Qualitative theory of dynamical systems and bifurcations; algebraic methods applied to bifurcations with symmetry; involutions associated with fold singularities and discrete dynamical systems; cyclicity and criticality, the focus–center problem, limit cycles, and piecewise continuous dynamical systems.

This research area addresses several central themes in Commutative Algebra and their applications to Algebraic Geometry and the Theory of Complex Singularities. Studies concentrate on Commutative Noetherian Rings and Finitely Generated Modules, with an emphasis on Homological Algebra applied to these modules. The research includes characterizations of special classes of modules and rings, such as Cohen-Macaulay Modules, Regular Rings, and Gorenstein Rings. Other central topics encompass the investigation of Auslander-Reiten Conjectures, the calculation and properties of Multiplicities of ideals and modules, and the study of Modules of Differentials and Kähler Differentials. The connections of these topics with Algebraic Geometry and the Theory of Complex Singularities are actively explored. The area also covers the analysis of Rees Algebras, Associated Graded Algebras, and their algebraic structures, and their consequences in singularity theory, in addition to the complex relationships between  Theory of Local Blow-ups , and Ramification Theory, including their applications in valuation theory.

Toric actions, Euler obstruction, and Brasselet number; Topology of stable maps and real singularities; Singularities at infinity and global fibrations of polynomial maps; Milnor fibrations of real singularities and regularity at infinity; Cobordism between Morse and generic maps; Differential geometry and singularities.

Lipschitz geometry and the bi-Lipschitz theory of singularities; Multiple-point spaces of map germs; Functions defined on singular varieties; Topological and differentiable invariants of singularities; Matrix singularities and determinantal varieties.