The Division of Topology was established in 1978.

Scientific interests of the division belong to fundamental research in some areas of topology and dynamical systems. The research is focused around the following topics:

Study of certain classes of low dimensional discrete chaotic dynamical systems. Particular interest is in a better understanding of the dynamics of hyperbolic and partially hyperbolic maps on surfaces, the dynamics of families of homeomorphisms on surfaces (with homoclinic tangencies) and the topological properties of strange attractors in chaotic dynamical systems. With these goals, we explore conjugacy invariants, methods of symbolic dynamics and kneading theory, topological entropy, rotational theory.

Discrete dynamical systems generated by diffeomorphisms in R^n and C^n and their bifurcations. We investigate analytic germs of diffeomorphisms with asymptotic expansions in scales of powers and (iterated) logarithms (in particular, the Dulac maps). Objective: Contribution to the theory of formal and analytic classification of diffeomorphisms and families of such mappings. Reading the intrinsic properties of generating functions, such as multiplicity, formal and analytic class, from the epsilon-neighborhoods of orbits of associated discrete dynamical systems. That is, from fractal properties of their orbits. Applications to continuous dynamic systems within the 16^th Hilbert's problem and cyclicity.

Fractal zeta functions of fractal sets and complex dimensions as a generalization of box dimensions and the notion of Minkowski content. Application to dynamical systems. The interest of research comes from the Riemann hypothesis and Weyl-Berry's conjecture. Objective: to explore various types of singularities of zeta functions of fractal sets and to associate them with other geometric properties of these sets. To examine the types of complex dimensions that occur in orbits and trajectories of dynamical systems. Understanding the properties of dynamical system generators from zeta functions and complex dimensions of the orbit. Investigation of complex dimensions of orbits that undergo bifurcations.

Topology and computability. Investigation of the theory of computable metric and topological spaces and, in general, computable analysis and topology. Objective: Finding conditions under which a set is computable or contains computable points, determining properties of computability structures on metric spaces, studying relationships between computability, topology, geometry and analysis.

Some of the courses at the undergraduate, graduate and doctoral studies of mathematics held by the members of the Division of Topology are as follows: Metric Spaces, General Topology, Algebraic Topology, Differential Topology, Nonlinear Dynamic Systems, Discrete Dynamic Systems, Geometry and Topology.