Department of Non-Linear Analysis and Applied Topology

Welcome to the website of the Department of Non-Linear Analysis and Applied Topology at the Faculty of Mathematics and Computer Science of Adam Mickiewicz University in Poznan.

The Department of Non-Linear Analysis and Applied Topology was established on 1 January 2020.

The department’s staff work on problems in the fields of non-linear analysis, convex analysis, and applied topology.

Research is done on problems in non-linear analysis, broadly defined. This includes topics related to non-linear differential and integral equations (e.g. Hammerstein integral equations, Volterra integral equations or fractional equations) in terms of the existence and the uniqueness of solutions in different classes (e.g. in the classes of almost periodic functions of different types – Bohr’s, Bochner’s, Stepanov’s, Levitan’s types – or in the classes of bounded variation functions in the sense of Jordan and Young) and the topological properties of solution sets (particularly Aronszajn-type theorems). Attention is also paid to some issues of operator theory (particularly superposition operators) and formal analysis, and to questions related to the theory of fixed points of functions and multifunctions and selected issues of general topology (hyperconvex spaces, R-trees, measures of non-compactness, etc.).

The second area of research concerns problems related to the algebraic, analytic, geometric and topological properties of pairs of convex and closed sets (e.g. the property of translation, set shadowing, Salle’s sets, etc.), of minimal pairs (in terms of existence and uniqueness up to translation) and of Minkowski–Radström–Hörmander spaces over any linear-topological Hausdorff space, as well as quasidifferential calculus (e.g. sublinear functions and their differences). It also includes applications of Minkowski–Radström–Hörmander spaces – for example, to the theory of multifunctions or to the description of crystal growth.

Since 2012, a group led by Professor Marzantowicz has carried out research in applied topology. This concerns in particular the different versions of the Bourgin–Yang theorem and related topics, as well as the theory of topological complexity as a geometric model of robot motion. We have introduced and examined the following generalizations of topological complexity: invariant complexity, effective complexity, and efficient complexity. Other issues related to applied topology addressed in the group’s research are optimization, group calculus, computer algebra, high-dimensional manifold topology, surgery theory and covariant surgery theory, knot symmetry, and the use of computer methods to solve problems of theoretical topology. Another area of interest for the group is applications of Reeb graphs to algebra and analysis.