynamics, and geometric topology demonstrated in Mather's work continues to shape core mathematical research. His results are characterized by mathematical rigor, conceptual clarity, and far-reaching impact, constructing essential tools and frameworks that facilitate ongoing advances in analysis, geometry, and dynamics.