Research topics

• Representation theory of algebras
  

• • Homological and categorical aspects of representation theory:
    

• • • Galois covering theory for module and functor categories.
    • Krull-Gabriel dimension and Prest Conjecture.
    • Homological problems for classes of Artin algebras defined by properties of the Auslander-Reiten quiver and structure of the component quiver.

• • Structure of module category for selfinjective algebras:

• • • Classification of selfinjective algebras of finite representation type over an arbitrary field.
    • Weighted surface algebras and symmetric periodic algebras of period 4.
    • Classification of tame selfinjective algebras for classes defined by properties of the bounded Gabriel quiver.
    • Deformed preprojective algebras of generalized Dynkin type.

• • Derived categories and equivalences of triangulated categories:
    

• • • Derived categories of gentle algebras and their equivalences.
    • Mutations of symmetric periodic algebras.
    • Categories of nilpotent operators with flags of invariant subspaces vs. coherent sheaves over weighted projective lines.
      

• • Algorithmic and combinatorial aspects of representation theory:
    

• • • Recovering information on structure and properties of an exact category from the level of its Grothendieck group.
    • Algorithms of constructing indecomposable matrix representations.
    • Birkhoff Problem, nilpotent subspaces, counting of submodule filtrations and Hall polynomials.
    • Constructions of combinatorial invariants determining the shape of bounded Gabriel quiver for distinguished classes of algebras.

• Selected topics of algebraic geometry
  

• • Geometry of module varieties:
    

• • • Global properties of module varieties and their irreducible components.
    • Degeneration order between module orbits depending on the algebra properties.
    • Local geometric properties of orbit closures in module varieties.
    • Classification of selected classes of singularities in the orbit closures of modules.
    • Tangent spaces, generators of the zero ideals and transversal slices for orbit closures of modules.

• • Degenerations of algebras:

• • • Profiled degeneration processes.
    • Description of the geometric degeneration scheme inside distinguished classes of algebras.

• Derivations, rings of constants and Jacobian Conjecture

• • • Square-free and radical factorizations vs. Jacobian Conjecture.
    • Jacobian Conjecture in positive characteristic.
    • Jacobian conditions for polynomials over unique factorization domains.
    • Algebraic structures on generalized sets.

• Applications of algebraic methods in mathematical physics, molecular biology and data analysis
  

• • • Poisson structures and symplectic manifolds.
    • Representations of gentle algebras and topological quantum field theory.
    • Hermiticity-preserving superoperators in quantum information theory.
    • Binding polynomials in context of the quantitative analysis of protein-ligand interactions.
    • Persistence modules and homologies in topological data analysis.