Partial differential equations with special attention to mathematical models in applied sciences such as engineering, physics, biology, chemistry, etc. The research covers: Function spaces (distributions, Sobolev spaces, etc); existence, uniqueness, regularity and singularity properties of solutions; numerical treatment by finite element, finite difference and boundary element methods; modification of mathematical models. The study of dense singularities of solutions of nonlinear partial differential equations with emphasis on Lie semigroups of noninvertible transformations of solutions; abstract differential geometry of algebras of generalised functions and de Rham cohomology; space-time foam structures with dense singularities