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.

Aluffi algebras and algebraic structure; gluing and its consequences in singularity theory; the relationships between toric geometry, local blow-up theory, and ramification theory, and 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.